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2024/Finite Volume Features, Global Geometry Representations, and Residual Training for Deep Learning-based CFD Simulation/full.md

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+# Finite Volume Features, Global Geometry Representations, and Residual Training for Deep Learning-based CFD Simulation
+
+Loh Sher En Jessica  $^{*12}$  Naheed Anjum Arafat  $^{*1}$  Wei Xian Lim  $^{1}$  Wai Lee Chan  $^{13}$  Adams Wai Kin Kong  $^{12}$
+
+# Abstract
+
+Computational fluid dynamics (CFD) simulation is an irreplaceable modelling step in many engineering designs, but it is often computationally expensive. Some graph neural network (GNN)-based CFD methods have been proposed. However, the current methods inherit the weakness of traditional numerical simulators, as well as ignore the cell characteristics in the mesh used in the finite volume method, a common method in practical CFD applications. Specifically, the input nodes in these GNN methods have very limited information about any object immersed in the simulation domain and its surrounding environment. Also, the cell characteristics of the mesh such as cell volume, face surface area, and face centroid are not included in the message-passing operations in the GNN methods. To address these weaknesses, this work proposes two novel geometric representations: Shortest Vector (SV) and Directional Integrated Distance (DID). Extracted from the mesh, the SV and DID provide global geometry perspective to each input node, thus removing the need to collect this information through message-passing. This work also introduces the use of Finite Volume Features (FVF) in the graph convolutions as node and edge attributes, enabling its message-passing operations to adjust to different nodes. Finally, this work extends the use of residual training to improve flow field prediction for a GNN scenario with immersed object, when low resolution data is available. Experimental results on two datasets with five differ
+
+*Equal contribution  $^{1}$ Rolls-Royce@NTU Corporate Lab, Nanyang Technological University, Singapore  $^{2}$ College of Computing and Data Science, Nanyang Technological University, Singapore  $^{3}$ School of Mechanical & Aerospace Engineering, Nanyang Technological University, Singapore. Correspondence to: Adams Wai Kin Kong <adamskong@ntu.edu.sg>, Wai Lee Chan <chan.wl@ntu.edu.sg>.
+
+Proceedings of the  $41^{st}$  International Conference on Machine Learning, Vienna, Austria. PMLR 235, 2024. Copyright 2024 by the author(s).
+
+ent state-of-the-art GNN methods for CFD indicate that SV, DID,FVF and residual training can effectively reduce the predictive error of current GNN-based methods by as much as  $41\%$  Our codes and datasets are available at https: //github.com/toggled/FvFGeo
+
+# 1. Introduction
+
+Computational fluid dynamics (CFD) is a branch of fluid dynamics in which physical phenomena involving fluid flow are modelled mathematically as partial differential equations (PDEs), like the Navier-Stokes (NS) equations, and solved computationally via numerical analysis. CFD is applied to a wide range of scientific and engineering problems that requires the flow of the fluid and its interaction with surfaces to be simulated, including aircraft aerodynamic optimisation (Martins, 2022), combustion engine design (Vijayashree & Ganesan, 2018), marine hydrodynamics prediction (Demirel, 2021), microfluidic device evaluation (Chaves et al., 2020), and urban planning (Zhang et al., 2021). Despite its versatility, CFD simulation is generally slow and/or costly due to the need for both high spatial and temporal resolutions to solve the governing PDEs accurately.
+
+Researchers have exploited deep learning to accelerate CFD simulation. Multilayer perceptron (MLPs) and convolutional neural networks (CNNs) have been considered. However, they both do not fit industrial requirements in many cases because of their restrictions in the input fields and architectures. MLPs cannot handle high dimensional input, which would increase the number of training parameters dramatically and cause overlearning. Thus, the current MLP methods such as PINN consider each spatial location separately and ignore their relationship (Raissi et al., 2019). In other words, they do not have an explicit scheme for information exchange between nodes. It should be emphasised that flow of a fluid in a particular location would influence that of a neighbour region. Although CNNs allow this information exchange, they can apply only to the flow field represented on a fixed, regular grid. In many industrial simulations, the computational domain may contain objects with
+
+a complex geometry, such as a turbine blade, and hence may have an irregular mesh, so CNNs are not suitable. To fit the industrial requirements and bypass the limitations, researchers have recently employed graph neural networks (GNN) for CFD problems, such as laminar and turbulent flow prediction and geometry optimisation (Liu et al., 2020; Baque et al., 2018). Some of these studies have considered physical properties or constraints in their GNN. For example, Horie & Mitsume (2022) designed boundary encoders to impose Dirichlet boundary conditions and Bonnet et al. (2022a) separated the nodes on airfoils and other nodes in their objective for computing drag and lift coefficients.
+
+The current GNN methods neglect the cell characteristics in the mesh, such as cell volume, face surface area, and face centroid, which are core components of the finite volume method, a widely adopted CFD method in industry (Moukalled et al., 2016). CFD simulators based on the finite volume method in general take three steps to compute the flow of the fluid. First, the simulation domain will be discretised into a finite number of small volumes known as cells. Next, the fluid properties, flow models, boundary conditions of the domain, and initial conditions for the flow will be prescribed. Finally, through Gauss's divergence theorem and assuming a constant solution in each cell, the governing PDEs can be written in an integral form, discretised by numerical approximation, and solved as a system of algebraic equations through numerical methods. The cell characteristics play a critical role in the discretisation process because they are used to model the flux between a certain cell and its neighbours for the purpose of conservation of mass, momentum, and energy. However, the current GNN methods do not consider this information.
+
+Another weakness of the current GNN methods is that their input node features, such as signed distance function (SDF), spatial coordinates, and inlet velocity, only provide very limited information about any embedded object and its surrounding environment to the nodes (Bonnet et al., 2022a; Belbute-Peres et al., 2020). Thus, the nodes need to collect this information through message-passing between neighbouring nodes. In fact, this weakness also appears in the traditional finite volume simulators, which propagate object boundary conditions only locally.
+
+To address these weaknesses, in this paper, we make the following three contributions regarding three aspects of graph neural network training for CFD:
+
+- Input layer. We propose Shortest Vector (SV) and Directional Integrated Distance (DID) for enhancing the performance of GNN-based methods. The SV and DID extracted from the mesh provide a global geometry perspective to each input node, thus easing the learning by removing the need to collect this information through message-passing.
+
+- Graph convolution. We propose the Finite Volume Features (FVF), including cell volume, face area normal vector, and face centroid, to be used as node and edge attributes in graph convolution, such that the convolution filters can be adjusted based on the cell characteristics. A theorem is presented to show that the input mesh can be reconstructed from the FVF.
+
+- Training scheme. While existing GNN methods (Belbute-Peres et al., 2020) exploit low-resolution data as prior knowledge, and the "learned correction" approach (Kochkov et al., 2021) corrects the error of very low-resolution simulations without an immersed object using a CNN, we demonstrate that residual training reduces the prediction error of a GNN by helping the model focus more on regions around and downstream from the geometries, where the low-resolution data tends to be less accurate.
+
+The experimental results show that the (i) combined effect of the proposed geometric features and finite volume features reduces predictive errors of MeshGraphNet (Pfaff et al., 2021), BSMS-GNN (Cao et al., 2023), Chen-GCNN (Chen et al., 2021) and Graph U-Net (Bonnet et al., 2022a) by as much as  $41\%$ , as well as reduces the predictive error of CFDGCN (Belbute-Peres et al., 2020) by about  $24\%$ , and (ii) additional usage of residual training increases the reduction of the error of CFDGCN to  $41\%$ . Further investigation of Chen-GCNN and MeshgraphNet models reveals that the residual training reduces the models' predictive errors by  $25\%$  and  $45\%$ , respectively.
+
+# 2. Preliminaries and Related Work
+
+Finite volume method. Consider the integral form of the steady state, incompressible turbulent Navier-Stokes equation for  $x$  direction over a control volume, which is given by
+
+$$
+\begin{array}{l} \int \nabla \cdot \left(\bar {u} _ {x} \overline {{\boldsymbol {U}}} - (\nu + \nu_ {t}) \nabla \bar {u} _ {x}\right) + \frac {\partial \bar {p}}{\partial x} \mathrm {d V} \\ = \oint \left(\bar {u} _ {x} \overline {{\boldsymbol {U}}} - (\nu + \nu_ {t}) \nabla \bar {u} _ {x}\right) \cdot \mathrm {d} \boldsymbol {S} + \int \frac {\partial \bar {p}}{\partial x} \mathrm {d V} = 0, \\ \end{array}
+$$
+
+where  $\overline{U} = [\bar{u}_x,\bar{u}_y]^T$  is the velocity vector,  $\nu$  and  $\nu_{t}$  are the dynamic and turbulent viscosities respectively, and  $\bar{p}$  is the normalised pressure. The second expression of Equation 1 is obtained from Gauss's theorem, where S denotes the area normal vector on the surface of the control volume and points outwards by convention (Moukalled et al., 2016). The finite volume method discretises the control volume into cells, as shown in Figure 1(a). For sufficiently small cell volume V like in Figure 1(b), all variables within a cell or along each of its faces are approximately constant (Moukalled
+
+et al., 2016), so Equation 1 becomes
+
+$$
+\begin{array}{l} \sum_ {f} \left(\bar {u} _ {x} \bar {\mathbf {U}} - (\nu + \nu_ {t}) \nabla \bar {u} _ {x}\right) _ {f} \cdot \boldsymbol {S} _ {f} + \left(\frac {\partial \bar {p}}{\partial x}\right) \mathbf {V} \tag {2} \\ = \sum_ {f} \boldsymbol {\Phi} _ {f} \cdot \boldsymbol {S} _ {f} + \Omega \mathrm {V} = 0, \\ \end{array}
+$$
+
+where  $f$  is a counter for the discrete faces of the cell. The first and second terms of the first expression of Equation 2 are known as the flux  $(\Phi)$  and source  $(\Omega)$  terms, respectively (Moukalled et al., 2016), thus constituting the second expression.
+
+In typical finite volume simulators, variables like  $\overline{U}$  and  $\bar{p}$  are solved at the cell centroids such as  $x_{i}$  and  $x_{j}$  in Figure 1(b). However, from Equation 2, the flux term needs face centroid values, which can be approximately interpolated from the cell centroids by (Tasri & Susilawati, 2021)
+
+$$
+\boldsymbol {\Phi} _ {f} \approx \left(\boldsymbol {\Phi} _ {i} \frac {\left| \left| \boldsymbol {c} _ {f , i j} - \boldsymbol {x} _ {i} \right| \right|}{\left| \left| \boldsymbol {x} _ {j} - \boldsymbol {x} _ {i} \right| \right|} + \boldsymbol {\Phi} _ {j} \frac {\left| \left| \boldsymbol {c} _ {f , i j} - \boldsymbol {x} _ {j} \right| \right|}{\left| \left| \boldsymbol {x} _ {j} - \boldsymbol {x} _ {i} \right| \right|}\right) . \tag {3}
+$$
+
+The interpolated flux term will then be projected to its respective face area normal vector,  $S_{f,ij}$ . Note that Equation 3 implies that the face centroid at  $c_{f,ij}$  lies along the line connecting  $x_i$  and  $x_j$ , which is not necessarily true because cells in a CFD simulation need not be regular and can have different sizes and shapes, as shown in Figure 1(b). In general, cells with small volumes are used in sensitive regions, for instance, the close vicinity of an object or wake region where flow variables may change drastically due to boundary conditions. Therefore, Equation 3 will incur an error that corresponds to the deviation from the face centroid, though a spatial correction scheme can be implemented as mitigation (Tasri & Susilawati, 2021). Also, the error will reduce with smaller cell volume, which is the main reason for the high computational cost of CFD. On the other hand, the source term weighted by the cell volume will be represented exactly by  $\Omega_j \mathbf{V}_j$ .
+
+Graph construction. In GNN-based CFD methods, both the inputs and outputs of the model are often graphs. The CFD simulation mesh  $M$  is represented as a graph  $G = (V, E)$ , where  $V$  and  $E$  represent a set of nodes and edges, respectively. There are two methods to do so.
+
+Using a 2D CFD mesh for illustration, the first method, as shown in Figure 2(a), is to directly represent the mesh nodes as graph nodes  $i \in V$ , and the faces between them as edges  $(i,j)$ ,  $(j,i) \in E$ . We refer to this as the mesh node-based method, and it has been used by other researchers (Bonnet et al., 2022a; Belbute-Peres et al., 2020; Chen et al., 2021; Pfaff et al., 2021). An alternate approach proposed by the authors is the following: the graph node  $i \in V$  represents the mesh cell centroid  $m_i \in M$ , and the bi-directional edge
+
+![](images/1ae7181be6a546db13586f0c30acec87fc80db451e22d74df75436666e596a8a.jpg)  
+(a)
+
+![](images/10233c09d748d9bae5170c5aa9d82ed158341313b2ee4877f44d571c700fa00a.jpg)  
+(b)
+
+![](images/078b75e314692f23b441d7ef7d7444d0f4f88829bd8a13eb81727f951373d72d.jpg)
+
+![](images/224600bf84717a2236daa52caf29cf1fd37c57a7cbbc43cdf261aab00341ba1a.jpg)  
+Figure 1. (a) CFD mesh with an airfoil body surrounded by different sizes of cells. (b) Illustration of cell characteristics, namely cell centroids,  $\pmb{x}_i$  and  $\pmb{x}_j$ , face centroid,  $c_{f,ij}$ , face area normal vector,  $S_{f,ij}$ , and cell volumes,  $V_i$  and  $V_j$ .
+
+![](images/dce9cc392f6f9ed853473a005897cd51be54eb63144a509adb0329dbb3333d14.jpg)  
+(a)  
+(b)
+
+![](images/25cacc518bd61b170ad6241dfb640a09d202ff32afb13d44af0b0ac5d2316516.jpg)  
+Figure 2. Two graph construction methods: (a) Mesh node-based: Graph nodes represent mesh nodes and graph edges represent mesh faces. (b) Cell centroid-based: Graph nodes represent either cell centroids (white) or boundary face centroids (black), and graph edges represent the adjacency of the cell centroids with one another or with boundary faces.
+
+$(i,j),(j,i)\in E$  indicates that cells  $i$  and  $j$  are adjacent, i.e., share a face. We refer to this approach as the cell centroid-based method, which is illustrated in Figure 2(b). This method is key to the use of FVF as described in  $\S 3$ .
+
+Note that in this method, the centroids of the boundary faces are represented as graph nodes as well, despite not being shared by two cells, to capture flow characteristics at the boundary. This representation allows message-passing from and towards boundary faces to be captured by the edges between these nodes and that of the cell adjacent to the boundary face.
+
+Global geometry representations. In a steady simulation, a GNN is trained to predict the velocity vector and pressure for each node. To train a GNN to predict target flow characteristics at each node, the current methods encode some features into the input nodes. The most common approaches include variants of the binary representation (Chen et al.,
+
+![](images/9a102c3746c903856272099d91a1f5c2709f0b7221c3793d30726bae7de21bc9.jpg)  
+(a) SDF
+
+![](images/a6719abc767962f5f8aaf4abb573fa6104ebf6a9b2d01b83fe487c8badfc6e5b.jpg)  
+(b) SV  
+Figure 3. Illustration of three geometry representations: (a) SDF value only indicates presence of the closest boundary point somewhere along the circle's circumference. (b) SV provides both distance and direction from the nearest boundary point. (c) DID gives the average distance of all boundary within several difference angle ranges.
+
+![](images/b6731a68214d3897385a0b66a9846f9f65a7eb015edcee82ba2e22b2f864a405.jpg)  
+(c) DID
+
+2021) and the Signed Distance Function (SDF) (Bonnet et al., 2022a; Belbute-Peres et al., 2020; Guo et al., 2016). In binary representations, nodes are given discrete values such as 0 and 1, depending on whether they are on the geometry boundary or not. The SDF, proposed by Guo et al. (2016) for CNNs, is defined as
+
+$$
+S D F \left(\boldsymbol {x} _ {i}\right) = \min  _ {\boldsymbol {x} _ {b} \in B} \| \boldsymbol {x} _ {i} - \boldsymbol {x} _ {b} \| h \left(\boldsymbol {x} _ {i}\right), \tag {4}
+$$
+
+where  $\boldsymbol{x}_i$  and  $\boldsymbol{x}_b$  denote an internal node and its closest boundary node, respectively.  $h(\boldsymbol{x}_i)$  is equal to 1, -1, and 0 if  $\boldsymbol{x}_i$  is outside, inside, and on the object boundary, respectively. The SDF representation was shown to be more effective than the simple binary representations in the CNN case (Guo et al., 2016). Since graphs generated from meshes do not have nodes on the inside of the object, the SDF for GNNs is in fact just the shortest Euclidean distance between  $\boldsymbol{x}_i$  and the object, estimated in a discrete case. However, it provides each node with very limited information about the object. The SDF value only indicates the existence of an object at the distance of  $SDF(\boldsymbol{x}_i)$ . No information about the object's size, shape, or direction from the node is given by the SDF, even though all these factors will affect the flow at the node. The boundary of the circle in Figure 3(a) indicates the uncertainty due to incomplete information of  $SDF(\boldsymbol{x}_i)$ . Although some other properties about the nodes and the flow, such as the spatial location of the nodes, inlet, angle of attack, and Mach number (Belbute-Peres et al., 2020), can also be provided as input node features, they cannot substitute the missing geometric information, which remains to be acquired by the nodes through message-passing. Note that this weakness is also true of typical CFD simulators where each mesh cell carries no global geometry information and the object boundary condition is transmitted only locally.
+
+Prior studies. The GNN-based CFD methods employ on-shell GNNs and exploit the physical knowledge in different ways. Some of them embed physical constraints and properties into the architectures and objective functions and
+
+some others directly use numerical simulators as a part of their methods. Belbute-Peres et al. (2020) combined a differentiable CFD simulator and graph convolution network (GCN) for fluid flow prediction. Liu et al. (2020) exploited graph attention network (GAT) for turbulent flow prediction without any physics prior. Ogoke et al. (2020) applied GraphSAGE (Hamilton et al., 2017) to predict drag forces around airfoils of different shapes and angles of attack under laminar flow. They demonstrated that GraphSAGE outperforms non-graph based MLP and CNN methods. Battaglia et al. (2018) proposed a general method called graph network (GN) blocks that can handle graphs with local features such as node features, edge features, and global graph-level features. A GN block first passes messages from nodes to edges through an edge convolution kernel before updating the edge features. The updated edge features are then aggregated to the nodes through sum or other permutation invariant operations as the edge messages. A node convolution kernel then takes the old node features and edge messages to obtain the updated node features. Finally, the updated node features and edge features are aggregated to compute global graph-level features. Sanchez-Gonzalez et al. (2020) used GN blocks to predict the future roll-out of physical systems of particles, including fluids, rigid solids, and deformable materials. Pfaff et al. (2021) used a GNN to predict future roll-outs of unsteady flow and showed better performance than CNN on various simulation scenarios in their proposed MeshGraphNet. More recently, their method was built upon by Libao et al. (2023) to include an RNN-based state encoding and physics loss terms. Bonnet et al. (2022a) released a large-scale high-resolution two-dimensional (2D) Reynolds-averaged Navier-Stokes (RANS) simulation datasets on airfoils and demonstrated good generalisation capabilities of GraphSAGE (Hamilton et al., 2017) and Graph U-Net (Gao & Ji, 2019) to different physical conditions and airfoil geometries. DiscretizationNet (Ranade et al., 2021) used finite volume discretisation to approximate spatio-temporal partial derivatives in its CNN encoder-decoder training. It is a non-data driven method that cannot generalise to new scenarios, requiring it to be trained on every new instance to obtain a solution. Also, its convolutions did not leverage cell characteristics as the finite volume method does. Chen et al. (2021) proposed their own permutation-invariant edge-convolution layer and smoothing layer to predict laminar flow on obstacles of different shapes. Their proposed architecture, which is referred to as Chen-GCNN in this paper, showed better performance than the standard U-net model (Ronneberger et al., 2015). Finally, similar concepts behind the MeshGraphNet and Chen-GCNN were further enhanced with multi-scalability in the MS-MGN by Fortunato et al. (2022), MultiScaleGNN by Lino et al. (2021), and BSMS-GNN by Cao et al. (2023), with the latter two using a GU-Net style processor for faster information propagation performance. All these GNN works neglect the cell
+
+![](images/65c82ee1021215ba9048e497d3d8a7c534fec80a72766960acbb4dd44f50bf93.jpg)  
+Figure 4. Illustration of the computation of DID. Each angle range is represented by the weighted average of the distance to the boundary (shown by the darker segments) and the  $\infty$  value for the directions where there is no boundary (shown by the lighter segments). Note that  $\infty$  is a predefined number in the implementation.
+
+characteristics, which, in the finite volume method, are used to discretise the governing equations and model the flow among the cells.
+
+# 3. Proposed Features and Training Scheme
+
+# 3.1. Global Geometry Perspective
+
+To give each node global geometric perspective about the object and its surrounding environment, we propose two features: Shortest Vector (SV), and Directional Integrated Distance (DID). SV is the vector formed by an internal cell centroid node  $\pmb{x}_i$  and its closest boundary face node  $\pmb{x}_b$  (Figure 3(b)), and is defined as
+
+$$
+\phi \left(\boldsymbol {x} _ {i}\right) = \boldsymbol {x} _ {i} - \boldsymbol {x} _ {b} \text {s . t .} \min  _ {\boldsymbol {x} _ {b} \in B} \left\| \boldsymbol {x} _ {i} - \boldsymbol {x} _ {b} \right\|. \tag {5}
+$$
+
+SV is related to SDF in that the length of SV,  $\| \phi(\boldsymbol{x}_i) \| = SDF(\boldsymbol{x}_i)$ . However, SV has more discriminative power than SDF because two nodes,  $\boldsymbol{x}_i$  and  $\boldsymbol{x}_j$ , can have the same SDF values (i.e.,  $SDF(\boldsymbol{x}_i) = SDF(\boldsymbol{x}_j)$ ) despite the vectors  $\phi(\boldsymbol{x}_i)$  and  $\phi(\boldsymbol{x}_j)$  not being the same.
+
+One limitation of SV is that only the closest node on the object's boundary is represented. Information in other directions is still missing. DID defined on an angular range is a generalisation of SV. DID has different angular segments to handle different directions. Figure 4 shows four overlapping angular segments, each with a range of  $2\pi /3$ . To clearly present the concept of DID in the following description, we consider a continuous object boundary. For the  $j^{th}$  segment of DID,  $\theta_{j}$  and  $\theta_j^\prime$  represent the starting and ending angles of the segment, respectively, where  $\theta_{j}\leq \theta_{j}^{\prime}$ . The continuous version of DID is defined as
+
+$$
+D I D \left(\boldsymbol {x} _ {i}, \theta_ {j}, \theta_ {j} ^ {\prime}, B _ {c}\right) = \int_ {\theta_ {j}} ^ {\theta_ {j} ^ {\prime}} w _ {j} (\theta) g \left(\boldsymbol {x} _ {i}, B _ {c}, \theta\right) d \theta , \tag {6}
+$$
+
+where  $w_{j}(\theta)$  is a suitable weightage function such as a Gaussian function centred at  $(\theta_{j} + \theta_{j}^{\prime}) / 2$ ,  $B_{c}$  is the continuous boundary of the object, and  $g(\pmb{x}_i,B_c,\theta)$  is the distance between the object and a node  $\pmb{x}_i$  in the direction  $\theta$ . If there
+
+is no object boundary at the direction  $\theta$ , an appropriately large constant, as denoted in Figure 3(c) as  $\infty$ , is given to  $g(\pmb{x}_i, B_c, \theta)$ . In this continuous version of DID,  $B_c$  is represented by a parametric equation rather than a set of nodes. Each DID value provides a weighted average distance between the object and the node in a particular angle range. While shorter angular segments make a more accurate description of the object, it increases the number of input features to train the network with. In this work, we use SV and DID to provide relative global geometry information of the object and its environment to the input nodes. Hence, this information will not have to be collected during propagation, easing the learning process. The algorithmic details of our discrete, non-parametric DID implementation can be found in the appendix.
+
+As stated in the implementation details of DID in Appendix B.1, the weight function used is a uniform distribution over segment range  $(\theta_{j},\theta_{j}^{\prime})$ . This is better suited for our choice of angle segments, where each segment overlaps with half of the segment preceding it. As a result, every point on the mesh is represented by two successive overlapping segments. In this case, the exact selection of regions is unlikely to significantly change the results, as every point on the boundary is represented the same number of times and with the same weight. On the other hand, if the chosen angle segments meant that some points (in highly overlapping regions) were represented more times than other points (in less overlapping regions), the user may choose to put more weight on the points that are represented fewer times.
+
+# 3.2. Finite Volume Graph Convolution
+
+As discussed in Section 2 with Equations 2-3 and the accompanying figure 1, the finite volume method uses cell characteristics, such as the cell  $(\boldsymbol{x}_i,\boldsymbol{x}_j)$  and face centroids  $(c_{f,i j})$  face area normal vector  $(S_{f})$ , and cell volume  $(\mathrm{V}_{i},\mathrm{V}_{j})$ , extensively. Motivated by the finite volume method, we hence embed these characteristics in GNN. To ease the following discussion, we will take the spatial graph convolution net
+
+work (SGCN) as an example. First introduced and evaluated on graphs from chemical compounds (Danel et al., 2020), the SGCN has two key features: (i) use of the spatial location of graph nodes as node attributes and (ii) use of multiple filters. One limitation is that the node attributes only consider node positions and ignore other useful information. Hence, we generalise the convolution to be able to take in node attributes,  $p_j$ , and edge attributes,  $q_{ij}$ . Given node feature  $h_i \in \mathbb{R}^{d_{in}}$  at the  $i^{th}$  node and its spatial location  $\boldsymbol{x}_i \in \mathbb{R}^t$ , a convolution using FVF is defined as
+
+$$
+\bar {h} _ {i} (U, b, N _ {i}) = \sum_ {j \in N _ {i}} R e L U \left(U ^ {T} \left(q _ {i j}\right) + b\right) \odot \left(h _ {j} \oplus p _ {j}\right), \tag {7}
+$$
+
+where  $U\in \mathbb{R}^{3t\times (d_{in} + 1)}$  and  $b\in \mathbb{R}^{d_{in} + 1}$  are trainable parameters,  $t$  is the spatial dimension of the CFD simulation, and  $N_{i}$  is an index set indicating the neighbourhood of the node  $i$ ,  $\odot$  is the element-wise multiplication, and  $\oplus$  is concatenation (Danel et al., 2020). In our model, the node attributes are its associated cell volume, denoted as  $p_j = \mathrm{V}_j\in \mathbb{R}^1$ , and the edge attributes are its associated face area normal vector and the relative spatial location of its face centroid to the nodes, denoted as  $q_{ij} = \mathbf{S}_{f,ij}\oplus (\pmb {c}_{f,ij} - \pmb {x}_i)\oplus (\pmb {c}_{f,ij} - \pmb {x}_j)\in \mathbb{R}^{3t}$ . Note that cell centroids  $\pmb{x}_i$  and  $\pmb{x}_j$  are used as reference points when we use face centroid,  $\pmb{c}_{f,ij}$ , as with the finite volume method. Finally, as with CNN operations, multiple filters are used and their outputs are concatenated such as
+
+$$
+\hat {h} _ {i} (\theta , N _ {i}, k) = \bar {h} _ {i} \left(U _ {1}, b _ {1}, N _ {i}\right) \oplus \dots \oplus \bar {h} _ {i} \left(U _ {k}, b _ {k}, N _ {i}\right), \tag {8}
+$$
+
+where  $\theta = \{U_1,\dots ,U_k,b_1,\dots ,b_k\}$  are trainable parameters. Finally, an MLP is applied on  $\hat{h}_i$  and the final output of node  $i$  is obtained, whose dimension is the same as the output dimension of the MLP.
+
+When used directly, we refer to this method as Finite Volume Graph Convolution (FVGC) and  $\mathrm{V}_i$ ,  $\mathbf{S}_{f,ij}$ ,  $(c_{f,ij} - x_i)$ ,  $(c_{f,ij} - x_j)$  as Finite Volume Features (FVF). Alternatively, the same principles can be incorporated into other graph convolution types indirectly, for instance by using the FVGC as the aggregation function of the SAGE convolution. In convolutions like the invariant edge convolution (Chen et al., 2021), which already employ multiple filters and edge features, just the use of FVF as node and edge attributes in each convolution has to be implemented.
+
+Any common 2D mesh typically used for CFD simulations, such as those with triangular or quadrilateral cells, can be reconstructed from its prescribed FVF. More specifically,
+
+Theorem 3.1 (The Completeness of FVF). Let  $M$  be a 2D mesh such that the cells along its farfield have no more than 2 boundary faces each, and the faces of each of its cells enclose a singular volume. If a graph  $G = (V, E)$  is the cell centroid-based graph representation of  $M$ , the relative positions of all the nodes and faces of  $M$  can be uniquely
+
+deduced given  $\mathbf{V}_i$ ,  $\forall i \in V$ , and  $S_{f,ij}$ ,  $(c_{f,ij} - x_i)$  and  $(c_{f,ij} - x_j)$ ,  $\forall (i,j) \in E$ .
+
+The implementation details of the FVF and proof of this theorem are provided in the appendix.
+
+# 3.3. Residual Training
+
+In the previous subsections, we propose geometric features and FVF to improve the learning performance. In this subsection, we exploit low-resolution data and residual learning further improving the learning performance. Residual training is a well-known approach in image superresolution (Zhang et al., 2018; Yang et al., 2019). The general idea is to train the network to predict the residual field  $F - Upsample(F_{LR})$ , where  $F_{LR}$  is the low-resolution field, instead of the original field  $F$  itself. However, the most common way of utilizing low-resolution data as reference in CFD-AI literature is to concatenate  $Upsample(F_{LR})$  to one of the intermediate convolution layers (Belbute-Peres et al., 2020). The prior knowledge from the low-resolution flow field could be further utilised through the residual training scheme. Instead of minimising the loss  $\mathcal{L}\left(F_{GT},\widehat{F}\right)$ , where  $\widehat{F}$  is the predicted flow variable,  $F_{GT}$  is the corresponding ground truth, and  $\mathcal{L}$  is an arbitrary loss criterion, the network minimises the residual loss  $\mathcal{L}\left(F_{GT},\widehat{F}_r + Upsample(F_{LR})\right)$ . Here,  $\widehat{F}_r$  is the predicted residual and the prediction is  $\widehat{F} = \widehat{F}_r + Upsample(F_{LR})$ . Since the low-resolution field is an approximation of the ground truth, much of the residual field will be close to zero. Thus, training the model to predict the residual field eases the learning, and helps the model to focus on the more nuanced areas where the low-resolution fields tend to be inaccurate. For instance, the areas around and downstream from internal geometries.
+
+Kochkov et al. (2021) utilised a similar training scheme in the method of "learned correction", aimed at correcting a very low-resolution field using a CNN, without internal geometries, to an output of the same resolution. In the next section, this work intends to support the use of residual training to improve GNN performance for flows around immersed objects on a much finer mesh.
+
+# 4. Experiments
+
+We evaluate our proposed geometric and finite volume features on two databases, 2DSHAPES (Viquerat & Hachem, 2020; Chen et al., 2022) and AirfRANS (Bonnet et al., 2022a), and five state-of-the-art GNN methods and their respective training schemes in the CFD-AI literature, MeshGraphNet (Pfaff et al., 2021), BSMS-GCNN (Cao et al., 2023), Chen-GCNN (Chen et al., 2021), Graph U-Net (Bonnet et al., 2022a), and CFDGCN (Belbute-Peres et al., 2020).
+
+While all state-of-the-art methods utilise mesh node-based graphs, we adapt both datasets to cell centroid-based graphs to train the models with FVF. Models that do not use FVF are trained on mesh node-based graphs to maintain consistency with the baselines. The effectiveness of residual training is demonstrated only on CFDGCN with the AirfRANS dataset because CFDGCN is the only method that takes in low-resolution data.
+
+2DSHAPES is a collection of 2000 random shapes where each shape is generated by connected Bézier curves along with their steady-state velocity vector  $[\bar{u}_x, \bar{u}_y]^T$  and pressure  $\bar{p}$ , at Reynolds number  $= 10$ . The training set, validation set, and test set consist of 1600, 200, and 200 shapes, respectively. We use this dataset to demonstrate the effectiveness of our features applied to the Chen-GCNN model.
+
+AirfRANS is a high fidelity aerodynamics dataset of airfoil shapes. Their (RANS) solutions range from Reynolds numbers 2 to 6 million and angle-of-attacks  $-5^{\circ}$  to  $+15^{\circ}$  degrees. In their scarce data regime, the training set, validation set, and test set consist of respectively 180, 20, and 200 airfoils along with their steady-state velocity vector  $[\bar{u}_x,\bar{u}_y]^T$ , pressure  $\bar{p}$ , and turbulent viscosity  $\nu_{t}$ . This dataset is used as-is to demonstrate the effectiveness of our features applied to the Graph U-Net. An amended dataset, which excludes the turbulent viscosity field, is likewise used for comparing BSMS-GNN and CFDGCN with and without the proposed features. Similarly, the dataset is amended to further exclude the pressure fields for the MeshGraphNet models. These fields are excluded to more closely match those predicted by the original methods. Additionally, we use the authors' codebase (Bonnet et al., 2022b) to generate RANS solutions of the same set of airfoils on coarser meshes ( $\sim$  20 times fewer nodes than the original), which we refer to as coarse-AirfRANS and utilise as a database of low-resolution reference flow fields to evaluate the effectiveness of residual training on CFDGCN.
+
+Performance metrics. Our objective is to determine if incorporating the proposed features and residual training into the state-of-the-art methods significantly improves performance. We evaluate different methods with different measures to correspond with the metrics in their respective studies. To compare Chen-GCNN with our models derived from it, we use the MAE loss function, which is the summation of mean-absolute error of flow variables averaged over the test set. For Graph U-Net from Bonnet et al. (Bonnet et al., 2022a), the following three types of metrics are employed:
+
+1. Test loss: Loss function by equation 3 in Bonnet et al. (2022a) evaluated on and averaged over the test set.  
+2. Volume MSE: Mean-squared errors of normalised flow field  $([\bar{u}_x,\bar{u}_y]^T,\bar{p},\nu_t)$  predictions at internal nodes.
+
+3. Surface MSE: Mean-squared error of normalised flow field predictions at boundary nodes. Due to boundary constraints, only pressure  $(\bar{p})$  is non-zero at airfoil surfaces.
+
+Finally, we evaluate the models derived from MeshGraphNet and CFDGCN using the MSE loss function, which is the mean-squared error of flow field prediction averaged over the test set, and the models derived from BSMS-GNN with the RMSE or root mean-squared error.
+
+# 4.1. Results
+
+MeshGraphNet (Pfaff et al., 2021). The MeshGraphNet model uses an encoder-processor-decoder architecture. The original convolution types were maintained in the experiments. Table 1 shows that the adoption of the geometric features (Geo) and the Finite Volume Features (FVF) reduced the MSE by  $\sim 41\%$  from the baseline.
+
+Chen-GCNN (Chen et al., 2021). On the Chen-GCNN model using invariant edge convolutions, Table 2 shows that the adoption of the geometric features and finite volume features reduces MAE by  $\sim 27\%$  from the baseline. We also tested GCNN with SAGE convolution and obtained  $\sim 82\%$  reduction in MAE with respect to the baseline.
+
+Table 1. Performance evaluation using MeshGraphNet on the AirffRANS dataset.  
+
+<table><tr><td>MODELS</td><td>MSE ×10-2</td></tr><tr><td>MESHGRAPHNET (BASELINE)</td><td>5.7571</td></tr><tr><td>MESHGRAPHNET W/ GEO</td><td>3.4683</td></tr><tr><td>MESHGRAPHNET W/ FVF W/ GEO</td><td>3.3811</td></tr></table>
+
+Table 2. Performance evaluation using Chen-GCNN on the 2DSHAPES dataset.  
+
+<table><tr><td>MODELS</td><td>CONV TYPE</td><td>MAE ×10-2</td></tr><tr><td>CHEN-GCNN (BASELINE)</td><td>INvariant</td><td>1.1590</td></tr><tr><td>CHEN-GCNN w/ GEO</td><td>EDGE</td><td>0.9727</td></tr><tr><td>CHEN-GCNN w/ FVF w/ GEO</td><td>CONVOLUTION</td><td>0.8491</td></tr><tr><td>CHEN-GCNN (BASELINE)</td><td>SAGE</td><td>6.7103</td></tr><tr><td>CHEN-GCNN w/ GEO</td><td>CONVOLUTION</td><td>3.8041</td></tr><tr><td>CHEN-GCNN w/ FVF w/ GEO</td><td></td><td>1.1982</td></tr></table>
+
+Table 3. Performance evaluation using BSMS-GNN on the Air- FRANS dataset.  
+
+<table><tr><td>MODELS</td><td>RMSE ×10-2</td></tr><tr><td>BSMS-GNN (BASELINE)</td><td>7.7589</td></tr><tr><td>BSMS-GNN w/ GEO</td><td>6.7498</td></tr><tr><td>BSMS-GNN w/ FVF w/ GEO</td><td>6.1919</td></tr></table>
+
+Table 4. Performance evaluation using Graph U-Net on the AirfRANS dataset.  
+
+<table><tr><td rowspan="2">MODELS</td><td rowspan="2">CONV TYPE</td><td rowspan="2">TEST LOSS ×10-2</td><td colspan="4">VOLUME MSE ×10-2</td><td>SURFACE MSE ×10-1</td></tr><tr><td>u_x</td><td>u_y</td><td>p</td><td>νt</td><td>p</td></tr><tr><td>GRAPH U-NET (BASELINE)</td><td>SAGE</td><td>1.816</td><td>1.140</td><td>1.429</td><td>2.190</td><td>2.492</td><td>0.932</td></tr><tr><td>GRAPH U-NET w/ GEO</td><td>CONVOLUTION</td><td>1.786</td><td>1.224</td><td>0.599</td><td>2.028</td><td>3.286</td><td>0.828</td></tr><tr><td>GRAPH U-NET w/ FVF w/ GEO</td><td></td><td>1.441</td><td>1.061</td><td>0.515</td><td>1.368</td><td>2.724</td><td>0.549</td></tr><tr><td>GRAPH U-NET (BASELINE)</td><td>VANILLA</td><td>15.310</td><td>19.249</td><td>12.845</td><td>14.041</td><td>15.190</td><td>4.633</td></tr><tr><td>GRAPH U-NET w/ GEO</td><td>GRAPH</td><td>12.709</td><td>17.408</td><td>11.256</td><td>10.106</td><td>12.146</td><td>3.631</td></tr><tr><td>GRAPH U-NET w/ FVF w/ GEO</td><td>CONVOLUTION</td><td>4.544</td><td>2.968</td><td>2.286</td><td>4.546</td><td>8.287</td><td>1.505</td></tr></table>
+
+BSMS-GNN (Cao et al., 2023). The Bi-Stride MultiScale GNN (BSMS-GNN) model also has an encoder-processor-decoder architecture, similar to the MeshGraphNet. Likewise, only the original convolution types were used in the experiment. The BSMS-GNN is a multi-scale model, and hence the FVF implementations were only applied to the encoder, decoder, and first and last convolutions of the processor, where the graph is at full resolution. As can be seen in Table 3, the adoption of our methods results in a  $\sim 20\%$  reduction in the RMSE.
+
+Graph U-Net (Bonnet et al., 2022a). We trained Graph U-Net architecture (baseline) with the same experimental set-up described in Appendix L of Bonnet et al. (Bonnet et al., 2022a). The reason for choosing Graph U-Net is that it is the best-performing model reported. We evaluated this baseline with two convolution types: SAGE convolution and vanilla graph convolution. In Table 4, we observe that, on SAGE convolution, the adoption of the geometric and finite volume features reduces the loss on the test set by about  $21\%$  as indicated by the baseline method's loss on the test set  $(1.816 \times 10^{-2})$  and that of Graph U-Net w/ FVF w/ Geo  $(1.441 \times 10^{-2})$ . We also observe that our features improve the volume MSE of three out of four flow variables as well as the surface MSE of pressure. Do note that each flow variable was not handled separately by the training loss, but rather the MSE of all four as a whole. We also tested Graph U-Net with vanilla graph convolution and obtained  $71\%$  reduction in the loss on the test set w.r.t the baseline.
+
+CFDGCN (Belbute-Peres et al., 2020). CFDGCN uses a CFD simulator in the training and testing scheme to generate low-resolution data and vanilla graph convolution to enhance learning on a high-resolution mesh with varying initial physical conditions, e.g., angle of attack and Mach number. Although CFDGCN generalises well to unknown physical conditions, it requires the geometry to remain fixed for the purpose of optimising the coarse mesh. This makes the model unable to generalise to different geometries and limits the method's practical functionality. Hence, in our
+
+Table 5. Performance evaluation using CFDGCN on the AirfRANS dataset.  
+
+<table><tr><td>MODELS</td><td>CONV TYPE</td><td>MSE ×10-2</td></tr><tr><td>CFDGCNN (BASELINE)</td><td>VANILLA</td><td>0.1211</td></tr><tr><td>CFDGCN w/ GEO</td><td>GRAPH</td><td>0.1093</td></tr><tr><td>CFDGCN w/ FVF w/ GEO</td><td>CONVOLUTION</td><td>0.0918</td></tr><tr><td>CFDGCN w/ RESIDUAL TRAINING W/ FVF w/ GEO</td><td></td><td>0.0719</td></tr><tr><td>CFDGCN (BASELINE)</td><td>SAGE</td><td>0.1342</td></tr><tr><td>CFDGCNN w/ GEO</td><td></td><td>0.1092</td></tr><tr><td>CFDGCNN w/ FVF w/ GEO</td><td>CONVOLUTION</td><td>0.0631</td></tr><tr><td>CFDGCNN w/ RESIDUAL TRAINING W/ FVF w/ GEO</td><td></td><td>0.0628</td></tr></table>
+
+Table 6. Computational time of SV, DID and FVF for two mesh resolutions.  
+
+<table><tr><td rowspan="2"></td><td colspan="2">AVERAGE COMPUTATION TIME (S)</td></tr><tr><td>COARSE-AIRFRANS</td><td>AIRFRANS</td></tr><tr><td>SV</td><td>0.001</td><td>0.018</td></tr><tr><td>DID</td><td>0.031</td><td>4.276</td></tr><tr><td>FVF</td><td>0.001</td><td>0.032</td></tr></table>
+
+adaption, we removed their differentiable CFD simulator component from the training and testing loop and directly passed the coarse-AirfRANS solutions for upsampling. As the experimental set-up assumed that a low-resolution flow field is available, we adopted and evaluated the proposed residual training scheme in addition to Geo and FVF. In Table 5, we observe that the predictive error reduces by about  $41\%$ , as indicated by the baseline method's MSE  $(0.1211 \times 10^{-2})$  and that of CFDGCN w/ residual training w/ FVF w/ Geo  $(0.0719 \times 10^{-2})$ . Incorporating our proposed features and residual training scheme one by one consistently reduces the error, suggesting that they are generally effective. We also tested CFDGCN with SAGE convolution and obtained a  $53\%$  reduction in MSE w.r.t the baseline.
+
+# 4.2. Computation Time of the Geometric Features
+
+We present the running time of computing the proposed features on Coarse-AirfRANS and (Fine) AirfRANS datasets, shown in Table 6. The average computation times of SV and
+
+Table 7. Impact of Residual training on Chen-GCNN (IVE convolution) and MeshGraphNet using AirrRANS dataset.  
+
+<table><tr><td>MODELS</td><td>MSE ×10-2</td></tr><tr><td>CHEN-GCNN W/ FVF W/ GEO</td><td>5.6104</td></tr><tr><td>CHEN-GCNN W/ RESIDUAL TRAINING W/ FVF W/ GEO</td><td>4.2166</td></tr><tr><td>MESHGRAPHNET W/ FVF W/ GEO</td><td>3.3811</td></tr><tr><td>MESHGRAPHNET W/ RESIDUAL TRAINING W/ FVF W/ GEO</td><td>1.8855</td></tr></table>
+
+FVF on coarse and fine meshes are quite reasonable. The reason the computational time of DID is relatively higher than that of other features is because the values of each angle segment are calculated consecutively rather than in parallel due to memory constraints. Note that the computational time can be reduced by calculating them in parallel or by using fewer angle segments.
+
+# 4.3. Effectiveness of Residual Training
+
+In order to evaluate the effectiveness of residual training more thoroughly, we have conducted experiments using the AirfRANS dataset with MeshGraphNet and Chen-GCNN with IVE convolution. The results are shown in Table 7. We observe that the mean-squared error reduces significantly due to the incorporation of residual training in both models. To be precise, we observe a  $25\%$  and  $45\%$  reduction in error of Chen-GCNN and MeshGraphNet models, respectively. These observations suggest that the residual training generally improves the learning performance.
+
+# 5. Conclusion
+
+This work presents two novel geometric representations, SV and DID, and the use of FVF in graph convolutions. The SV and DID provide a more complete representation of the geometry to each node. Moreover, the FVF enable the graph convolutions to more closely model the finite volume simulation method. Their effectiveness at reducing prediction error has been shown across two datasets, as well as five different state-of-the-art methods and training scenarios using various types of graph convolution. Additionally, this paper demonstrates the ability of residual training to further improve accuracy in scenarios with low-resolution data.
+
+Limitations. The experiments assumed the availability of sufficient but not overwhelming data to an extent. The benefits of the proposed features over the baseline are likely to decrease as the amount of training data increases. This is because, with a large dataset, the networks can learn all necessary information from the data and thus rely less on the prior knowledge given in the proposed features. However, it should be emphasized that such a database would be extremely costly to simulate. Generalising the geometric
+
+features to a 3D case and testing our methods on models designed to predict turbulent flows have been left for future work. Finally, the effectiveness of residual training assumes the availability of additional data from low-resolution simulations as an approximate solution.
+
+# Acknowledgement
+
+The research, undertaken in the Rolls-Royce Corporate Lab @ Nanyang Technological University, is supported by the Singapore Government (Industry Alignment fund IAF-ICP Grant - I1801E0033). We thank Yi Wang and Bryce Conduit from Rolls-Royce plc. for their feedback on this work. We also thank the authors of BSMS-GNN and Chen-GCNN for sharing their codes and data.
+
+# Impact Statement
+
+This study introduces approaches to enhance AI predictions in CFD simulations. Similar to conventional CFD solvers, these methods find application in the design process of diverse components, encompassing those relevant to weaponry. The team will not actively pursue collaboration or knowledge-sharing with defence or military agencies.
+
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+
+# Technical Appendix
+
+# A. Relation Between a CFD Mesh and its FVF
+
+In this section, we show that any common 2D mesh can be reconstructed from the FVF. Specifically, assuming that in a 2D mesh,
+
+1. the cells along the farfield have no more than two boundary faces each, and  
+2. the faces of each cell enclose a singular volume,
+
+the relative positions of all the nodes and the faces of a mesh can be uniquely deduced given
+
+1. the volumes of all cells  $\mathbf{V}_i$  
+2. the face area normal vector  $\mathbf{S}_{f,ij}$  of all internal and geometry boundary faces, and  
+3. the vector from the cell centroids to the face centroids  $(\pmb{c}_{f,ij} - \pmb{x}_i)$  and  $(\pmb{c}_{f,ij} - \pmb{x}_j)$  of all internal and geometry boundary faces.
+
+By the "relative positions"  $\pmb{a}_i'$  of mesh nodes that have the absolute positions  $\pmb{a}_i$ , we mean that there exists some constant  $\alpha$  such that  $\pmb{a}_i = (\pmb{a}_i' - \alpha), \forall i$ . By the principle of translational invariance, the absolute positions of the mesh points are inconsequential to the flowfield as long as their relative positions remain the same. We make no further notational distinction between the two in this section.
+
+Lemma A.1. Given the (relative) position of the face centroid  $c_{f,ij}$  and face area normal vector  $S_{f,ij} = (S_x, S_y)$  of a face in a 2D mesh, the unique (relative) positions of the mesh nodes  $\mathbf{a}, \mathbf{b}$  of the face, and the existence of the face, can be deduced.
+
+Proof. From the (relative) position of the face centroid  $c_{f,ij}$ , we know that  $(\pmb{a} + \pmb{b}) / 2 = c_{f,ij}$ . Likewise, we can assume without loss of generality that  $(\pmb{b} - \pmb{a})$  is the face area normal vector  $\mathbf{S}_{f,ij}$  rotated by a right angle clockwise, or  $(\pmb{b} - \pmb{a}) = (S_y, - S_x)$ . The unique (relative) positions of the mesh nodes  $\pmb{a}, \pmb{b}$  can be solved from this system of two linear equations. The existence of a face shared by them is obvious.
+
+Lemma A.2. Given the (relative) position of a cell's centroid  $\pmb{x}_i$  and the (relative) positions of all but one of its mesh nodes  $\pmb{b}_1, \dots, \pmb{b}_{n-1}$ , the unknown (relative) position of its remaining mesh node  $\pmb{a}$  can be found to be  $n\pmb{x}_i - \sum_{j=1}^{n-1}\pmb{b}_j$ .
+
+Proof. From the (relative) position of the cell centroid  $\pmb{x}_i$ , we know that  $\frac{1}{n}\left(\pmb{a} + \sum_{j=1}^{n-1}\pmb{b}_j\right) = \pmb{x}_i$ . The rest is obvious.
+
+Lemma A.3. If and only if  $\pmb{x}_i$  is the (relative) position of a cell centroid with the mesh nodes of (relative) positions  $\pmb{b}_1, \dots, \pmb{b}_{n-1}$ , then  $\pmb{x}_i = n\pmb{x}_i - \sum_{j=1}^{n-1}\pmb{b}_j$ .
+
+Proof.
+
+$$
+\begin{array}{l} \frac {1}{n - 1} \sum_ {j = 1} ^ {n - 1} \boldsymbol {b} _ {j} = \boldsymbol {x} _ {i} \Leftrightarrow \sum_ {j = 1} ^ {n - 1} \boldsymbol {b} _ {j} = (n - 1) \boldsymbol {x} _ {i} \\ \Leftrightarrow \sum_ {j = 1} ^ {n - 1} \boldsymbol {b} _ {j} = n \boldsymbol {x} _ {i} - \boldsymbol {x} _ {i} \Leftrightarrow \boldsymbol {x} _ {i} = n \boldsymbol {x} _ {i} - \sum_ {j = 1} ^ {n - 1} \boldsymbol {b} _ {j} \\ \end{array}
+$$
+
+![](images/666c8094d6abd8628aa6b82153268d664e9e5cc0845ed84ec97197d8bc2776e9.jpg)
+
+Theorem A.4 (The Completeness of FVF). Let  $M$  be a 2D mesh such that the cells along its farfield have no more than 2 boundary faces each, and the faces of each of its cells enclose a singular volume. If a graph  $G = (V,E)$  is the cell centroid-based graph representation of  $M$ , the relative positions of all the nodes and faces of  $M$  can be uniquely deduced given  $\mathbf{V}_i$ ,  $\forall i \in V$ , and  $S_{f,ij}$ ,  $(\pmb{c}_{f,ij} - \pmb{x}_i)$  and  $(\pmb{c}_{f,ij} - \pmb{x}_j)$ ,  $\forall (i,j) \in E$ .
+
+Proof. By the cell-centroid based graph construction method, each cell of the input mesh corresponds to a node  $i \in V$ , and each internal face and geometry boundary face of the input mesh corresponds to an edge  $(i,j) \in E$ .
+
+The vectors  $(\pmb{c}_{f,ij} - \pmb{x}_i)$  and  $(\pmb{c}_{f,ij} - \pmb{x}_j)$  are given in the edge attributes of the FVF. It is obvious that the positions of all face centroids  $\pmb{c}_{f,ij}$  and cell centres  $\pmb{x}_i, \pmb{x}_j$  relative to one another in a connected mesh can be deduced from this. The face area vector  $\mathbf{S}_{f,ij}$  is also included as an edge attribute. Hence, from Lemma A.1, all internal faces, geometry adjacent boundary faces, and their respective mesh nodes can be found, shown in Figure 5. This already accounts for the significant majority of the original mesh. All that remains are the edges along the farfield boundary, as well as the relative position of any mesh nodes that did not abut an internal face (such as those at corners).
+
+As mentioned before, the relative positions of the cell centroids  $\pmb{x}_i$  can be deduced by the edge attributes. Assuming that these cells do not have more than 2 boundary faces along the farfield each, they will have at most 1 mesh node not associated with an internal edge. This is a fair assumption, considering the typical meshes with triangular or quadrilateral cells, commonly used for 2D CFD simulations. If there is a node with an unknown relative position, by Lemma A.2, it can be found as  $n\pmb{x}_i - \sum_{j=1}^{n-1}\pmb{b}_j$  where  $b_1, \ldots, b_{n-1}$  are the known mesh positions. Alternatively, if  $n\pmb{x}_i - \sum_{j=1}^{n-1}\pmb{b}_j$  is equal to the cell centroid as in Lemma A.3, it can be concluded that  $\{b_1, \ldots, b_{n-1}\}$  is the complete set of mesh nodes. Finally, the unique faces connecting these mesh nodes along the farfield can be deduced from the assumption that the faces of each cell along the farfield should enclose a singular volume. This is illustrated in Figure 6.
+
+![](images/6ffd304f665dcb6b6eb44cad0363813f3bd0df6fa9aebc8364c71701de9fb595.jpg)  
+Figure 5. Obtaining the mesh nodes and faces of all internal and geometry boundary faces.
+
+![](images/7004ae8295ee5da76ec4c2337670daa2872fbe889ecd0442f1367d13b9ec76b0.jpg)
+
+![](images/2296daf4266f958333c9696e3cf2433fba16d71d5b74ce1e5ebdb989c4642577.jpg)
+
+![](images/10c6a3bc79665624d803abb3545596313797138b1e3ea28d98e82eb5703d0038.jpg)  
+Figure 6. Obtaining corner mesh nodes and farfield boundary faces.
+
+![](images/4b914d812a797dde850ed72f1e7b62c74b199ece18ad4455e11711100fd8ea25.jpg)
+
+![](images/241efb5d744d57bff86895eca52631dfea02ac24da89c23d42217765cc46a4e1.jpg)
+
+# B. Implementation Details
+
+# B.1. Numerical DID Implementation
+
+While the continuous DID has already been defined, the geometry boundary had to be represented as a parametric equation, which may not always be available. Instead, we calculate a discrete DID using numerical integration, described in detail in Algorithm 1. Alternatively, the numerical DID calculation of an angle range can be simplified as:
+
+1. Obtain the mean distance to the node from every unobstructed boundary point:  $d$ .  
+2. Find the weighted proportion of the angle segment which faces the object boundary:  $w_{\theta}$ .  
+3. The DID value is calculated to be the weighted average of the mean distance to the boundary and the  $\infty$  parameter:  $w_{\theta} \cdot d + (1 - w_{\theta}) \cdot \infty$ .
+
+It can be seen from Algorithm 1 that the process runs in  $\mathcal{O}(|K|*|V|)$  time, where  $K$  is the set of boundary nodes and  $V$  is the set of all nodes. However, as  $|K| <   <   |V|$  in most practical CFD scenarios, the computation cost is almost linear to  $|V|$ .
+
+Also note that a singular geometry was assumed. For multiple-geometry scenarios, it has to be considered that  $w_{\theta}$  may not be represented by a single continuous angle range like  $(\theta_{min}, \theta_{max})$ . The rest of the algorithm may remain unchanged.
+
+In all experiments, the starting and ending angles of the DID segments were  $\left(\theta_{j},\theta_{j}^{\prime}\right) = \left(-\frac{\pi}{4},\frac{\pi}{4}\right),\left(0,\frac{\pi}{2}\right),\left(\frac{\pi}{4},\frac{3\pi}{4}\right),\left(\frac{\pi}{2},\pi\right),$ $\left(\frac{3\pi}{4},\frac{5\pi}{4}\right),\left(\pi ,\frac{3\pi}{2}\right),\left(\frac{5\pi}{4},\frac{7\pi}{4}\right),\left(\frac{3\pi}{2},2\pi\right)$ . These were chosen to give 8 overlapping arcs each spanning  $\frac{\pi}{2}$  degrees, centred at  $\frac{\pi}{4}$  intervals. The respective weight functions  $w_{j}(\theta)$  were uniform distributions over the  $(\theta_{j},\theta_{j}^{\prime})$  range. Finally, 4 was used as the large number  $\infty$ , as this was larger than any finite distance from a point in the mesh to the geometry boundary in any direction.
+
+Algorithm 1 Calculation of the DID of a field  
+1: Input: Set of nodes  $V$ , positions of each node  $pos = [(x_i, y_i) : i \in V]$  and boundary indices  $bd = [k \in V : k \text{ is on the geometry boundary}]$ , starting and ending angles  $\left[(\theta_j, \theta_j') : 0 \leq j < J\right]$ , weight functions  $[w_j : 0 \leq j < J]$ , and large value  $\infty$ .  
+2: Initialise  $DID \gets []$   
+3: for  $j \in [0, \dots, J-1]$  do  
+4: Initialise  $DID_j \gets []$   
+5: for  $i \in V$  do  
+6: Initialise  $\theta_{min}, \theta_{max} \gets \theta_j', \theta_j$   
+7: Initialise  $DID_i \gets 0$   
+8: Initialise  $w \gets 0$   
+9: for  $k \in bd$  do  
+10: Initialise  $\theta_{i,k} \gets \tanh((y_j - y_i) / (x_k - x_i))$   
+11: if  $(\theta_j < \theta_{i,k} < \theta_j')$  and  $(k \text{ is unobstructed from } i)$  then  
+12:  $DID_i \gets DID_i + w_j(\theta_{i,k}) * \min(||(x_i, y_i) - (x_k, y_k)||, \infty)$   
+13:  $w \gets w + w_j(\theta_{i,k})$   
+14:  $\theta_{min} \gets \theta_{i,k}$  if  $\theta_{i,k} < \theta_{min}$   
+15:  $\theta_{max} \gets \theta_{i,k}$  if  $\theta_{i,k} > \theta_{max}$   
+16: end if  
+17: end for  
+18:  $w_\theta \gets \left(\int_{\theta_{min}}^{\theta_{max}} w_j(\theta) d\theta\right) / \left(\int_{\theta_j}^{\theta_j'} w_j(\theta) d\theta\right)$   
+19:  $DID_i \gets DID_i / w$   
+20:  $DID_i \gets w_\theta * DID_i + (1 - w_\theta) * \infty$   
+21: end for  
+22:  $DID_j$ .append(DID_i)  
+23: end for  
+24: DID.append(DID_j)  
+25: Return: DID
+
+# B.2. FVF implementation
+
+Vanilla Graph Convolution and SAGE Convolution For models using vanilla graph convolutions, the convolutions were entirely replaced by the FVGC earlier described when FVF was used. For models using SAGE convolutions, FVF was used by implementing the FVGC as the aggregation function of the SAGE convolution. In both cases, the hidden dimension, or the number of filters, used in each convolution was always 3 for all convolutions in all experiments.
+
+Invariant Edge Convolution Following the notation of the original paper, the Invariant Edge (IVE) convolutions used by Chen et al. (2021) update the edge features  $\pmb{x}_e$  and node features  $\pmb{x}_v$  into  $\pmb{x}_e'$  and  $\pmb{x}_v'$  respectively, according to the following rules
+
+$$
+\boldsymbol {x} _ {e} ^ {\prime} = f _ {e} \left(\frac {\boldsymbol {x} _ {v _ {1}} + \boldsymbol {x} _ {v _ {2}}}{2}, \frac {\left| \boldsymbol {x} _ {v _ {1}} - \boldsymbol {x} _ {v _ {2}} \right|}{2}, \boldsymbol {x} _ {e}\right), \tag {9}
+$$
+
+$$
+\boldsymbol {x} _ {v} ^ {\prime} = f _ {v} \left(\boldsymbol {x} _ {v}, \sum_ {e _ {i} \in N (v)} \boldsymbol {x} _ {e _ {i}} ^ {\prime}\right), \tag {10}
+$$
+
+where  $v_{1}$  and  $v_{2}$  are the two nodes connected by edge  $e$ ,  $N(v)$  is the set of neighbouring edges around node  $v$ , and  $f_{e}$  and  $f_{v}$  are both MLPs with 1 hidden layer of 128 neurons. The number of neurons in the output layer, and hence output dimension of the kernel, can be customised. This is similar to the use of multiple filters in FVGC. Hence, when using FVF with IVE convolutions, only the node and edge attributes have to be implemented, resulting in the following rules
+
+$$
+\boldsymbol {x} _ {e} ^ {*} = \boldsymbol {q} _ {e} \oplus \boldsymbol {x} _ {e}, \quad \boldsymbol {x} _ {v} ^ {*} = \boldsymbol {x} _ {v} \oplus \boldsymbol {p}, \tag {11}
+$$
+
+$$
+\boldsymbol {x} _ {e} ^ {\prime} = f _ {e} \left(\frac {\boldsymbol {x} _ {v _ {1}} ^ {*} + \boldsymbol {x} _ {v _ {2}} ^ {*}}{2}, \frac {\left| \boldsymbol {x} _ {v _ {1}} ^ {*} - \boldsymbol {x} _ {v _ {2}} ^ {*} \right|}{2}, \boldsymbol {x} _ {e} ^ {*}\right), \tag {12}
+$$
+
+$$
+\boldsymbol {x} _ {v} ^ {\prime} = f _ {v} \left(\boldsymbol {x} _ {v} ^ {*}, \sum_ {e _ {i} \in N (v)} \boldsymbol {x} _ {e _ {i}} ^ {\prime}\right), \tag {13}
+$$
+
+where  $\pmb{p}$  is the node attributes and  $q_{e}$  is the edge attributes of the edge  $e$ . Note that the new edge and node features are still  $\pmb{x}_{e}^{\prime}$  and  $\pmb{x}_{v}^{\prime}$ , while  $\pmb{x}_{e}^{*}$  and  $\pmb{x}_{v}^{*}$  are just intermediate calculations.
+
+Unlike in vanilla graph convolutions or SAGE convolutions, the hidden dimension is not fixed at 3, but will instead follow the number of intermediate edge features used in the original work as described in the next section.
+
+MeshGraphNet Convolution Still following the notation of the original paper, the MeshGraphNet processor designed by Pfaff et al. (2021) used convolutions that updated the edge features  $e^{M}$  and node features  $v$  into  $e^{\prime M}$  and  $v^{\prime}$  according to the following
+
+$$
+\boldsymbol {e} _ {i j} ^ {\prime M} = f ^ {M} \left(\boldsymbol {e} _ {i j} ^ {M}, \boldsymbol {v} _ {i}, \boldsymbol {v} _ {j}\right),
+$$
+
+$$
+\boldsymbol {v} _ {i} ^ {\prime} = f ^ {V} \left(\boldsymbol {v} _ {i}, \sum_ {j} e _ {i j} ^ {\prime M}\right). \tag {14}
+$$
+
+Where  $v_{i}$  and  $v_{j}$  are the features of two nodes connected by an edge with edge features  $e_{ij}^{M}$ , and  $f^{M}$  and  $f^{V}$  are MLPs with 2 hidden layers of size 128 each. In all MeshGraphNet implementations, they have an output of size 128 as well. When using FVF, it was changed to the following
+
+$$
+\boldsymbol {e} _ {i j} ^ {* M} = \boldsymbol {e} _ {i j} ^ {M} \oplus \boldsymbol {q} _ {i j}, \quad \boldsymbol {v} _ {i} ^ {*} = \boldsymbol {v} _ {i} \oplus \boldsymbol {p} _ {i},
+$$
+
+$$
+\boldsymbol {e} _ {i j} ^ {\prime M} = f ^ {M} \left(\boldsymbol {e} _ {i j} ^ {* M}, \boldsymbol {v} _ {i} ^ {*}, \boldsymbol {v} _ {j} ^ {*}\right), \tag {15}
+$$
+
+$$
+\boldsymbol {v} _ {i} ^ {\prime} = f ^ {V} (\boldsymbol {v} _ {i} ^ {*}, \sum_ {j} \boldsymbol {e} _ {i j} ^ {\prime M}) .
+$$
+
+We also appended the FV node attributes and FV edge attributes to the inputs of the encoder and decoder convolutions accordingly.
+
+BSMS-GNN Convolution Likewise, the original BSMS-GNN processor designed by Cao et al. (2023) used convolutions to update the edge features  $\pmb{e}_l^s$  and node features  $\pmb{v}_l$  at level  $l$  for a problem involving  $S$  edge sets like so
+
+$$
+\boldsymbol {e} _ {l, i j} ^ {s} = f _ {l} ^ {s} \left(\Delta \boldsymbol {x} _ {l, i j}, \boldsymbol {v} _ {l, i}, \boldsymbol {v} _ {l, j}\right), \quad s = 1, \dots , S,
+$$
+
+$$
+\boldsymbol {v} _ {l, i} ^ {\prime} = \boldsymbol {v} _ {l, i} + f _ {l} ^ {V} \left(\boldsymbol {v} _ {l, i}, \sum_ {j} \boldsymbol {e} _ {l, i j} ^ {1}, \dots , \sum_ {j} \boldsymbol {e} _ {l, i j} ^ {S}\right). \tag {16}
+$$
+
+Where  $\Delta \pmb{x}_{l,ij} = \pmb{x}_i - \pmb{x}_j$  is the relative positions of the nodes  $i$  and  $j$ , and  $f_{l}^{s}$  and  $f_{l}^{V}$  are MLPs with 2 hidden layers of a hidden dimension of 128. For all BSMS-GNN implementations, an output of size 128 was used as well. With the FVF, it was changed into
+
+$$
+\boldsymbol {v} _ {l, i} ^ {*} = \boldsymbol {v} _ {l, i} \oplus p _ {i},
+$$
+
+$$
+\boldsymbol {e} _ {l, i j} ^ {s} = f _ {l} ^ {s} \left(\Delta x _ {l, i j}, \boldsymbol {v} _ {l, i} ^ {*}, \boldsymbol {v} _ {l, j} ^ {*}, q _ {i j}\right), \quad s = 1, \dots , S, \tag {17}
+$$
+
+$$
+\boldsymbol {v} _ {l, i} ^ {\prime} = \boldsymbol {v} _ {l, i} ^ {*} + f _ {l} ^ {V} (\boldsymbol {v} _ {l, i} ^ {*}, \sum_ {j} \boldsymbol {e} _ {l, i j} ^ {1}, \dots , \sum_ {j} \boldsymbol {e} _ {l, i j} ^ {S})
+$$
+
+As mentioned before, this was only done for the first and last convolutions of the processor where the graph is at full resolution. Similar to the MeshGraphNet convolution implementation, the FV node attributes were appended to the inputs of the encoder and decoder convolutions as well.
+
+Table 8. Sizes of the MeshGraphNet models compared.  
+
+<table><tr><td>MODELS</td><td>NUMBER OF TRAINABLE PARAMETERS</td></tr><tr><td>MESHGRAPHNET (BASELINE)</td><td>2282370</td></tr><tr><td>MESHGRAPHNET W/ GEO</td><td>2283650</td></tr><tr><td>MESHGRAPHNET W/ FVF W/ GEO</td><td>2302470</td></tr><tr><td>MESHGRAPHNET W/ RESIDUAL TRAINING W/ FVF W/ GEO</td><td>2302470</td></tr></table>
+
+Table 9. Sizes of the BSMS-GNN models compared.  
+
+<table><tr><td>MODELS</td><td>NUMBER OF TRAINABLE PARAMETERS</td></tr><tr><td>BSMS-GNN (BASELINE)</td><td>2578947</td></tr><tr><td>BSMS-GNN w/ GEO</td><td>2580227</td></tr><tr><td>BSMS-GNN w/ FVF w/ GEO</td><td>2582787</td></tr></table>
+
+# B.3. Model Architectures and Hyperparameter Choices
+
+MeshGraphNet The MeshGraphNet used an encoder-processor-decoder structure. The encoder consisted of two MLPs, for encoding node and edge features respectively. The input node and edge dimensions were both 3 for the baseline model. The number of input node channels increased to 13 if SV and DID were being used. Likewise, the input node channels increased again to 14, and the input edge features to 9, if FVF was implemented. It had a hidden and output size of 128. The decoder consisted of just one MLP to decode the node features only, with an input and hidden size of 128. If FVF was being used, the input size increased to 129. The output size was 2, for the velocity fields.
+
+The processor had 15 layers of the MeshGraphNet convolutions as earlier described. The input node and edge sizes were 128 unless FVF was implemented when the number of node channels increased to 129, and likewise, the edge channels increased to 134. The number of output nodes and edge channels was consistently 128. All activations used in the model were ReLU. The resulting number of trainable parameters in each model is shown in Table 8.
+
+Each model was trained for 250 epochs on half-precision, using a mean squared error (MSE) training loss with a learning rate of 0.0001 and a batch size of 1.
+
+BSMS-GNN The BSMS-GNN also had an encoder-processor-decoder structure. The encoder only consisted of one MLP of input size 1 that represented the node type. When SV and DID were used, this increased to 11. Likewise, when FVF was implemented, it increased again to 12. It had two hidden layers with a hidden size of 128 and an output of size 128 as well. The decoder was similarly an MLP with an input of size 128, or 129 if FV was used, and had 2 hidden layers with hidden size 128 and an output size of 3 for the velocity and pressure fields.
+
+The processor had an architecture similar to that of the Graph U-Net, with 9 different scales or levels and only 1 convolution per level. The number of nodes would decrease progressively from the bi-stride pooling method explained in the original paper. The input and output node and edge feature dimensions were 128 each, except when FVF was implemented, and the input node and edge features increased to 129 and 134, respectively. All activations used in the model were ReLU. The resulting number of trainable parameters in each model is shown in Table 9.
+
+Each model was trained on half-precision, using a mean squared error (MSE) training loss with a starting learning rate of 0.0001 that decayed exponentially at a decay rate of 0.7943 until the learning rate hit a minimum of 0.000001. All BSMS-GNN models were trained for 200 epochs with a batch size of 1.
+
+Chen-GCNN The input node features to the baseline model consisted of a boolean representation of the geometry and the spatial coordinates. For models using IVE convolution, there were also input edge features, which were simply the average of the node features of each node in the edge. Thus, the input node features and input edge features both have a size of 3. When geometry features were used, SV and DID replaced the boolean representation, making the feature sizes 12 instead.
+
+The model itself had eight convolution and smoothing layers, and finally a  $1 \times 1$  convolution output layer. When IVE convolutions were used, the intermediate edge and node feature dimensions in each layer were (4,8,16,32,64,64,32,16)
+
+Table 10. Sizes of the Chen-GCNN models compared.  
+
+<table><tr><td>MODELS</td><td>CONV TYPE</td><td>NUMBER OF TRAINABLE PARAMETERS</td></tr><tr><td>CHEN-GCNN (BASELINE)</td><td>INVARIANT</td><td>217853</td></tr><tr><td>CHEN-GCNN w/ GEO</td><td>EDGE</td><td>222461</td></tr><tr><td>CHEN-GCNN w/ FVF w/ GEO</td><td>CONVOLUTION</td><td>227185</td></tr><tr><td>CHEN-GCNN (BASELINE)</td><td>SAGE</td><td>20193</td></tr><tr><td>CHEN-GCNN w/ GEO</td><td>CONVOLUTION</td><td>20337</td></tr><tr><td>CHEN-GCNN w/ FVF w/ GEO</td><td></td><td>58067</td></tr></table>
+
+Table 11. Sizes of the Graph U-Net models compared.  
+
+<table><tr><td>MODELS</td><td>CONV TYPE</td><td>NUMBER OF TRAINABLE PARAMETERS</td></tr><tr><td>GRAPH U-NET (BASELINE)</td><td>SAGE</td><td>65820</td></tr><tr><td>GRAPH U-NET w/ GEO</td><td rowspan="2">CONVOLUTION</td><td>66396</td></tr><tr><td>GRAPH U-NET w/ FVF w/ GEO</td><td>160465</td></tr><tr><td>GRAPH U-NET (BASELINE)</td><td>VANILLA</td><td>107667</td></tr><tr><td>GRAPH U-NET w/ GEO</td><td>GRAPH</td><td>108243</td></tr><tr><td>GRAPH U-NET w/ FVF w/ GEO</td><td>CONVOLUTION</td><td>105201</td></tr></table>
+
+and (8, 16, 32, 64, 64, 32, 16, 8) respectively, as they were in the original work. When SAGE convolutions were used, there were no intermediate edge feature dimensions. Instead, the number of filters used in the convolutions was fixed at 3, and the node feature dimensions remained (8, 16, 32, 64, 64, 32, 16, 8) as in the IVE convolutions. The final output layer produced 3 node features representing predicted velocity and pressure. There are also skip connections from the input graph to the output of every convolutional and smoothing block, where the spatial coordinates of the nodes were concatenated to the node features. This is similar to the node attributes concatenated to the node features in the convolutions using FVF, although the cell volume was not used as a node attribute in the original work.
+
+Table 10 shows the number of trainable parameters in each model. The model depth and width were kept the same as the original Chen-GCNN model (Chen et al., 2021) as it was already relatively small.
+
+In training, the loss function chosen was the mean absolute error (MAE). We implemented early stopping, where training terminated if the MAE on the validation set did not improve after 60 epochs, or if it reached the maximum of 1000 epochs. A batch size of 64 was used, and the initial learning rate and decay rate were both 0.002 to match the original work, with the decay schedule also following the same formula. Further details can be found in the original codebase<sup>1</sup>. The only difference in our training scheme implementation would be that we used half-precision training, and clipped the gradients to 5.
+
+Graph U-Net Our implementation of the Graph U-Net was adapted from the original codebase<sup>2</sup>. The baseline model started and ended with an MLP encoder and decoder of layer sizes [7, 64, 64, 8] and [8, 64, 64, 4] respectively. The models that used geometry features had an encoder with layer sizes [16, 64, 64, 8] instead, as the SDF input feature was replaced with SV and DID.
+
+For the U-Net itself, the models that did not use FVF both followed the original Graph U-Net architecture exactly. The downward and upward passes had five scales each. At each scale of the downward pass, the number of node features doubled, while the graph nodes were down-sampled by half to create radius graphs of radii  $5\mathrm{cm}$ ,  $20\mathrm{cm}$ ,  $50\mathrm{cm}$ ,  $1\mathrm{m}$  and  $10\mathrm{m}$  respectively. In the upward pass, skip connections concatenated the node features of the respective scale in the downward pass to that of the previous scale. In models that used FVF, however, no down-sampling or up-sampling was done to preserve the mesh structure and its corresponding mesh characteristics. Nevertheless, the depth of the model and the presence of skip-links remained the same.
+
+Table 11 shows the number of trainable parameters in each model. Models that used SAGE convolutions kept the original
+
+Table 12. Sizes of the CFDGCN models compared.  
+
+<table><tr><td>MODELS</td><td>CONV TYPE</td><td>MODEL WIDTH</td><td>#TRAINABLE PARAMETERS</td></tr><tr><td>CFDGCN (BASELINE)</td><td>VANILLA</td><td>512</td><td>1057283</td></tr><tr><td>CFDGCN w/ GEO</td><td>GRAPH</td><td>512</td><td>1061891</td></tr><tr><td>CFDGCN w/ FVF w/ GEO</td><td>CONVOLUTION</td><td>284</td><td>1021780</td></tr><tr><td>CFDGCN w/ RESIDUAL TRAINING w/ FVF w/ GEO</td><td></td><td>284</td><td>1021780</td></tr><tr><td>CFDGCN (BASELINE)</td><td></td><td>512</td><td>2112003</td></tr><tr><td>CFDGCN w/ GEO</td><td>SAGE</td><td>512</td><td>2121219</td></tr><tr><td>CFDGCN w/ FVF w/ GEO</td><td>CONVOLUTION</td><td>315</td><td>2153241</td></tr><tr><td>CFDGCN w/ RESIDUAL TRAINING w/ FVF w/ GEO</td><td></td><td>315</td><td>2153241</td></tr></table>
+
+Graph U-Net model width of 8 (Bonnet et al., 2022a), causing the model sizes to increase from the use of SV, DID, and FVF. This is to be consistent with the experimental practices of Bonnet et al. (2022a), who allowed their model sizes to vary. On the other hand, when vanilla graph convolutions were used, the width of the models without FVF was increased to 15 to make the model sizes similar to the model with FVF. This is because the models with only vanilla graph convolution without FVF were too small to train effectively when using the original model parameters.
+
+The learning rate for the experiments was set with a one-cycle cosine rate capped at 0.001, with the number of epochs fixed at 1600 as in the scarce data regime of the original work (Bonnet et al., 2022a). Half precision was used, and gradients clipped to 5.
+
+For models that did not use FVF, mesh node-based graphs were used. However, the total number of nodes was fixed at 32000 via random subsampling of the input. Edges were formed between nodes within  $5\mathrm{cm}$  of each other, with each node having a maximum of 64 neighbours. In training, each input graph was only subsampled once, and the training loss found against the subsampled ground-truth. In testing, subsampling was done repeatedly till a value was found for all the nodes in the original graph, and nodes with multiple values were assigned the average, before the test losses found against the unsampled ground-truth. On the other hand, on FVF models, an unsampled cell centroid-based graph was used in both training and testing. As before, this was to maintain the mesh's structure and characteristics represented in the FVF.
+
+CFDGCN As explained earlier, to adapt the CFDGCN model to become generalisable to different geometries, we used the coarse mesh of each flow scenario in the data as a fixed input to the model, rather than as a trainable feature of the model. On all other aspects, however, the architecture of the CFDGCN was largely preserved.
+
+The coarse mesh was upsampled once using squared distance-weighted, k-nearest neighbours interpolation to size of the fine mesh. The input node features of the baseline model were the spatial coordinates, angle of attack, mach number and SDF. If geometry features were used, the SDF was replaced by SV and DID. The graph was passed through 3 graph convolutions before the upsampled coarse mesh was appended to the output of the 3rd layer, and another 3 convolutions was performed to generate the final prediction.
+
+All convolutions in models that did not use FVF had 512 hidden channels, just as the original CFDGCN did (Belbute-Peres et al., 2020). However, the number of hidden channels was adjusted for the FV and residual models to keep model size similar for comparability. Table 12 shows the number of trainable parameters in each model. A batch size of 1 was used on all experiments, and half precision was used. All other training parameters were kept the same as the original work. More details can be found from the codebase<sup>3</sup>.
+
+# C. Coarse AirfRANS Dataset
+
+AirfRANS (Bonnet et al., 2022a) provides 1000 simulated airfoil cases but with a consistent mesh resolution. Having simulations of the same airfoil geometry but from a lower resolution is important for learning tasks such as super-resolution, which is needed for CFDGCN (Belbute-Peres et al., 2020). Since the residual training scheme is intended, albeit not limited, to demonstrate superior performance on super-resolution task, a lower-resolution variant of the AirfRANS cases were needed. The AirfRANS authors released the mesh generation script, which made it possible.
+
+Table 13. The new values of different cell grading for coarse AirfRANS dataset.  
+
+<table><tr><td>CELL GRADING</td><td>NEW VALUES</td></tr><tr><td>YGRADING</td><td>1000</td></tr><tr><td>YUGRADING</td><td>1000</td></tr><tr><td>YDGRADING</td><td>1000</td></tr><tr><td>XUUGRADING</td><td>10</td></tr><tr><td>XDUGRADING</td><td>10</td></tr><tr><td>XUMAEROGRADING</td><td>2</td></tr><tr><td>XDMAEROGRADING</td><td>2</td></tr><tr><td>XMGRADING</td><td>1</td></tr><tr><td>XDTRAILGRADING</td><td>0.0001</td></tr><tr><td>XUDGRADING</td><td>0.5</td></tr><tr><td>XDDGRADING</td><td>0.5</td></tr><tr><td>LEADUGRADING</td><td>0.05</td></tr><tr><td>LEADDGRADING</td><td>0.05</td></tr></table>
+
+Table 14. Extrapolation and full data results of the CFDGCN models with vanilla graph convolution  
+
+<table><tr><td>MODELS</td><td>DATA
+REGIME</td><td>MSE
+×10-2</td></tr><tr><td>CFDGCN (BASELINE)</td><td></td><td>0.3210</td></tr><tr><td>CFDGCN w/ GEO</td><td>REYNOLD&#x27;S</td><td>0.0889</td></tr><tr><td>CFDGCN w/ FVF w/ GEO</td><td>NUMBER</td><td>0.0678</td></tr><tr><td>CFDGCN w/ RESIDUAL TRAINING W/ FVF w/ GEO</td><td></td><td>0.0661</td></tr><tr><td>CFDGCN (BASELINE)</td><td></td><td>0.9098</td></tr><tr><td>CFDGCN w/ GEO</td><td>AOA</td><td>0.1846</td></tr><tr><td>CFDGCN w/ FVF w/ GEO</td><td></td><td>0.1772</td></tr><tr><td>CFDGCN w/ RESIDUAL TRAINING W/ FVF w/ GEO</td><td></td><td>0.1595</td></tr><tr><td>CFDGCN (BASELINE)</td><td></td><td>0.0737</td></tr><tr><td>CFDGCN w/ GEO</td><td>FULL</td><td>0.0668</td></tr><tr><td>CFDGCN w/ FVF w/ GEO</td><td></td><td>0.0655</td></tr><tr><td>CFDGCN w/ RESIDUAL TRAINING W/ FVF w/ GEO</td><td></td><td>0.0499</td></tr></table>
+
+The majority of settings in the original AirfRANS dataset remain unchanged, except for the mesh resolution. In this modified dataset, the number of cells in all directions has been reduced to one-quarter of the original settings. Additionally, the gradings of the cell expansion have been reset to ensure a smooth transition of the cell thickness as listed in Table 13.
+
+# D. Computational Environment
+
+All experiments are run on a server with 32 Intel(R) Xeon(R) Silver 4110 CPU @ 2.10GHz processors, 256 GB RAM, and 3 dedicated Nvidia V100 GPU cards each with 32GB memory. All models are trained on a single GPU.
+
+# E. Additional Results
+
+# E.1. Extrapolation and Full Data Regimes
+
+While the experiments using the AirfRANS dataset were done using the scarce data regime, more CFDGCN models were trained using the Reynold's Number and AoA extrapolation regimes and the full data regime. The results are shown in Table 14. They demonstrate that the proposed methods remain effective under a variety of scenarios with different data availability.
+
+![](images/28282574d954731114de23d3c051c100226efbf3f837fcd5cf49b0084ec9b4ed.jpg)  
+Figure 7. Velocity x-component (u)
+
+![](images/40fc94b1145870294251101c1d255626622d5669aaa6a26942f0eefb49575353.jpg)  
+Figure 8. Velocity y-component (v).
+
+# E.2. Rationale for residual training
+
+As mentioned earlier, the residual field helps the model to focus on the more nuanced areas where the low-resolution fields tend to be inaccurate. In Figures 7 and 8, we illustrate this phenomenon to emphasise the importance of residual training. We show ground truth velocity components (u,v) and pressure (p) corresponding to the ground truth flow field on the left, while the corresponding residual fields are on the right. In most of the regions, the residual is close to zero. Hence, the model has less difficulty learning flow field values in these regions, which are coloured in darker shades.
+
+# F. Predictive Error Visualisation
+
+Figure 10 shows a comparison between the predicted velocity x-component flow fields,  $u$ , of the baseline method Chen-GCNN and Chen-GCNN w/ FVF w/ Geo. Both models used invariant edge convolutions. We observe that the predictions of both methods capture the flow features present in the Ground truth,  $\mathrm{GT}(u)$ . A close inspection of the absolute error of the predicted flow field in figure 11 reveals that Chen-GCNN w/ FVF w/ Geo produces relatively small regions with high error, whereas there are many regions with high error in the flow field predicted by Chen-GCNN. This observation indicates that incorporating our proposed features reduces the predictive error across different regions in the domain.
+
+![](images/d99fe492bd23bd1bf44229080a0c87163a4853ea9dcc612530bd3c8a1e66199a.jpg)  
+Figure 9. Pressure (p).
+
+![](images/a6efe1b87f867d70eeaac8e13cbd94ed208a896a1e09c720ac0991d19ae5e6ad.jpg)  
+Figure 10. Velocity x-component (u) visualisation.
+
+# G. Boundary Layer Profile
+
+We analyse the velocity profile at the boundary layer in Figure 12. We plot the normalised distance from the airfoil boundary  $(y / c)$  vs the normalised velocity x-component  $u / U_{\infty}$ . Figure 12 shows that GUNet w/ Geo predicts velocity  $x$ -component near the airfoil boundary more accurately compared to the baseline GUNet.
+
+![](images/56982ff2a37fcd16ba3dacd46254e5113b43f8de18c0931a34d4c642408f5d8c.jpg)  
+Figure 11. Absolute error of predictions.
+
+![](images/f6dfe080c5f8e3f2890cf71a5de73ee9c9306fa8ae7264758dfb2572d2feb387.jpg)  
+$|\hat{U} - U|$
+
+![](images/706ad8683ceb128b6ab24ec11e7786cde9f63cb27fef9d58ce515f6981d35f99.jpg)  
+Figure 12. Comparison of the predicted boundary layers profiles on two random test airfoils at four different abscissas in the scarce data regime with respect to the true ones. Each row of plots represents a different airfoil, and each column of plots represents a different abscissa. The  $x$  component of the velocity is denoted by u, and the inlet velocity is denoted by  $U_{\infty}$ .

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2024/First-Order Manifold Data Augmentation for Regression Learning/full.md

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+# First-Order Manifold Data Augmentation for Regression Learning
+
+Ilya Kaufman<sup>1</sup> Omri Azencot<sup>1</sup>
+
+# Abstract
+
+Data augmentation (DA) methods tailored to specific domains generate synthetic samples by applying transformations that are appropriate for the characteristics of the underlying data domain, such as rotations on images and time warping on time series data. In contrast, domain-independent approaches, e.g. mixup, are applicable to various data modalities, and as such they are general and versatile. While regularizing classification tasks via DA is a well-explored research topic, the effect of DA on regression problems received less attention. To bridge this gap, we study the problem of domain-independent augmentation for regression, and we introduce FOMA: a new data-driven domain-independent data augmentation method. Essentially, our approach samples new examples from the tangent planes of the train distribution. Augmenting data in this way aligns with the network tendency towards capturing the dominant features of its input signals. We evaluate FOMA on in-distribution generalization and out-of-distribution robustness benchmarks, and we show that it improves the generalization of several neural architectures. We also find that strong baselines based on mixup are less effective in comparison to our approach. Our code is publicly available at https://github.com/azencot-group/FOMA
+
+# 1. Introduction
+
+Classification and regression problems primarily differ in their output's domain. In classification, we have a finite set of labels, whereas in regression, the range is an infinite set of quantities—either discrete or continuous. In classical work (Devroye et al., 2013), classification is argued to be "easier" than regression, but more generally, it is agreed by
+
+$^{1}$ Department of Computer Science, Ben-Gurion University of the Negev, Beer-Sheva, Israel. Correspondence to: Ilya Kaufman <ilyakau@post.bgu.ac.il>.
+
+Proceedings of the  $41^{st}$  International Conference on Machine Learning, Vienna, Austria. PMLR 235, 2024. Copyright 2024 by the author(s).
+
+many that classification and regression problems should be treated differently (Muthukumar et al., 2021). Particularly, the differences between classification and regression are actively explored in the context of regularization. Regularizing neural networks to improve their performance on new samples has received a lot of attention in the past few years. One of the main reasons for this increased interest is that most of the recent successful neural models are overparameterized. Namely, the amount of learnable parameters is significantly larger than the number of available training samples (Allen-Zhu et al., 2019a;b), and thus regularization is often necessary to alleviate overfitting issues. Recent studies on overparameterized linear models identify conditions under which overfitting is "benign" in regression (Bartlett et al., 2020), and uncover the relationship between the choice of loss functions in classification and regression tasks (Muthukumar et al., 2021). Still, the regularization of deep neural regression networks is not well understood.
+
+In this work, we focus on regularizing deep models via Data Augmentation (DA), where data samples are artificially generated and used during training. In general, DA techniques can be categorized into domain-dependent (DD) methods and domain-independent (DI) approaches. The former are specific for a certain data modality such as images or text, whereas the latter typically do not depend on the data format. Numerous DD- and DI-DA approaches are available for classification tasks (Shorten & Khoshgoftaar, 2019; Shorten et al., 2021), and many of them consistently improve over non-augmented models. Unfortunately, DI-DA for regression problems is underexplored. Recent works on linear models study the connection between the DA policy and optimization (Hanin & Sun, 2021), as well as the generalization effects of linear DA transformations (Wu et al., 2020). We contribute to this line of work by proposing and analyzing a new domain-independent data augmentation method for nonlinear deep regression, and by testing our approach on in-distribution generalization and out-of-distribution robustness tasks (Yao et al., 2022).
+
+Many strong data augmentation methods were proposed in the past few years. Particularly relevant to our study is the family of mixup-based techniques that are commonly used in classification applications. The original method, mixup (Zhang et al., 2017), produces convex combinations of training samples, promoting linear behavior for
+
+in-between samples. The method is domain-independent and data-agnostic, i.e., it is indifferent to the given data samples. Mixup was shown to solve the Vicinal Risk Minimization (VRM) problem instead of the usual Empirical Risk Minimization (ERM) problem. In comparison, our approach can also be viewed as solving a VRM problem, and it is domain-independent and data-driven, namely, augmentations depend on the given data distribution. In our extensive evaluations, we will show that mixup-based methods are less effective for regression in comparison to our approach.
+
+Contributions. Challenged by the differences between classification and regression and motivated by the success of domain-independent methods such as mixup, we propose a simple, domain-independent and data-driven DA routine, termed First-Order Manifold Augmentation (FOMA). Let  $X, Y$  be the input and output mini-batch tensors, respectively, and let  $Z_{l} = g_{l}(X)$  be the hidden code at layer  $l$ . Our method produces new training samples  $Z_{l}(\lambda), Y(\lambda)$  from the given ones by scaling down their small singular values by a random  $\lambda \in [0,1]$ . At its core, FOMA incorporates into training the assumption that data with similar dominant components of the train set should be treated as true samples. Our implementation of FOMA is fully differentiable, and thus it is applicable to any layer of a given network.
+
+We detail FOMA in Sec. 3, motivating our design choices and illustrating its effect on data. We analyze our approach using perturbation theory and introduce its associated vicinal risk minimization (Sec. 4). Our experimental evaluation focuses on in-distribution generalization (Sec. 5.1) and on out-of-distribution robustness (Sec. 5.2), where we empirically demonstrate the superiority of FOMA. We offer a potential explanation to the success of our method (Sec. 3, App. B). Finally, an ablation study is performed, justifying our design choices (Sec. 5.3).
+
+# 2. Related Work
+
+Deep neural networks regularization is an established research topic with several existing works (Goodfellow et al., 2016). Common regularization approaches include weight decay, dropout (Srivastava et al., 2014), batch normalization (Ioffe & Szegedy, 2015), and data augmentation (DA). Here, we categorize DA techniques to be either domain-dependent or domain-independent. Domain-dependent DA was shown to be effective for, e.g., image data (LeCun et al., 1998) and audio signals (Park et al., 2019), among other domains. However, adapting these methods to new data formats is typically challenging and often infeasible. While several works focused on automatic augmentation (Lemley et al., 2017; Cubuk et al., 2019; Lim et al., 2019; Tian et al., 2020; Cubuk et al., 2020), there is concurrently an increased interest on domain-independent DA methods, allowing to regularize
+
+neural networks when only basic data assumptions are allowed (Naiman et al., 2023). We focus in what follows on domain-independent techniques that were proposed in the context of classification and regression problems.
+
+DA for classification. Zhang et al. (2017) proposed to perform convex mixing of input samples as well as one-hot output labels during training. The new training procedure, named mixup, minimizes the Vicinal Risk Minimization (VRM) problem instead of the typical Empirical Risk Minimization (ERM). Many extensions of mixup were proposed, including mixing latent features (Verma et al., 2019), same-class mixing (DeVries & Taylor, 2017), among other extensions (Guo et al., 2019; Hendrycks et al., 2019; Yun et al., 2019; Berthelot et al., 2019; Greenewald et al., 2021; Lim et al., 2021). ISDA (Wang et al., 2019) formulates a new cross-entropy loss for DA-based training using the per-class covariance matrix.
+
+DA for regression. Significantly less attention has been drawn to designing domain-independent data augmentation for regression tasks. A recent survey (Wen et al., 2020) on DA for time series data lists a few basic augmentation tools. Dubost et al. (2019) propose to recombine samples for regression tasks with countable outputs, and thus their method can not be directly extended to the uncountable regime. RegMix (Hwang & Whang, 2021) developed a meta learning framework based on reinforcement learning for mixing samples in their neighborhood. A recent work (Yao et al., 2022) showed that applying vanilla mixup with adjusted sampling probabilities based on label similarity can improve generalization on regression tasks. Another work (Schneider et al., 2023) suggests DA based on Anchor regression (Rothenhausler et al., 2021) which allows mixing multiple samples based on their cluster that encodes a homogeneous group of observations.
+
+# 3. First-Order Manifold Augmentation
+
+A learning task is typically described as a function which maps inputs to outputs. In this view, a learning model is approximating that function using e.g., a neural network, and it is formulated via  $f: \mathcal{X} \to \mathcal{Y}$ , denoting the input and output domains by  $\mathcal{X}$  and  $\mathcal{Y}$ , respectively. A regression problem is such that the output domain is (un)countable, e.g.,  $\mathcal{Y} \subset \mathbb{N}^m$  or  $\mathcal{Y} \subset \mathbb{R}^m$ . We consider the general setting where  $\mathcal{X} \subset \mathbb{R}^n$ ,  $\mathcal{Y} \subset \mathbb{R}^m$ . During training, the learning model is provided with a training set  $\mathcal{D} = \{(x_i, y_i)\}_{i=1}^N$ , sampled from  $(x_i, y_i) \sim \mathcal{P}$ . Our method extends  $\mathcal{D}$  by producing a new training distribution as we describe below.
+
+To generate new samples, we consider the singular value decomposition (SVD) of a matrix  $M \in \mathbb{R}^{q \times r}$ ,  $q \geq r$  which is given by  $M = USV^T$ .  $U, V$  are orthogonal, and  $S$
+
+```python
+def scale(A, k, lam):
+    U, s, Vt = torch.linalg.svd(A, full_matrices=False)
+    lamrepeat = lamrepeat(s.shape[-1] - k)
+    lamonoes = torch.ones(k)
+    s = s * torch.cat((lamonoes, lamrepeat))
+    A = U @ torch.diag(s) @ Vt
+    return A
+# X, Y are in batch x feats
+for (X, Y) in loader:
+    lam = beta.Beta(alpha, alpha).sample()
+    A = torch.catenate((X, Y), axis=1)
+    A = scale(A, k, lam)
+    X, Y = A[:, :, n], A[:, n:]
+    optimizer.zero_grad()
+    loss(net(X), Y).backward()
+    optimizer step()
+```
+
+![](images/038d50f03e6933fd8d9511928eee9ec530f6a3b4e2ae06937bf85e49f338036c.jpg)  
+Figure 1. We show the pseudocode for FOMA at the input level,  $l = 0$  (left). We demonstrate the effect of a few DA methods on 2D data whose intrinsic dimension is one (right).
+
+is diagonal consisting of the singular values ordered by  $\sigma_{1} \geq \sigma_{2} \geq \dots \geq \sigma_{r} \geq 0$ . SVD is intimately related to principal component analysis (PCA) which in turn is heavily studied in manifold learning and dimensionality reduction (Ma & Fu, 2012). It is well known that the best rank  $k$  approximation of  $M$  is given by omitting the last  $(r - k)$  singular values, i.e.,  $M_{k} = \sum_{j=1}^{k} \sigma_{j} u_{j} v_{j}^{T}$  (Eckart & Young, 1936). The matrix  $M_{k}$  preserves the  $(k)$  dominant components in  $M$ , and discards the rest. Further, SVD is also known to yield a first-order approximation of the data manifold (Singer & Wu, 2012; Kaufman & Azencot, 2023). Our key insight is that scaling the small singular values produces training samples that are in close proximity to the manifold, and thus to the true data distribution  $\mathcal{P}$ .
+
+In particular, based on the manifold hypothesis (Cayton et al., 2008), we assume that the data samples  $\mathcal{D}$  live on or close to a manifold  $\mathcal{M} \subset \mathcal{X} \times \mathcal{Y}$ . We denote by  $\mathcal{T}(x,y)$  the tangent plane of the data manifold  $\mathcal{M}$  at the point  $(x,y) \in \mathcal{M}$ . Namely,  $\mathcal{T}(x,y)$  is the linear approximation of  $\mathcal{M}$  at  $(x,y)$ . Below, we utilize SVD to approximate  $\mathcal{T}(x,y)$ , and to generate artificial samples by considering pairs  $(\tilde{x},\tilde{y})$  in the tangent plane of the  $(x,y)$ .
+
+Let the input and output mini-batch tensors  $X \in \mathbb{R}^{b \times n_0}$  and  $Y \in \mathbb{R}^{b \times m}$ , respectively, where w.l.o.g  $b \geq n_0 + m$  is the batch size. We denote the network by  $f(X) = f_l(g_l(X))$ ,  $Z_l \coloneqq g_l(X)$  where  $g_l$  maps inputs to latent representations  $Z_l \in \mathbb{R}^{b \times n_l}$  at layer  $l \in [0, L]$ , and  $f_l$  maps latent vectors to outputs. Let  $\lambda \sim \mathrm{Beta}(\alpha, \alpha)$  for  $\alpha \in (0, \infty)$  and  $k \in [1, n_l + m]$  be the index of the singular value after which we scale down. Then, the new artificial samples
+
+$Z_{l}(\lambda ,k),Y(\lambda ,k)$  are defined via
+
+$$
+A := \left[ Z _ {l}, Y \right] = U S V ^ {T} \in \mathbb {R} ^ {b \times \left(n _ {l} + m\right)},
+$$
+
+$$
+A (\lambda , k) := U S (\lambda , k) V ^ {T} \in \mathbb {R} ^ {b \times (n _ {l} + m)},
+$$
+
+$$
+Z _ {l} (\lambda , k) := A (\lambda , k) _ {1: n _ {l}} \in \mathbb {R} ^ {b \times n _ {l}},
+$$
+
+$$
+Y (\lambda , k) := A (\lambda , k) _ {n _ {l} + 1: n _ {l} + m} \in \mathbb {R} ^ {b \times m},
+$$
+
+where  $[\cdot, \cdot]$  concatenates along columns,  $A_{1:i}$  stands for the first  $i$  column vectors in  $A$ , and  $S(\lambda, k)$  is the diagonal matrix of scaled down singular values. Namely, we compute  $S(\lambda, k) = \mathrm{diag}(\sigma_1, \ldots, \sigma_k \mid \lambda \sigma_{k+1}, \ldots, \lambda \sigma_{n_l + m})$ . We propose two methods for choosing the parameter  $k$ , one is based on the intrinsic dimension of the data, denoted by  $\mathrm{FOMA}$  and the second is based on the explained variance of the data denoted by  $\mathrm{FOMA}_{\rho}$ .
+
+Intrinsic dimension. We set the value of  $k$  to be equal the intrinsic dimension (ID) of  $A$ , i.e., the minimal number of features needed to represent the data with little information loss (Facco et al., 2017). The ID can be estimated for the entire dataset at the input level, or alternatively, approximated per batch during the training process. Setting  $k$  to equal the ID is motivated by our analysis below in Sec. 4, where we show that FOMA is equivalent to sampling from the tangent space of the manifold. In practice, we use  $k$  dominant singular vectors to approximate the tangent space.
+
+Explained variance. In addition to the ID, we also consider the value  $k$  to depend on the hyper-parameter  $\rho \in [0,1]$  that represents the "amount" of signal to keep unchanged, i.e.,
+
+$$
+k = \arg \max  _ {\tilde {k}} \sum_ {j = 1} ^ {\tilde {k}} \sigma_ {j} / \sum_ {j} \sigma_ {j} \leq \rho .
+$$
+
+Similar to mixup (Zhang et al., 2017), our method recovers the original dataset  $\mathcal{D}$  as  $\alpha \to 0, \forall k$ .
+
+![](images/71f519a6b031ac77ab4e124acad1654ca9a8ba6c45134fd7cb91bfe546f102b6.jpg)  
+Figure 2. Training stability and overfitting. (a) RMSE loss on the train set. (b) RMSE loss on the test set. (c) Generalization gap: the difference between test error and train error
+
+![](images/73948c6cb7e269bafb28d424368152fae4df0d0f6e6722d47f68963f107080e5.jpg)
+
+![](images/5cb9ebdc467aeea64b1e82a8263fdfaffa836601d9e8b4b75b5a3661dd55ca77.jpg)
+
+The loss function associated with FOMA is
+
+$$
+\begin{array}{l} \mathcal {L} (f) = \mathbb {E} _ {(X, Y)} \mathbb {E} _ {\lambda} \mathbb {E} _ {l} c _ {\lambda} \left[ \left(f _ {l}, 1\right) \circ \chi \left(g _ {l} (X), Y\right)\right) ], \\ \begin{array}{r l} \mathrm {s . t .} & (X, Y) \sim \mathcal {D}, \lambda \sim \mathcal {H} (\sigma), l \sim [ 0, L ], \end{array} \\ \end{array}
+$$
+
+where  $c_{\lambda}:\mathbb{R}^{b\times m}\times \mathbb{R}^{b\times m}\to [0,\infty)$  is a cost function, typically mean squared error (MSE). The scale transform  $\chi$  takes a pair of tensors  $g_{l}(X),Y$  , and it scales down by  $\lambda$  the last  $(n_l + m - k)$  singular values of their concatenation. A key attribute of FOMA is that it is fully differentiable since the singular value decomposition can be backpropagated (Ionescu et al., 2015). We provide an example PyTorch pseudocode in Fig. 1 (left). The computational complexity of FOMA is governed by SVD calculation which has a complexity of  $\mathcal{O}(\min (qr^2,rq^2))$  for a  $q\times r$  matrix.
+
+Design choices. For certain  $\lambda$  values, the new sample  $Z_{l}(\lambda ,k),Y(\lambda ,k)$  may be too far from  $\mathcal{P}$ . With this in mind, we explored the option of scaling down the loss function  $c(\cdot ,\cdot)$  by a parameter  $\mu (\lambda)$  in addition to modifying the singular values. However, we tested various profiles  $\mu (\lambda)$  and discovered the most consistent models are obtained when no scaling of loss occurs, see Sec. 5.3. Importantly, this means that our approach adopts a different ansatz in comparison to mixup-based methodologies. While mixup incorporates uncertainty into the model training using "inbetween" samples and labels, our method uses the new data as if it was sampled from the true distribution, since we do not scale  $c$ . An alternative option which would be conceptually closer to mixup is to scale the large singular
+
+values as well as the loss term. We show in Sec. 5.3 that this choice is usually inferior to FOMA.
+
+Batch selection. Given a sample  $(x,y) \in \mathcal{X} \times \mathcal{Y}$ , we aim to produce training samples that are in close proximity to the manifold  $\mathcal{M}$ , and thus to the true data distribution  $\mathcal{P}$ . However, we do not know the structure of  $\mathcal{M}$ , and thus we can not sample from it directly, unfortunately. To overcome this challenge, we sample from a linear approximation of the  $\mathcal{M}$  at  $(x,y)$  given by the tangent plane  $\mathcal{T}(x,y)$ . In practice, we need a set of points  $N_{(x,y)} = \{(x_i,y_i)\}$  in the neighborhood of  $(x,y)$  to estimate  $\mathcal{T}(x,y)$  via SVD. The proximity of the points  $N_{(x,y)}$  to  $(x,y)$  has a direct effect on the quality of the approximation of the tangent plane. In practice, we construct batches of points that are close to each other as measured by the Euclidean distances between the labels in  $\mathcal{V}$ . We hypothesize that better approximations should generate samples closer to the data manifold, improving empirical test results. To validate our claim, we tested two batch selection methods for training 1) randomly selecting batches (random); and 2) constructing batches using samples of points that are close to each other (close). For each setting we trained several models and selected the model that achieved the best performance on the validation set. In Table 1, we report the results of batch selection methods on the test set. It is notable that constructing batches of close points outperforms random batch selection.
+
+Table 1. Results for different batch selection methods FOMA.  
+
+<table><tr><td>Dataset</td><td>Airfoil ↓</td><td>NO2 ↓</td><td>Exchange ↓</td><td>Electricity ↓</td><td>Echo ↓</td><td>RCF ↓</td><td>Crimes ↓</td><td>Poverty ↑</td><td>DTI ↑</td></tr><tr><td>FOMA- random</td><td>2.159</td><td>0.515</td><td>0.014</td><td>0.059</td><td>5.248</td><td>0.175</td><td>0.144</td><td>0.433</td><td>0.491</td></tr><tr><td>FOMA- close</td><td>1.646</td><td>0.521</td><td>0.013</td><td>0.058</td><td>5.215</td><td>0.171</td><td>0.132</td><td>0.492</td><td>0.496</td></tr><tr><td>FOMAρ - random</td><td>2.012</td><td>0.514</td><td>0.014</td><td>0.060</td><td>5.224</td><td>0.163</td><td>0.134</td><td>0.409</td><td>0.508</td></tr><tr><td>FOMAρ - close</td><td>1.471</td><td>0.512</td><td>0.013</td><td>0.059</td><td>5.512</td><td>0.159</td><td>0.128</td><td>0.488</td><td>0.503</td></tr></table>
+
+![](images/96e5c3146c2efc2ebc69949a382bcdc9b754b3a5e44f4272cf1847faf6c58684.jpg)  
+Figure 3. Evaluating a non-augmented model and a model trained with FOMA on train data whose small singular values are scaled down for different values of  $\lambda$  (left). We show on the right panel the probability density function of the original data (green), and its modifications using  $\lambda = 0$  (blue), and  $\lambda = 0.5$  (orange).
+
+![](images/e04fc71795a27bb3f58c955b89b0ea7f0bf1f704198f1d63f7218849872eca04.jpg)
+
+Stability and overfitting We trained two models on the NO2 dataset, one without using our method (ERM) and one with our method (FOMA) and report the results in Fig. 2. We observe that (a) the training process of FOMA is stable with respect to ERM, i.e., the loss is monotonically decreasing with relatively small fluctuations. Furthermore, FOMA exhibits better test performance (b) and smaller generalization gap (c).
+
+Computational resources The computational complexity of FOMA is governed by SVD calculation which has a complexity of  $\mathcal{O}(\min(qr^2, rq^2))$  for a  $q \times r$  matrix. In Table 2, we compare the average epoch time across 50 epochs in seconds to provide an estimate for the empirical computational cost of FOMA. The results are obtained with a single RTX3090 GPU. Note that the runtime of FOMA is very dependant on the batch size. Larger batch size results in less SVD computations, and depending on implementation, SVD of larger matrices can be evaluated faster.
+
+Table 2. Training times using FOMA (in seconds).  
+
+<table><tr><td>DATASET</td><td>AIRFOIL</td><td>ELECTRICITY</td><td>DTI</td></tr><tr><td>ERM</td><td>0.058</td><td>1.406</td><td>33.374</td></tr><tr><td>FOMA- INPUT</td><td>0.148</td><td>3.101</td><td>43.433</td></tr><tr><td>FOMA- LATENT</td><td>0.286</td><td>2.837</td><td>51.399</td></tr><tr><td>FOMA- BOTH</td><td>0.321</td><td>4.376</td><td>63.071</td></tr><tr><td>FOMAρ - INPUT</td><td>0.119</td><td>2.806</td><td>41.527</td></tr><tr><td>FOMAρ - LATENT</td><td>0.149</td><td>3.021</td><td>43.985</td></tr><tr><td>FOMAρ - BOTH</td><td>0.262</td><td>4.315</td><td>57.35</td></tr></table>
+
+The effect of FOMA on data and learning. We generated a 2D point cloud whose intrinsic dimension is one (shown in blue, Fig. 1), and we applied different DA methods on this data. The three panels in the figure show in orange
+
+the augmented data when using additive noise, mixup, and FOMA with  $\lambda = 0.5$  and over the original point cloud colored in light blue. Injecting noise alters each point in its neighborhood, whereas mixup draws the points towards the center of their convex hull. In contrast, FOMA aligns the new samples along the dominant component of the original data. Notably, our approach may increase the span of training data, and thus it can improve estimation in regression as was recently shown in (Wu et al., 2020).
+
+We argue that training on samples created with our method encourages the inherent tendency of the network to model the dominant parts of the data better (Naiman & Azencot, 2023). To demonstrate this phenomenon, we trained a three-layer fully connected network with and without FOMA on the NO2 dataset. The trained models are evaluated on the test dataset modified using a 100 varying  $\lambda \in [0,1]$  values, see Fig. 3 (left). Namely, we modify the singular values of every batch in the dataset for each  $\lambda$ , and feed the resulting data for inference. Surprisingly, the non-augmented model (blue curve) performs better on the unseen modified samples, yielding the minimum at  $\lambda \approx 0.5$ . Further, we note that for the majority of  $\lambda$  values, the test MSE is lower than the error obtained for the original data  $\mathcal{D}$  (i.e., for  $\lambda = 1$ ). In comparison, the regularized network attains a qualitatively similar plot in terms of the minimizing  $\lambda$  and test MSE profile, however, the MSE is lower for all  $\lambda$ . This example shows that the (non-augmented and augmented) models generalize better to data projected to the manifold  $\mathcal{M}$ , except for a few low  $\lambda$  values. We conclude that deep regression models may benefit from altering their training procedure by using samples closer to the data manifold  $\mathcal{M}$ . Finally, this behavior was found to be consistent across several architectures and datasets, see App. B.
+
+Table 3. Comparison of in-distribution generalization tasks. Bold values represent the best results and underlined values are second best. We report the average RMSE and MAPE over three seeds. Full results with standard deviation can be found in App F.  
+
+<table><tr><td rowspan="2"></td><td colspan="2">Airfoil</td><td colspan="2">NO2</td><td colspan="2">Exchange-Rate</td><td colspan="2">Electricity</td><td colspan="2">Echo</td></tr><tr><td>RMSE ↓</td><td>MAPE (%)↓</td><td>RMSE ↓</td><td>MAPE (%)↓</td><td>RMSE ↓</td><td>MAPE (%)↓</td><td>RMSE ↓</td><td>MAPE (%)↓</td><td>RMSE ↓</td><td>MAPE (%)↓</td></tr><tr><td>ERM†</td><td>2.901</td><td>1.753</td><td>0.537</td><td>13.615</td><td>0.0236</td><td>2.423</td><td>0.0581</td><td>13.861</td><td>5.402</td><td>8.700</td></tr><tr><td>Mixup†</td><td>3.730</td><td>2.327</td><td>0.528</td><td>13.534</td><td>0.0239</td><td>2.441</td><td>0.0585</td><td>14.306</td><td>5.393</td><td>8.838</td></tr><tr><td>Mani Mixup†</td><td>3.063</td><td>1.842</td><td>0.522</td><td>13.382</td><td>0.0242</td><td>2.475</td><td>0.0583</td><td>14.556</td><td>5.482</td><td>8.955</td></tr><tr><td>C-Mixup*</td><td>2.748</td><td>1.645</td><td>0.516</td><td>13.069</td><td>0.024</td><td>2.456</td><td>0.057</td><td>13.349</td><td>5.362</td><td>8.868</td></tr><tr><td>ADA*</td><td>2.357</td><td>1.377</td><td>0.515</td><td>13.128</td><td>0.022</td><td>2.250</td><td>0.059</td><td>13.58</td><td>-</td><td>-</td></tr><tr><td>FOMA</td><td>1.646</td><td>0.963</td><td>0.521</td><td>13.23</td><td>0.013</td><td>1.280</td><td>0.058</td><td>14.614</td><td>5.215</td><td>8.331</td></tr><tr><td>FOMAρ</td><td>1.471</td><td>0.816</td><td>0.512</td><td>12.894</td><td>0.013</td><td>1.262</td><td>0.059</td><td>13.437</td><td>5.512</td><td>8.742</td></tr><tr><td>C-Mixup†</td><td>2.717</td><td>1.610</td><td>0.509</td><td>12.998</td><td>0.020</td><td>2.041</td><td>0.057</td><td>13.372</td><td>5.177</td><td>8.435</td></tr><tr><td>ADA†</td><td>2.360</td><td>1.373</td><td>0.515</td><td>13.128</td><td>0.021</td><td>2.116</td><td>0.059</td><td>13.464</td><td>-</td><td>-</td></tr></table>
+
+Inspecting the data distribution and its modifications, reveals the differences between the data generated when select  $\lambda$  values are used. The distribution of the original data (green) shown in Fig. 3 (right) is bimodal. In comparison, the blue curve  $(\lambda = 0)$  for which the small singular values change to zeros, has a unimodal distribution. The orange curve  $(\lambda = 0.5)$  for which both the non-augmented model and a model trained with FOMA attained the minimum loss, has a smoother transition between the major mode and the minor mode. From the analysis above, we conclude the following. First, the network prefers data whose distribution is simpler (orange) than the original distribution (blue), yet not too simple (green). Second, our regularization encourages this tendency by providing the model with such data, leading to improved MSE profiles. To the best of our knowledge, the above analysis is novel on deep regression models.
+
+Notably, while it may argued that the behavior in Fig. 3 (left) is natural and intuitive as the model "simply" performs better on denoised signals, we argue differently. In particular, this plot somewhat contradicts our understanding of overfitting which occurs in high probability for tiny datasets such as NO2 (the entire dataset is composed of 500 entries) using multiple weights network such as the fully connected network we used with around  $20k$  parameters. Specifically, since the data is highly likely to be overfit by the network, we expect the MSE value to be lowest for  $\lambda = 1$ , and MSE value equal or higher for any  $\lambda < 1$ . Thus, we advocate that the above analysis may reveal a characteristic feature of regression neural networks. Our analysis is reinforced further as other datasets and architectures follow a similar pattern (App. B). Importantly, we are unaware of a similar experiment in the literature of deep regression neural networks.
+
+# 4. Analysis
+
+Relation to additive noise. In what follows, we would like to answer the following question: Does applying FOMA is merely a variant of injecting additive noise? To this end, we analyze FOMA from a perturbation theory viewpoint.
+
+Specifically, we would like to understand how a random data perturbation affects the singular values of the data matrix  $A \in \mathbb{R}^{q \times r}$ ,  $q \geq r$ . We denote by  $\sigma_1 \geq \sigma_2 \geq \dots \geq \sigma_r$  the singular values of  $A$ . The perturbed matrix and its singular values set are denoted by  $\tilde{A} = A + E$  and  $\{\tilde{\sigma}_j\}_{j=1}^r$ , respectively. We write  $\inf_2(A)$  and  $|A|_2$  to denote the smallest and largest singular values of any matrix  $A$ . The following classical result provides an estimated bound for the perturbed singular values (Stewart, 1979; 1998).
+
+Theorem 1. Let  $P$  be the orthogonal projection onto the column space of  $A$ . Let  $P_{\perp} = I - P$ . Then
+
+$$
+\tilde {\sigma} _ {j} ^ {2} = \left(\sigma_ {j} + \gamma_ {j}\right) ^ {2} + \eta_ {j} ^ {2}, j = 1 \ldots , r,
+$$
+
+where  $|\gamma_j| \leq |PE|_2$  and  $\inf_{2}(P_{\perp}E) \leq \eta_j \leq |P_{\perp}E|_2$ .
+
+Following (Stewart, 1979), we make two observations with respect to Thm. 1. First, if  $\sigma_{j} \gg |E|_{2}$  then it dominates the bound and we have  $\tilde{\sigma}_{j} \cong \sigma_{j} + \gamma_{j}$ . Second and more important to our setting, when  $\sigma_{j}$  is of order  $|E|_{2}$ , the term  $\eta_{j}$  will tend to dominate. Indeed, in these cases the term  $\eta_{j}$  increases the singular value  $\sigma_{j}$ . We conclude that random perturbations to  $A$  tend to increase its small singular values. In contrast, FOMA typically decreases the small singular values, while leaving the large  $\sigma_{j}$  unchanged. Thus, FOMA is in effect a complementary approach to injecting additive noise, allowing a finer control over the resulting new samples. Finally, we note that for a certain choice of hyper-parameters, our approach can be viewed as injecting noise per the above analysis. For example, taking  $\lambda \sim \mathrm{Uniform}(1.0, \alpha)$  for  $\alpha > 1.0$  will increase all the small singular values of  $A$  by a factor of  $\lambda \in [1.0, \alpha]$ , where Uniform is the random uniform distribution.
+
+FOMA as a Vicinal Risk Minimization (VRM). Given a cost function  $c: \mathcal{V} \times \mathcal{V} \to \mathbb{R}^+$ , the learning problem aims at minimizing the expectation of the loss  $c(f(x), y)$  over the distribution  $\mathcal{P}(x, y)$ ,  $x \in \mathcal{X}, y \in \mathcal{Y}$ . A fundamental challenge, shared by most real-world scenarios, is that the true distribution of the data is unfortunately unknown. The alternative is to minimize over the empirical distribution of
+
+a train set  $\{(x_i,y_i)\}_{i = 1}^n$  given by
+
+$$
+\mathrm {d} \mathcal {P} _ {\text {e m p}} (x, y) = \frac {1}{n} \sum_ {i} \delta_ {x _ {i}} (x) \delta_ {y _ {i}} (y).
+$$
+
+The resulting scheme is the common training procedure of modern neural networks, formally known as the Empirical Risk Minimization (ERM) (Vapnik, 1991).
+
+While  $\mathcal{P}_{\mathrm{emp}}$  provides a basic approximation of the true  $\mathcal{P}$ , it was suggested (Chapelle et al., 2001) that other density estimates  $\mathrm{d}\mathcal{P}_{\mathrm{est}}$  that take into account the vicinity of  $(x_{i},y_{i})$  should be considered. The recent mixup approach (Guo et al., 2019) exploits this idea by proposing a Vicinal Risk Minimization (VRM) procedure that is based on the vicinal distribution estimate  $\frac{1}{n}\sum_{i,j}\delta_{\tilde{x}_{ij}(\lambda)}(x)\delta_{\tilde{y}_{ij}(\lambda)}(y)$ , defined using convex combinations  $\tilde{z}_{ij}(\lambda) = \lambda z_i + (1 - \lambda)z_j$  for  $z\in \{x,y\}$  and  $\lambda \sim \mathrm{Beta}(\alpha ,\alpha)$ . In this context, the main difference between FOMA and mixup is in the definition of vicinity as we describe below.
+
+We denote by  $\mathcal{T}(x,y)$  the tangent plane of the data manifold  $\mathcal{M}$  at the point  $(x,y)\in \mathcal{M}\subset \mathcal{X}\times \mathcal{Y}$ . Namely,  $\mathcal{T}(x,y)$  is the linear approximation of  $\mathcal{M}$  at  $(x,y)$ . For every pair  $(x,y)$ , we define a new density distribution  $\mathcal{P}_{\mathrm{tan}}$  which considers all pairs  $(a,b)$  in the tangent plane of  $(u,v)\in \mathcal{M}$ . Formally,
+
+$$
+\mathrm {d} \mathcal {P} _ {\tan} (x, y) = \int_ {\mathcal {M}} \int_ {\mathcal {T} (u, v)} \delta_ {a} (x) \delta_ {b} (y) \mathrm {d} a b \mathrm {d} u v.
+$$
+
+Then, FOMA approximates the latter expression by generating an estimate of the tangent plane  $\mathcal{T}_{\mathrm{est}}$  via SVD, yielding the following vicinal estimate
+
+$$
+\mathrm {d} \mathcal {P} _ {\text {e s t}} (x, y) = \frac {1}{n} \sum_ {i} \frac {1}{k _ {i}} \sum_ {j} \delta_ {x _ {j}} (x) \delta_ {y _ {j}} (y),
+$$
+
+$$
+\left(x _ {j}, y _ {j}\right) \in \mathcal {T} _ {\text {e s t}} \left(x _ {i}, y _ {i}\right), k _ {i} = \left| \mathcal {T} _ {\text {e s t}} \left(x _ {i}, y _ {i}\right) \right|.
+$$
+
+# 5. Experiments
+
+# 5.1. In-Distribution Generalization
+
+In this section, we assess the effectiveness of FOMA and compare it to previous methods in terms of its ability to generalize within the given distribution. We utilize the datasets used in the study conducted by Yao et al. (2022) and closely replicate their experimental setup.
+
+Datasets. We use the following five datasets to evaluate the performance of in-distribution generalization. Two tabular datasets: Airfoil Self-Noise (Airfoil) (Brooks et al., 2014) and NO2 (Aldrin, 2004). Airfoil includes the aerodynamic and acoustic test results of airfoil blade sections and NO2 predicts the level of air pollution at specific locations. Two time series datasets: Exchange-Rate and Electricity (Lai
+
+et al., 2018), where Exchange-Rate provides a collection of daily exchange rates and Electricity is utilized for predicting the hourly electricity consumption. Finally, Echocardiogram Videos is designed for predicting the ejection fraction. It consists of a collection of videos that provide visual representations of the heart from various perspectives. See App. C for a detailed description of the datasets.
+
+Experimental settings. We conduct a comparative analysis between our approach, FOMA, and several existing strong baseline methods, namely Mixup (Zhang et al., 2017), Manifold-Mixup (Verma et al., 2019), C-Mixup (Yao et al., 2022), Anchor Data Augmentation (ADA) (Schneider et al., 2023), and classical expected risk minimization (ERM). Importantly, we denote by C-Mixup* and ADA* the results for these methods as reproduced in our environment, whereas C-Mixup† and ADA† represent the results as reported in (Yao et al., 2022; Schneider et al., 2023), respectively. For a fair comparison, we compare our results with respect to the starred models, as we were unable to reproduce the reported results in the original papers; this observation also appeared in (Schneider et al., 2023) regarding C-Mixup. Further, to maintain consistency with the methodology outlined in (Yao et al., 2022), we adopt the same model architectures, using a fully connected three-layer network for tabular datasets, an LST-Attn (Lai et al., 2018) for time series data, and EchoNet-Dynamic (Ouyang et al., 2020) for predicting the ejection fraction. We use Root Mean Square Error (RMSE) and Mean Averaged Percentage Error (MAPE) as evaluation metrics. Detailed experimental settings and hyperparameters are provided in App. E.
+
+Results. We report the in-distribution generalization results in Table 3. In all settings, lower numbers are better. Per column, we mark in bold the best available result, and we underline the second best error measure. Table 3 shows that both variants of FOMA improve over standard training via ERM, often by large margins. Further, our approach achieves new state-of-the-art error measures in comparison to the other baseline approaches on all datasets and metrics, except for Electricity where we obtain on-par results to C-Mixup. In particular,  $\mathrm{FOMA}_{\rho}$  yields the best available results in most cases. Finally, we observe the most significant improvement occurs on two datasets: Airfoil and Exchange-Rate in which FOMA reduces the error of the state-of-the-art result by approximately  $37\%$  and  $38\%$ , respectively.
+
+# 5.2. Out-of-Distribution Robustness
+
+In this section, we assess the effectiveness of FOMA and compare it to previous methods on tasks involving out-of-distribution robustness. To this end, we consider the datasets used in the study (Yao et al., 2022), and we closely replicate their experimental setup.
+
+Table 4. Comparison of out-of-distribution robustness problems. Bold values represent the best results and underlined values are second best. We report the average RMSE across domains and the "worst within-domain" RMSE over three different seeds. For the DTI and PovertyMap datasets, we report average  $R$  and "worst within-domain"  $R$  . Full results with standard deviation can be found in App F.  
+
+<table><tr><td rowspan="2"></td><td>RCF (RMSE)</td><td colspan="2">Crimes (RMSE)</td><td colspan="2">DTI (R)</td><td colspan="2">PovertyMap (R)</td></tr><tr><td>Avg. ↓</td><td>Avg. ↓</td><td>Worst ↓</td><td>Avg. ↑</td><td>Worst ↑</td><td>Avg. ↑</td><td>Worst ↑</td></tr><tr><td>ERM†</td><td>0.164</td><td>0.136</td><td>0.170</td><td>0.483</td><td>0.439</td><td>0.80</td><td>0.50</td></tr><tr><td>Mixup†</td><td>0.159</td><td>0.134</td><td>0.168</td><td>0.459</td><td>0.424</td><td>0.81</td><td>0.46</td></tr><tr><td>ManiMixup†</td><td>0.157</td><td>0.128</td><td>0.155</td><td>0.474</td><td>0.431</td><td>-</td><td>-</td></tr><tr><td>C-Mixup*</td><td>0.153</td><td>0.131</td><td>0.166</td><td>0.474</td><td>0.441</td><td>0.803</td><td>0.516</td></tr><tr><td>ADA*</td><td>0.171</td><td>1.30</td><td>0.156</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>FOMA</td><td>0.171</td><td>0.132</td><td>0.164</td><td>0.496</td><td>0.430</td><td>0.776</td><td>0.492</td></tr><tr><td>FOMAρ</td><td>0.159</td><td>0.128</td><td>0.158</td><td>0.503</td><td>0.459</td><td>0.832</td><td>0.482</td></tr><tr><td>C-Mixup†</td><td>0.146</td><td>0.123</td><td>0.146</td><td>0.498</td><td>0.458</td><td>0.81</td><td>0.53</td></tr><tr><td>ADA†</td><td>0.175</td><td>0.130</td><td>0.156</td><td>0.493</td><td>0.448</td><td>0.794</td><td>0.522</td></tr></table>
+
+Table 5. Ablation study of FOMA over modifying data at the input or latent levels, different loss scaling profiles  $\mu \left( \lambda \right)$  ,and scaling down the small or large singular values.  
+
+<table><tr><td rowspan="2">mode</td><td rowspan="2">μ(λ)</td><td rowspan="2">scale</td><td colspan="2">NO2</td><td colspan="2">Electricity</td></tr><tr><td>RMSE↓</td><td>MAPE↓</td><td>RMSE↓</td><td>MAPE↓</td></tr><tr><td>input</td><td>1</td><td>small</td><td>0.52 ± 0.01</td><td>13.23 ± 0.09</td><td>5.83e-2± 1e-3</td><td>13.02 ± 0.07</td></tr><tr><td>input</td><td>λ</td><td>small</td><td>0.53 ± 0.01</td><td>13.23 ± 0.09</td><td>5.83e-2± 1e-3</td><td>13.02 ± 0.37</td></tr><tr><td>input</td><td>λ2</td><td>small</td><td>0.54 ± 0.01</td><td>13.53 ± 0.27</td><td>5.79e-2± 2e-4</td><td>13.50 ± 0.04</td></tr><tr><td>input</td><td>1</td><td>large</td><td>0.79 ± 0.02</td><td>19.22 ± 0.45</td><td>5.89e-2± 4e-4</td><td>13.89 ± 0.39</td></tr><tr><td>input</td><td>λ</td><td>large</td><td>0.76 ± 0.02</td><td>18.61 ± 0.37</td><td>5.86e-2± 9e-4</td><td>13.15 ± 0.05</td></tr><tr><td>input</td><td>λ2</td><td>large</td><td>0.74 ± 0.01</td><td>18.01 ± 0.28</td><td>5.88e-2± 1e-3</td><td>13.23 ± 0.13</td></tr><tr><td>latent</td><td>1</td><td>small</td><td>0.53 ± 0.01</td><td>13.21 ± 0.08</td><td>6.11e-2± 9e-4</td><td>15.58 ± 0.38</td></tr><tr><td>latent</td><td>1</td><td>large</td><td>0.65 ± 0.02</td><td>16.35 ± 0.56</td><td>7.16e-2± 1e-3</td><td>18.62 ± 0.70</td></tr><tr><td>input + latent</td><td>1</td><td>small</td><td>0.52 ± 0.01</td><td>13.19 ± 0.14</td><td>5.94e-2± 1e-3</td><td>14.78 ± 0.48</td></tr><tr><td>input + latent</td><td>1</td><td>large</td><td>0.65 ± 0.02</td><td>21.25 ± 0.48</td><td>7.97e-2± 1e-4</td><td>20.77 ± 0.41</td></tr></table>
+
+Datasets. We use the following four datasets to assess the performance of out-of-distribution robustness. 1) RCFashionMNIST (RCF) (Yao et al., 2022), which is a synthetic modification of Fashion-MNIST, modeling sub-population shifts, where the goal is to predict the angle of rotation for each object. 2) Communities and Crime (Crime) (Redmond, 2009) is a tabular dataset that focuses on predicting the total number of violent crimes per 100K population, where the objective is to develop a model that can generalize to states that were not included in the training data. 3) Drug-Target Interactions (DTI) (Huang et al., 2021) is aiming to predict out-of-distribution drug-target interactions where the year is the domain information. 4) PovertyMap (Koh et al., 2021) is a satellite image regression dataset that has been created with the goal of estimating asset wealth in countries that were not included in the training set. For further details on the various datasets, see App. C.
+
+Experimental settings. Similarly to Sec. 5.1, we compare FOMA to the same baseline approaches. In terms of metrics, we report the RMSE (lower is better) for RCF-MNIST and Crimes. For PovertyMap and DTI, we use  $R$  (higher is better) as the evaluation metric, originally proposed in the respective papers (Koh et al., 2021; Huang et al., 2021). Following Yao et al. (2022), we trained ResNet-18 for RCF-MNIST and PovertyMap datasets, three-layer fully connected networks for Crimes, and DeepDTA (Öztürk et al., 2018) for DTI. More details regarding hyper-parameters and the experimental setup appear in App. E.
+
+Results. We report both the average and worst-domain performance metrics for out-of-distribution tasks in Table 4. Dashed cells represent cases where results are not available. While the results for our approach are somewhat more mixed in comparison to the in-distribution challenge, we still find FOMA to be highly effective. In particular, FOMA improved over ERM in almost all cases, with the exception of RCF.
+
+We find our technique to obtain comparable results for RCF and Crimes with respect to the best baselines. Notably, FOMA presents a noticeable gap in DTI and PovertyMap in comparison to SOTA approaches such as C-Mixup and ADA. Finally, similarly to the in-distribution setting,  $\mathrm{FOMA}_{\rho}$  achieves better results in comparison to FOMA.
+
+# 5.3. Ablation study
+
+FOMA is a data augmentation method that scales down singular values of the data. However, there are several design choices to make. For example, we can choose to apply FOMA on the input data, on the learned representations, or apply it on both in succession. Another decision is what singular values we should scale down: the smaller ones, capturing less explained variance of the data, or the larger ones, which may create samples that are further away from the data manifold and thus expand the underlying distribution. Another choice to make is how to scale the singular values and potentially the loss function. The values  $\lambda \sim \mathrm{Beta}(\alpha, \alpha)$  by which we scale the singular values vary between batches, the higher the value of  $\lambda$ , the more noise is removed. By scaling the loss function differently for each batch, we can change the update rule of the optimizer, potentially leading to an improved behavior of FOMA.
+
+We detail in Table 5 the ablation results we obtained for NO2 and Electricity datasets while exploring the parameter spaces of the above design choices. Overall, applying our technique at the input level, without scaling the loss, and scaling the small singular values seems to consistently yield good results across datasets and tasks (see also Tables 8, 9). More specifically, we find in Table 5 that for the datasets NO2 and Electricity, scaling down the small singular values is preferred to scaling the larger ones and more generally, this observation holds for all datasets (see Table 8). On the other hand, when using  $\mathrm{FOMA}_{\rho}$ , allowing more freedom in selecting what singular values to scale, some datasets achieve better results when scaling the larger singular values, e.g., Airfoil and Exchange-rate. Furthermore, we have two observations regarding the loss scaling profiles  $\mu(\lambda)$ : 1) the effect of decreasing  $\mu$  is inversely proportional between scaling small singular values and large singular values. For instance, when scaling the small singular values of NO2, smaller  $\mu$  corresponds with better performance, whereas when scaling the large singular values, larger  $\mu$  corresponds to better performance. 2) The effect of  $\mu$  is inversely proportional between the datasets, decreasing  $\mu$  improves the performance on Electricity while achieving worse performance on NO2. Finally, we note that employing FOMA in the latent space or both in the input and latent spaces yields inconsistent results across datasets.
+
+# 6. Discussion
+
+We have proposed FOMA, a data-driven method for data augmentation of regression tasks. We showed that FOMA supports the network tendency of representing dominant components of its input signals by creating virtual examples sampled from the tangent planes of the original train set. Implementing FOMA is straightforward, and it is fully differentiable. Throughout an extensive evaluation, we have shown that FOMA improves the generalization error of neural models on regression benchmarks including in-distribution generalization and out-of-distribution tasks. We also ablated our model with respect to several design choices. Generally, we find our method to obtain highly competitive results and often surpass state-of-the-art approaches.
+
+We inspected the effect of the hyper-parameters  $\alpha$ ,  $\rho$  and whether to scale the small or large singular values. For FOMA, we observe a relatively consistent performance across tasks, whereas  $\mathrm{FOMA}_{\rho}$  presents mixed results (see Tabs. 8, 9). Given that FOMA and  $\mathrm{FOMA}_{\rho}$  perform similarly, we can not derive a specific guideline for choosing these hyper-parameters. Another limitation is that the time complexity of FOMA is governed by the SVD calculation, which may be restrictive for large train batches. Finally, we mention that if  $k = n_0 + m$ , i.e., the tangent space dimension equals that of the data rank, then our approach can be only applied to the dominant singular values.
+
+There are several exciting avenues for future exploration. First, is there a fundamental link between the vicinal distribution employed and the learned representation? While several existing works suggest that linearity yields better models (Azencot et al., 2020; Berman et al., 2023; Zeng et al., 2023), the model dependency on the specific definition of vicinity is still not well understood. Second, can similar methods to ours be useful in classification tasks? The adaptation of FOMA to classification is straightforward, however, several design choices which were tuned for regression may require change in a classification setting. In particular, the computational demands of SVD-based data augmentation are higher in comparison to mixup schemes. Improving these aspects by e.g., approximating the tangent space of the manifold may be highly impactful in regression as well as classification tasks. Another interesting avenue to explore is the relation between FOMA and generative modeling (Naiman et al., 2024). We plan to explore these questions in future work.
+
+# Acknowledgements
+
+This research was partially supported by the Lynn and William Frankel Center of the Computer Science Department, Ben-Gurion University of the Negev, an ISF grant 668/21, an ISF equipment grant, and by the Israeli Council
+
+for Higher Education (CHE) via the Data Science Research Center, Ben-Gurion University of the Negev, Israel.
+
+# Impact Statement
+
+This paper presents work whose goal is to advance the field of Machine Learning. There are many potential societal consequences of our work, none which we feel must be specifically highlighted here.
+
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+
+# A. Method overview
+
+In this section we provide details for different methods used to evaluate the intrinsic dimension and linear dimension which are used for FOMA and  $\mathrm{FOMA}\rho$  respectively.
+
+# A.1. Intrinsic dimension
+
+To estimate the intrinsic dimension of data, we use the TwoNN (Facco et al., 2017) ID estimator. The ID-estimator relies on the distances to only the two closest neighbors of each point, minimizing the influence of inconsistencies in the dataset during estimation.
+
+Let  $X = \{x_{1}, x_{2}, \dots, x_{N}\}$  be a set of points sampled uniformly on a manifold with intrinsic dimension  $d$ . For each point  $x_{i}$ , we calculate the two shortest distances  $r_{1}, r_{2}$  from other elements in  $X \setminus \{x_{i}\}$  and determine the ratio  $\mu_{i} = \frac{r_{2}}{r_{1}}$ . It has been proven that  $\mu_{i}, 1 \leq i \leq N$  are distributed according to a Pareto distribution with parameter  $d + 1$  on the interval  $[1, \infty)$ , specifically  $f(\mu_{i} | d) = d\mu_{i}^{-(d + 1)}$ . While  $d$  can be estimated by maximizing the likelihood:
+
+$$
+P \left(\mu_ {1}, \mu_ {2}, \dots \mu_ {N} \mid d\right) = d ^ {N} \prod_ {i = 1} ^ {N} \mu_ {i} ^ {- (d + 1)}. \tag {1}
+$$
+
+We adopt the approach suggested by Facco et al. (2017) using the cumulative distribution  $F(\mu) = 1 - \mu^{-d}$ . The method involves estimating the parameter  $d$  through linear regression on the empirical estimate of  $F(\mu)$ . To do this, we arrange the values of  $\mu$  in ascending order and define  $F^{emp}(\mu_i) \approx \frac{i}{N}$ . A linear regression is then performed on the set of points  $\{(\log \mu_i, -\log (1 - F_i^{emp}))\}_{i=1}^N$  where the slope of the fitted line is the estimated ID.
+
+# A.2. Linear dimension
+
+A common method for estimating the linear dimension of data is to perform principal component analysis (PCA) or SVD and count the number of components that should be included to describe some percentage of the variance in the data, usually above  $90\%$ . More formally:
+
+$$
+k = \arg \max  _ {\tilde {k}} \sum_ {j = 1} ^ {\tilde {k}} \sigma_ {j} / \sum_ {j} \sigma_ {j} \leq \rho .
+$$
+
+# B. Sequential models capture dominant components of data better
+
+Following the discussion in Sec. 3, we verify empirically that neural networks model the dominant parts of their data better. We repeat the experiment in Fig. 3 (left) using additional two datasets which are trained on different architectures. For evaluation, we use the dataset whose singular values are modified using varying values of  $\lambda$ . The results are presented in Fig. 4, where solid lines represent the results of the non-regularized model, and dashed lines are associated with models trained with FOMA. Remarkably, we observe a similar qualitative behavior as we reported in Sec. 3. In particular, the highest MSE values are obtained for both the baseline and regularized models for  $\lambda = 1$ , i.e., when the data is unchanged. Further, the model attain improved error measures as  $\lambda$  decreases, where the error profile is similar for the baseline and regularized models. Based on these results, we deduce that sequential models prefer to represent and compute the dominant components of data, reinforcing our choice for supplying such data to the network during training.
+
+# C. Dataset Description
+
+In this section, we provide detailed descriptions of datasets used in the experiments in this work.
+
+Airfoil Self-Noise (Brooks et al., 2014). The dataset comprises aerodynamic and acoustic test findings for various sizes of NACA 0012 airfoils, obtained at different wind tunnel speeds and angles of attack. Each input instance consists of five features: frequency, angle of attack, chord length, free-stream velocity, and suction side displacement thickness. The label represents the one-dimensional scaled sound pressure level. To normalize the input features, min-max normalization is applied. As per reference (Hwang & Whang, 2021), the training, validation, and test sets consist of 1003, 300, and 200 examples, respectively.
+
+![](images/be6539aa2212614d3394009becb523580f3b90492797247558331116c6d5032a.jpg)  
+Figure 4. Evaluating a non-augmented model (solid lines) and a model trained with FOMA (dashed lines) on train data whose small singular values are scaled down for different values of  $\lambda$  (see also Fig. 3, left). Communities and crime dataset trained on a three-layer full connected network (left). Electricity trained on LST-Attn (Lai et al., 2018) (right).
+
+![](images/d720f8325ff2ac03d8ba7c2b4b92aa9f478a05d357bfbdece519f8deedefbd1f.jpg)
+
+NO2 (Aldrin, 2004). The NO2 emission dataset originates from a study examining the relationship between air pollution near a road and traffic volume along with meteorological variables. Each input comprises seven features, including the logarithm of the number of cars per hour, temperature 2 meters above ground, wind speed, temperature difference between 25 and 2 meters above ground, wind direction, hour of the day, and the day number since October 1st, 2001. The hourly values of the logarithm of NO2 concentration, measured at Alnabru in Oslo between October 2001 and August 2003, serve as the response variable or label. As per reference (Hwang & Whang, 2021), there are 200 examples in the training set, 200 in the validation set, and 100 in the test set.
+
+Exchange-Rate (Lai et al., 2018). The exchange-rate dataset comprises a time-series collection of daily exchange rates for eight countries: Australia, Britain, Canada, Switzerland, China, Japan, New Zealand, and Singapore, spanning from 1990 to 2016. The total length of the time series is 7,588, with a daily sampling frequency. A sliding window size of 168 days is applied. The input dimension is  $168 \times 8$ , and the label dimension is  $1 \times 8$  data points. Following the methodology outlined in (Lai et al., 2018), the dataset has been divided into training  $(60\%)$ , validation  $(20\%)$ , and test  $(20\%)$  sets in chronological order.
+
+Electricity (Brooks et al., 2014). This dataset is a time-series collection obtained from 321 clients, representing electricity consumption in kWh recorded every 15 minutes from 2012 to 2014. The total length of the time series is 26,304, sampled hourly. As with the Exchange-Rate data, a window size of 168 is utilized, resulting in an input dimension of  $168 \times 321$  and a corresponding label dimension of  $1 \times 321$ . Similar to the methodology described in Lai et al. (Lai et al., 2018), the dataset is divided accordingly.
+
+Echo (Ouyang et al., 2020). The Echocardiogram Videos dataset comprises 10,030 labeled apical-4-chamber echocardiogram videos captured from various perspectives, accompanied by expert annotations for studying cardiac motion and chamber sizes. These videos were sourced from individuals undergoing imaging at Stanford University Hospital between 2016 and 2018. To delineate the left ventricle area, initial preprocessing involves frame-by-frame semantic segmentation of the videos. This preprocessing method generates video clips containing 32 frames of  $112 \times 112$  RGB images, which serve as input for predicting ejection fraction. The dataset is partitioned into training, validation, and test sets, with sizes of 7,460, 1,288, and 1,276, respectively.
+
+RCF. RCF-MNIST, where "RCF" stands for "Rotated-Colored-Fashion", is a dataset constructed with specific color and rotation attributes. In this dataset, the normalized RGB vector for red and blue is  $[1,0,0]$  and  $[0,0,1]$  respectively, and the normalized rotation angle (i.e., label) for each image is denoted as  $g$ , where  $g \in [0,1]$ . During the construction of the training set,  $80\%$  of the images are colored using the RGB value  $[g,0,1 - g]$ , while the remaining  $20\%$  are colored with  $[1 - g,0,g]$ . Consequently, there is a strong spurious correlation between color information and labels within the training set. To simulate distribution shift in the test set, this spurious correlation is reversed, with  $80\%$  of the images colored using RGB values  $[1 - g,0,g]$ , and the remaining  $20\%$  with  $[g,0,1 - g]$ . The impact of this spurious correlation on performance is evaluated by comparing the performance of the same test set with or without distribution shift. The results, presented in
+
+Table 11, demonstrate that the subpopulation shift induced by the spurious correlation indeed affects performance negatively, as anticipated.
+
+PovertyMap. This dataset is part of the WILDS benchmark (Koh et al., 2021), comprising satellite images sourced from 23 African countries, which are utilized for predicting the village-level real-valued asset wealth index. Each input consists of a  $224 \times 224$  multispectral LandSat satellite image with 8 channels, while the corresponding label represents the real-valued asset wealth index. The domains of the images encompass information regarding the country, urban, and rural areas. The dataset is divided into 5 distinct cross-validation folds, with all countries in these splits being disjoint to facilitate the out-of-distribution setting. All experimental configurations adhere to the methodology outlined by Koh et al. (Koh et al., 2021).
+
+Crime (Redmond, 2009). The Communities And Crimes dataset is a tabular compilation that integrates socio-economic data from the 1990 US Census, law enforcement data from the 1990 US LEMAS survey, and crime data from the 1995 FBI UCR. It encompasses 122 attributes believed to have some plausible connection to crime, such as median family income and the percentage of officers assigned to drug units. The target attribute for prediction is per capita violent crimes, which includes offenses such as murder, rape, robbery, assault, among others. To prepare the data, all numeric features are normalized using the decimal range 0.00 to 1.00 through an equal-interval binning method, and missing values are imputed with the average values of the corresponding attributes. Domain information is denoted by state identifications, resulting in a total of 46 domains. The dataset is partitioned into training, validation, and test sets containing 1,390, 231, and 373 instances, respectively. These sets consist of 31, 6, and 9 disjoint domains, respectively.
+
+DTI (Huang et al., 2021). The Drug-target Interactions dataset is designed to forecast the binding activity score between each small molecule and its corresponding target protein. Input features encompass information on both the drug and target protein, represented by one-hot vectors, while the output label denotes the binding activity score. The training and validation sets are curated from the years 2013 to 2018, while the test set spans the years 2019 to 2020. The "Year" attribute serves as the domain information.
+
+# D. Additional Experiments
+
+# D.1. Comparison with additive noise
+
+In Section 4 we show a relation of our method to additive noise. For completeness, we add a supplementary experiment that compares our method with additive noise as shown in Table. 6 and Table. 7. The experiment was conducted as follows: Given a sample  $x_{i} \in \mathbb{R}^{n}$  and  $y_{i} \in \mathbb{R}^{m}$  we sample  $\epsilon_{x} \sim \mathcal{N}(0, \sigma \times I_{n})$ ,  $\epsilon_{y} \sim \mathcal{N}(0, \sigma \times I_{m})$  and create the noised sample  $(\tilde{x}_{i} = x_{i} + \epsilon_{x}, y_{i} = \tilde{y}_{i} + \epsilon_{y})$ . Where  $\sigma$  is selected from  $\{0.1, 0.01, 0.001, 0.0001\}$  and the best model is chosen according to the performance on the validation set.
+
+Table 6. Comparison of in-distribution generalization tasks with additive noise.  
+
+<table><tr><td rowspan="2"></td><td colspan="2">Airfoil</td><td colspan="2">NO2</td><td colspan="2">Exchange-Rate</td><td colspan="2">Electricity</td></tr><tr><td>RMSE ↓</td><td>MAPE (%)↓</td><td>RMSE ↓</td><td>MAPE (%)↓</td><td>RMSE ↓</td><td>MAPE (%)↓</td><td>RMSE ↓</td><td>MAPE (%)↓</td></tr><tr><td>ERM†</td><td>2.901</td><td>1.753</td><td>0.537</td><td>13.615</td><td>0.0236</td><td>2.423</td><td>0.0581</td><td>13.861</td></tr><tr><td>ERM+noise</td><td>3.172</td><td>1.864</td><td>0.524</td><td>13.326</td><td>0.016</td><td>1.568</td><td>0.600</td><td>14.068</td></tr><tr><td>FOMA</td><td>1.646</td><td>0.963</td><td>0.521</td><td>13.23</td><td>0.013</td><td>1.280</td><td>0.058</td><td>14.614</td></tr><tr><td>FOMAρ</td><td>1.471</td><td>0.816</td><td>0.512</td><td>12.894</td><td>0.013</td><td>1.262</td><td>0.059</td><td>13.437</td></tr></table>
+
+Table 7. Comparison of out-of-distribution robustness problems with additive noise.  
+
+<table><tr><td rowspan="2"></td><td>RCF (RMSE)</td><td colspan="2">Crimes (RMSE)</td><td colspan="2">DTI (R)</td></tr><tr><td>Avg. ↓</td><td>Avg. ↓</td><td>Worst ↓</td><td>Avg. ↑</td><td>Worst ↑</td></tr><tr><td>ERM†</td><td>0.164</td><td>0.136</td><td>0.170</td><td>0.483</td><td>0.439</td></tr><tr><td>ERM + noise</td><td>0.180</td><td>0.136</td><td>0.166</td><td>0.492</td><td>0.442</td></tr><tr><td>FOMA</td><td>0.171</td><td>0.132</td><td>0.164</td><td>0.496</td><td>0.430</td></tr><tr><td>FOMAρ</td><td>0.159</td><td>0.128</td><td>0.158</td><td>0.503</td><td>0.459</td></tr></table>
+
+# E. Hyperparameters
+
+We list the hyperparameters for every dataset in Table 8 and Table 9 for the methods FOMA and FOMA, respectively. In our main results, we apply our method on the input space or on the latent space or both and report the one with best performance. All hyperparameters are selected by cross-validation, evaluated on the validation set. Some of the hyperparameters such as architecture and optimizer are not included in the table since they were not changed and were used as they appear in previous works (Yao et al., 2022).
+
+Table 8. Hyperparameter choices for the experiments using FOMA.  
+
+<table><tr><td>Dataset</td><td>Airfoil</td><td>NO2</td><td>Exchange-Rate</td><td>Electricity</td><td>Echo</td><td>RCF</td><td>Crimes</td><td>PovertyMap</td><td>DTI</td></tr><tr><td>Learning rate</td><td>5e-4</td><td>1e-3</td><td>1e-2</td><td>5e-4</td><td>1e-4</td><td>1e-4</td><td>5e-3</td><td>1e-3</td><td>5e-4</td></tr><tr><td>Batch size</td><td>32</td><td>64</td><td>16</td><td>128</td><td>10</td><td>128</td><td>8</td><td>32</td><td>64</td></tr><tr><td rowspan="2">Input/Latent Epochs</td><td>latent</td><td>both</td><td>input</td><td>both</td><td>latent</td><td>latent</td><td>input</td><td>latent</td><td>latent</td></tr><tr><td>200</td><td>100</td><td>200</td><td>150</td><td>20</td><td>250</td><td>200</td><td>50</td><td>200</td></tr><tr><td rowspan="2">Singular values α</td><td>small</td><td>small</td><td>small</td><td>small</td><td>small</td><td>small</td><td>small</td><td>small</td><td>large</td></tr><tr><td>0.8</td><td>0.9</td><td>0.2</td><td>1.1</td><td>1.1</td><td>1</td><td>0.6</td><td>1.5</td><td>0.5</td></tr></table>
+
+Table 9. Hyperparameter choices for the experiments using FOMA  $\rho$  .  
+
+<table><tr><td>Dataset</td><td>Airfoil</td><td>NO2</td><td>Exchange-Rate</td><td>Electricity</td><td>Echo</td><td>RCF</td><td>Crimes</td><td>PovertyMap</td><td>DTI</td></tr><tr><td>Learning rate</td><td>5e-4</td><td>1e-3</td><td>1e-3</td><td>1e-4</td><td>1e-4</td><td>1e-4</td><td>1e-3</td><td>5e-3</td><td>1e-2</td></tr><tr><td>Batch size</td><td>128</td><td>8</td><td>8</td><td>8</td><td>10</td><td>8</td><td>64</td><td>32</td><td>64</td></tr><tr><td rowspan="2">Input/Latent Epochs</td><td>input</td><td>input</td><td>input</td><td>input</td><td>latent</td><td>latent</td><td>both</td><td>latent</td><td>latent</td></tr><tr><td>100</td><td>250</td><td>150</td><td>200</td><td>20</td><td>250</td><td>100</td><td>50</td><td>250</td></tr><tr><td rowspan="2">Singular values α</td><td>large</td><td>small</td><td>large</td><td>large</td><td>small</td><td>small</td><td>large</td><td>small</td><td>large</td></tr><tr><td>1.4</td><td>0.3</td><td>1</td><td>0.7</td><td>1.1</td><td>1.5</td><td>0.6</td><td>1</td><td>0.6</td></tr><tr><td>ρ</td><td>0.975</td><td>0.95</td><td>0.8</td><td>0.875</td><td>0.85</td><td>0.95</td><td>0.875</td><td>0.875</td><td>0.825</td></tr></table>
+
+# F. Results with Standard Deviation
+
+In Tables 10, 11, we report the full results of in-distribution generalization and out-of-distribution robustness respectively.
+
+Table 10. Full results of in-distribution generalization. We compute the mean and standard deviation for results of three seeds.  
+
+<table><tr><td rowspan="2"></td><td colspan="2">Airfoil</td><td colspan="2">NO2</td><td colspan="2">Exchange-Rate</td></tr><tr><td>RMSE ↓</td><td>MAPE (%)↓</td><td>RMSE ↓</td><td>MAPE (%)↓</td><td>RMSE ↓</td><td>MAPE (%)↓</td></tr><tr><td>ERM</td><td>2.901 ± 0.067</td><td>1.753 ± 0.078</td><td>0.537 ± 0.005</td><td>13.615 ± 0.165</td><td>0.023 ± 0.003</td><td>2.423 ± 0.365</td></tr><tr><td>Mixup</td><td>3.730 ± 0.190</td><td>2.327 ± 0.159</td><td>0.528 ± 0.005</td><td>13.534 ± 0.125</td><td>0.023 ± 0.002</td><td>2.441 ± 0.286</td></tr><tr><td>Mani Mixup</td><td>3.063 ± 0.113</td><td>1.842 ± 0.114</td><td>0.522 ± 0.008</td><td>13.357 ± 0.214</td><td>0.024 ± 0.004</td><td>2.475 ± 0.346</td></tr><tr><td>C-Mixup*</td><td>2.739 ± 0.06</td><td>1.640 ± 0.069</td><td>0.516 ± 0.01</td><td>13.069 ± 0.294</td><td>0.024 ± 0.005</td><td>2.455 ± 0.629</td></tr><tr><td>ADA*</td><td>2.357 ± 0.118</td><td>1.377 ± 0.064</td><td>0.515 ± 0.006</td><td>13.128 ± 0.12</td><td>0.021 ± 0.006</td><td>2.250 ± 0.781</td></tr><tr><td>FOMA</td><td>1.646 ± 0.103</td><td>0.963 ± 0.056</td><td>0.521 ± 0.013</td><td>13.23 ± 0.289</td><td>0.013 ± 0.000</td><td>1.280 ± 0.037</td></tr><tr><td>FOMAρ</td><td>1.471 ± 0.047</td><td>0.816 ± 0.008</td><td>0.512 ± 0.008</td><td>12.894 ± 0.217</td><td>0.013 ± 0.000</td><td>1.262 ± 0.037</td></tr><tr><td>C-Mixup†</td><td>2.717 ± 0.067</td><td>1.610 ± 0.085</td><td>0.509 ± 0.006</td><td>12.998 ± 0.271</td><td>0.0203 ± 0.001</td><td>2.041 ± 0.134</td></tr><tr><td>ADA†</td><td>2.360 ± 0.133</td><td>1.373 ± 0.056</td><td>0.514 ± 0.007</td><td>13.127 ± 0.146</td><td>0.020 ± 0.006</td><td>2.115 ± 0.689</td></tr><tr><td rowspan="2"></td><td colspan="2">Electricity</td><td colspan="2">Echo</td><td rowspan="2"></td><td rowspan="2"></td></tr><tr><td>RMSE ↓</td><td>MAPE (%)↓</td><td>RMSE ↓</td><td>MAPE (%)↓</td></tr><tr><td>ERM</td><td>0.058 ± 0.001</td><td>13.861 ± 0.152</td><td>5.402 ± 0.024</td><td>8.700 ± 0.015</td><td></td><td></td></tr><tr><td>Mixup</td><td>0.058 ± 0.000</td><td>14.306 ± 0.048</td><td>5.393 ± 0.040</td><td>8.838 ± 0.108</td><td></td><td></td></tr><tr><td>Mani Mixup</td><td>0.058 ± 0.000</td><td>14.556 ± 0.057</td><td>5.482 ± 0.066</td><td>8.955 ± 0.082</td><td></td><td></td></tr><tr><td>C-Mixup*</td><td>0.057 ± 0.000</td><td>13.471 ± 0.15</td><td>5.483 ± 0.097</td><td>9.121 ± 0.208</td><td></td><td></td></tr><tr><td>ADA*</td><td>0.059 ± 0.001</td><td>13.578 ± 0.146</td><td>-</td><td>-</td><td></td><td></td></tr><tr><td>FOMA</td><td>0.058 ± 0.000</td><td>14.653 ± 0.166</td><td>5.215 ± 0.061</td><td>8.331 ± 0.088</td><td></td><td></td></tr><tr><td>FOMAρ</td><td>0.059 ± 0.000</td><td>13.437 ± 0.26</td><td>5.476 ± 0.01</td><td>8.742 ± 0.091</td><td></td><td></td></tr><tr><td>C-Mixup†</td><td>0.057 ± 0.001</td><td>13.372 ± 0.106</td><td>5.177 ± 0.036</td><td>8.435 ± 0.089</td><td></td><td></td></tr><tr><td>ADA†</td><td>0.058 ± 0.001</td><td>13.464 ± 0.296</td><td>-</td><td>-</td><td></td><td></td></tr></table>
+
+Table 11. Full results of out-of-distribution generalization. We compute the mean and standard deviation for results of three seeds.  
+
+<table><tr><td rowspan="2"></td><td>RCF (RMSE)</td><td colspan="2">Crimes (RMSE)</td><td colspan="2">DTI (R)</td><td colspan="2">PovertyMap (R)</td></tr><tr><td>Avg. ↓</td><td>Avg. ↓</td><td>Worst ↓</td><td>Avg. ↑</td><td>Worst ↑</td><td>Avg. ↑</td><td>Worst ↑</td></tr><tr><td>ERM</td><td>0.162 ± 0.003</td><td>0.134 ± 0.003</td><td>0.173 ± 0.009</td><td>0.464 ± 0.014</td><td>0.429 ± 0.004</td><td>0.80 ± 0.04</td><td>0.50 ± 0.07</td></tr><tr><td>Mixup</td><td>0.176 ± 0.003</td><td>0.128 ± 0.002</td><td>0.154 ± 0.001</td><td>0.465 ± 0.004</td><td>0.437 ± 0.016</td><td>0.81 ± 0.04</td><td>0.46 ± 0.03</td></tr><tr><td>ManiMixup</td><td>0.157 ± 0.020</td><td>0.128 ± 0.003</td><td>0.155 ± 0.009</td><td>0.474 ± 0.004</td><td>0.431 ± 0.009</td><td>-</td><td>-</td></tr><tr><td>C-Mixup*</td><td>0.153 ± 0.004</td><td>0.130 ± 0.003</td><td>0.161 ± 0.01</td><td>0.475 ± 0.013</td><td>0.440 ± 0.016</td><td>0.804 ± 0.03</td><td>0.517 ± 0.06</td></tr><tr><td>ADA*</td><td>0.171 ± 0.009</td><td>0.130 ± 0.003</td><td>0.156 ± 0.006</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>FOMA</td><td>0.171 ± 0.015</td><td>0.132 ± 0.002</td><td>0.164 ± 0.002</td><td>0.492 ± 0.003</td><td>0.442 ± 0.019</td><td>0.776 ± 0.03</td><td>0.492 ± 0.05</td></tr><tr><td>FOMAρ</td><td>0.159 ± 0.01</td><td>0.128 ± 0.004</td><td>0.158 ± 0.002</td><td>0.503 ± 0.008</td><td>0.459 ± 0.01</td><td>0.832 ± 0.04</td><td>0.482 ± 0.06</td></tr><tr><td>C-Mixup†</td><td>0.146 ± 0.005</td><td>0.123 ± 0.000</td><td>0.146 ± 0.002</td><td>0.498 ± 0.008</td><td>0.458 ± 0.004</td><td>0.81 ± 0.03</td><td>0.53 ± 0.07</td></tr><tr><td>ADA†</td><td>0.162 ± 0.014</td><td>0.129 ± 0.003</td><td>0.155 ± 0.006</td><td>0.492 ± 0.009</td><td>0.448 ± 0.009</td><td>0.793 ± 0.03</td><td>0.521 ± 0.06</td></tr></table>

2024/First-Order Manifold Data Augmentation for Regression Learning/images.zip

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2024/FlashST_ A Simple and Universal Prompt-Tuning Framework for Traffic Prediction/1591a92e-d519-49e7-ace8-85352ae1ed50_content_list.json

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+    {
+        "type": "text",
+        "text": "FlashST: A Simple and Universal Prompt-Tuning Framework for Traffic Prediction",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Zhonghang Li  $^{1,2}$  Lianghao Xia  $^{2}$  Yong Xu  $^{1,3}$  Chao Huang*",
+        "bbox": [
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+        "page_idx": 0
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+    {
+        "type": "text",
+        "text": "Abstract",
+        "text_level": 1,
+        "bbox": [
+            243,
+            239,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "The objective of traffic prediction is to accurately forecast and analyze the dynamics of transportation patterns, considering both space and time. However, the presence of distribution shift poses a significant challenge in this field, as existing models struggle to generalize well when faced with test data that significantly differs from the training distribution. To tackle this issue, this paper introduces a simple and universal spatio-temporal prompt-tuning framework-FlashST, which adapts pre-trained models to the specific characteristics of diverse downstream datasets, improving generalization in diverse traffic prediction scenarios. Specifically, the FlashST framework employs a lightweight spatio-temporal prompt network for in-context learning, capturing spatio-temporal invariant knowledge and facilitating effective adaptation to diverse scenarios. Additionally, we incorporate a distribution mapping mechanism to align the data distributions of pre-training and downstream data, facilitating effective knowledge transfer in spatio-temporal forecasting. Empirical evaluations demonstrate the effectiveness of our FlashST across different spatio-temporal prediction tasks using diverse urban datasets. Code is available at https://github.com/HKUDS/FlashST.",
+        "bbox": [
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+            258,
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1. Introduction",
+        "text_level": 1,
+        "bbox": [
+            89,
+            678,
+            215,
+            694
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Accurate traffic forecasting is a crucial objective in the domain of urban computing. It aims to precisely predict and analyze the dynamic transportation patterns within cities. The key goals of traffic prediction include supporting urban planning, enabling real-time monitoring and management of traffic, and contributing to the development of smart city",
+        "bbox": [
+            89,
+            704,
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+            794
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "applications (Guo et al., 2019; Han et al., 2021; Li et al., 2024). The primary challenge lies in effectively modeling the intricate spatial and temporal correlations among different time intervals and geographical locations.",
+        "bbox": [
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+        "page_idx": 0
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+    {
+        "type": "text",
+        "text": "Neural network techniques have become a prominent approach for spatio-temporal prediction tasks, thanks to their remarkable capabilities in feature representation. Initially, researchers leveraged Recurrent Neural Networks (RNNs) (Yu et al., 2017) to capture temporal dependencies, and Convolutional Neural Networks (CNNs) (Zhang et al., 2017; Yao et al., 2018) to model spatial correlations. Subsequently, Graph Neural Networks (GNNs) have emerged as effective models for capturing complex spatial dependencies, utilizing message passing mechanisms to learn the relationships among spatial units such as regions and road segments. In constructing the graph adjacency matrices, researchers have explored various factors. Some have considered static geographical distance (Li et al., 2018; Yu et al., 2018), while others have incorporated time-aware region correlations (Han et al., 2021; Li et al., 2022). Additionally, there have been efforts to learn the region-wise relevance in a data-driven manner (Wu et al., 2019; 2020a).",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Although the aforementioned methods have shown effectiveness, most current spatio-temporal prediction models struggle to generalize effectively when confronted with distribution shifts across diverse downstream datasets and tasks. In their methods, the assumption of an inconsistent distribution between training and testing data becomes a hindrance to accurate predictions in real-life urban scenarios with uncontrolled shifts from unseen spatio-temporal data. As illustrated in Figure 1, directly applying parameters learned from training on dataset A to test on dataset B can lead to suboptimal performance due to significant variations in spatio-temporal characteristics across different data distributions. Therefore, there is a need to enhance the generalization ability of spatio-temporal forecasting models by efficiently adapting them to handle such distribution shifts.",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "While there are potential benefits to enabling model adaptation for spatio-temporal prediction methods, several key challenges remain to be addressed. Firstly, C1: efficiently distilling specific and complex spatio-temporal contextual information from the downstream tasks is essential. How-",
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+    {
+        "type": "page_footnote",
+        "text": "$^{1}$ South China University of Technology  $^{2}$ University of Hong Kong  $^{3}$ PAZHOU LAB. Correspondence to: Chao Huang <chao-huang75@gmail.com>.",
+        "bbox": [
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+    {
+        "type": "page_footnote",
+        "text": "Proceedings of the  $41^{st}$  International Conference on Machine Learning, Vienna, Austria. PMLR 235, 2024. Copyright 2024 by the author(s).",
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+        "text": "1",
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+    {
+        "type": "image",
+        "img_path": "images/db884f595d7f7fcd4b850a8936bc81034dcf9ab370fe1f36937aa5473fc80490.jpg",
+        "image_caption": [
+            "Figure 1. Motivations behind FlashST. The left figure illustrates the diverse data distributions across various ST datasets, while the right figure demonstrates that the end-to-end model's parameters are overfit to training set A and fail to generalize to test set B."
+        ],
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+    {
+        "type": "text",
+        "text": "ever, equipping pre-trained models with the ability to understand and incorporate the spatial and temporal characteristics of new domain data that is only accessible during testing is a formidable challenge. Secondly, C2: there is often a significant distribution gap between the training and test datasets, especially when they are collected from different spatio-temporal scenarios and domains. Consequently, it is essential to enhance the model adaptation framework by enabling it to effectively bridge the distribution gap and capture spatio-temporal invariants. This enhancement will facilitate the successful knowledge transfer from the pretrained phase to the downstream prediction tasks.",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Contributions. To overcome the challenges mentioned above, we introduce a lightweight and innovative prompting framework (FlashST) that aims to achieve spatiotemporal in-context learning, enabling efficient and effective model adaptation across different spatio-temporal prediction tasks. Within our framework, we employ a spatio-temporal prompt network to enable in-context learning. To tackle challenge C1, we propose a context distillation mechanism that captures contextual signals from unseen data, facilitating adaptation to diverse spatio-temporal scenarios. Furthermore, a dependency modeling scheme is introduced to capture and understand the relationships between time and locations, enabling effective analysis of inter-dependencies among spatio-temporal elements. To overcome challenge C2, we enhance our FlashST framework by incorporating a unified distribution mapping mechanism, which bridges the distribution gap between pre-trained and downstream tasks. This mechanism facilitates the effective transfer of knowledge from the pre-trained insights to downstream spatio-temporal forecasting by aligning the data distributions through the regularization of prompt embeddings.",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We conduct extensive experiments on four distinct types of spatio-temporal data tasks to evaluate the effectiveness of our proposed framework. The results demonstrate that our framework substantially improves the generalization capabilities in downstream prediction tasks across different spatio-temporal datasets. The noteworthy enhancement in predictive performance, achieved with model efficiency, underscores the efficacy of our FlashST framework.",
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+    {
+        "type": "text",
+        "text": "2. Related Work",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Deep Spatio-Temporal Learning is a thriving research field that aims to model and understand the intricate spatiotemporal dynamics in real-world urban data. Various approaches have been proposed, including integrating recurrent neural networks (RNNs) (Yao et al., 2018; Zhao et al., 2020) or Transformers (Wang et al., 2020; Xu et al., 2020) to capture long-term patterns and short-term fluctuations. Attention mechanisms have been widely used to dynamically weigh spatial and temporal features (Zheng et al., 2020; Zhang et al., 2021). MLPs are also utilized as temporal encoders to capture temporal correlations (Shao et al., 2022a). Another important direction involves leveraging Graph Neural Networks (GNNs) (Yu et al., 2018; Li et al., 2018; Yu et al., 2018; Ye et al., 2021) to simultaneously capture spatial dependencies and temporal evolutions. These models employ graph structures to represent relationships between spatial entities, facilitating information propagation. However, most existing deep spatio-temporal models adopt an end-to-end approach, limiting their generalization across diverse urban forecasting tasks and domains.",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Pre-Training with Spatio-Temporal Data. There has been a recent surge of interest in using pre-training for acquiring comprehensive representations from spatio-temporal data through self-supervised learning. This approach aims to learn generalizable representations that capture the underlying patterns and dynamics. Two dominant research lines have emerged: i) integrating contrastive learning into the pre-training stage for spatio-temporal data (Zhang et al., 2023a), and ii) utilizing masked autoencoders (Shao et al., 2022b; Li et al., 2023) to reconstruct the masked input data to learn the spatio-temporal relationships.",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Prompt-Tuning is a technique that optimizes prompts or instructions during inference to fine-tune the model, resulting in more accurate and context-specific predictions (Shrivastava et al., 2023; Liu et al., 2023a). It has gained recognition as a promising approach to improve the performance and adaptability of pre-trained models in various domains. For example, in prompt-based language generation, it allows precise control over the generated output, aligning it with specific requirements or desired styles (Brown et al., 2020; Schick & Schütze, 2020; Zhang et al., 2023b). Similarly, in vision prompt learning, specific instructions guide the model's attention towards relevant visual features or concepts (Zhou et al., 2022b;a; Sohn et al., 2023). This study explores spatio-temporal prompt-tuning to enhance prediction models in urban computing.",
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+    {
+        "type": "text",
+        "text": "3. Preliminaries",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Representation of Spatial-Temporal Data: The spatial-temporal information is captured and encoded using a three-",
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+        "page_idx": 1
+    },
+    {
+        "type": "header",
+        "text": "Title Suppressed Due to Excessive Size",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "page_number",
+        "text": "2",
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+    {
+        "type": "text",
+        "text": "way tensor  $\\mathbf{X}$ , where  $\\mathbf{X} \\in \\mathbb{R}^{R \\times T \\times F}$ . In this representation,  $R$  represents the number of regions,  $T$  denotes the time slots, and  $F$  indicates the number of features. Each entry  $\\mathbf{X}_{r,t,f}$  of the tensor corresponds to the  $f$ -th feature of the  $r$ -th region at the  $t$ -th time slot. For example, in the context of traffic flow prediction, the tensor  $\\mathbf{X}$  captures traffic volume data, which quantifies the number of vehicles passing through a specific region within a fixed time interval (e.g., 5 minutes).",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Spatio-Temporal Prediction. In the context of spatiotemporal learning, a common scenario revolves around predicting future urban spatio-temporal conditions using historical records. This scenario can be described as follows:",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathbf {X} _ {t _ {K + 1}: t _ {K + P}} ^ {A} = g \\left(\\mathbf {X} _ {t _ {K - H + 1}: t _ {K}} ^ {A}\\right) \\tag {1}\n$$\n",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "In this scenario,  $\\mathbf{X}^A$  represents the spatio-temporal data from dataset A. The function  $g(\\cdot)$  corresponds to the spatiotemporal predictive function, which could be methods like spatio-temporal transformers or spatio-temporal graph neural networks (GNNs). These predictive methods generate forecasts for the next  $P$  time slots by leveraging the historical spatio-temporal data from the previous  $H$  time slots.",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathbf {X} _ {t _ {K + 1}: t _ {K + P}} ^ {B} = \\tilde {g} \\left(f \\left(\\mathbf {X} _ {t _ {K - H + 1}: t _ {K}} ^ {B}\\right)\\right) \\tag {2}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Furthermore, we introduce  $\\tilde{g} (\\cdot)$  as the spatio-temporal prediction model with fixed parameters, and  $\\mathbf{X}^B$  represents an unseen spatio-temporal dataset. The goal of this work is to enable swift adaptation of the  $\\tilde{g} (\\cdot)$  model to a new dataset while ensuring parameter efficiency. To achieve this, we propose incorporating a lightweight and effective prompt-tuning paradigm into the pre-trained  $\\tilde{g} (\\cdot)$  model.",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "4. Methodology",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "This section provides an in-depth explanation of the technical details of the proposed FlashST framework. We commence by outlining the foundational paradigm and design principles that guide the development of the FlashST framework. Subsequently, we present a comprehensive overview of the framework's main components: the Spatio-Temporal In-Context Learning and the Unified Distribution Mapping Mechanism. To provide a visual representation of the overall model architecture, please refer to Figure 2.",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "4.1. Spatio-Temporal In-Context Learning",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "Our in-context learning framework is implemented through a spatio-temporal prompt network, which comprises two primary components: (i) Spatio-temporal context distillation mechanism efficiently captures contextual signals that are aware of both time and location from the unseen data. By doing so, it enables the model to learn from the specific context of the data, facilitating effective adaptation to various spatio-temporal scenarios. (ii) Spatio-temporal dependency",
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+    },
+    {
+        "type": "text",
+        "text": "modeling scheme incorporates complex relationships across time and locations into the in-context spatio-temporal network. By capturing and modeling these dependencies, the network can effectively understand the interdependencies and interactions between different spatio-temporal elements.",
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+    {
+        "type": "text",
+        "text": "4.1.1. Spatio-Temporal Context Distillation",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Spatio-Temporal Data Projection. To initialize the representation of the spatio-temporal data, we employ a projection layer with two steps: normalization using the Z-Score function and augmentation through linear transformation:",
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+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathbf {E} _ {r, f} = \\bar {\\mathbf {X}} _ {r, f} \\mathbf {e} _ {0}; \\quad \\bar {\\mathbf {X}} _ {r, t, f} = \\mathbf {Z} \\left(\\mathbf {X} _ {r, t, f}\\right) = \\frac {\\mathbf {X} _ {r , t , f} - \\mu}{\\sigma} \\tag {3}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "$\\mathbf{E}_{r,f} \\in \\mathbb{R}^d$  represents the representation for the  $r$ -th region of the  $f$ -th feature, where  $d$  denotes the number of hidden units.  $\\bar{\\mathbf{X}}_{r,t,f} \\in \\mathbb{R}$  corresponds to the normalized spatiotemporal data feature obtained through the Z-Score  $(Z(\\cdot))$  function. The parameters  $\\mu$  and  $\\sigma$  represent the mean and standard deviation of the raw ST matrix  $\\mathbf{X}$ . The parametric vector  $\\mathbf{e}_0 \\in \\mathbb{R}^{T \\times d}$  denotes the unit embedding vector for the normalized spatio-temporal feature.",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Temporal Context Incorporation. In order to capture dynamic and periodic spatio-temporal patterns from diverse urban data, we introduce the time-aware context into our prompt network. This context is based on multi-resolution temporal features, specifically the hour of the day  $(\\mathbf{z}^{(d)} \\in \\mathbb{R}^T)$  and the day of the week  $(\\mathbf{z}^{(w)} \\in \\mathbb{R}^T)$ . To effectively extract the temporal contextual signals, we employ a process of transformation and concatenation (denoted as  $\\mathrm{CAT}[,]$ ), which can be summarized as follows:",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathbf {M} _ {t} = \\operatorname {C A T} \\left[ \\mathbf {z} _ {t} ^ {(d)} \\cdot \\mathbf {e} _ {1}, \\mathbf {z} _ {t} ^ {(w)} \\cdot \\mathbf {e} _ {2} \\right] \\tag {4}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Spatial Context Incorporation. To enhance the prompt network with geographical contextual information related to regional properties, we incorporate the citywide road network structure as an encoding feature that reflects spatial context. This process starts by formulating the normalized Laplacian matrix, which is defined as follows:",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\triangle = \\mathbf {I} - \\mathbf {D} ^ {- 1 / 2} \\mathbf {A D} ^ {- 1 / 2} \\tag {5}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Here,  $\\mathbf{I}$ ,  $\\mathbf{D}$ , and  $\\mathbf{A}$  denote the identity matrix, the degree matrix, and the adjacency matrix, respectively. The adjacency matrix is computed by considering the distances between areas and the road structure. Since Laplacian eigenvectors effectively preserve global graph structure information in Euclidean space, we perform eigenvalue decomposition to obtain  $\\triangle = \\mathbf{U} \\boldsymbol{\\Lambda} \\mathbf{U}^{\\top}$ . After extracting the eigenvalue matrix  $\\boldsymbol{\\Lambda}$  and the corresponding eigenvector matrix  $\\mathbf{U}$ , we derive the structure-aware node properties  $\\mathbf{C} \\in \\mathbb{R}^{R \\times d_r}$  by projecting  $\\mathbf{U}$  to obtain the  $d_r$  smallest nontrivial eigenvectors.",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "header",
+        "text": "Title Suppressed Due to Excessive Size",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "page_number",
+        "text": "3",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "image",
+        "img_path": "images/effd40696667807a4483bb063bad61144fabe5d2457046964e71dbf7b064d2c9.jpg",
+        "image_caption": [
+            "Figure 2. Our proposed FlashST framework adopts an architecture that integrates spatio-temporal in-context learning and a unified distribution mapping mechanism, offering an efficient and effective approach for spatio-temporal prompt-tuning across diverse scenarios."
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Acknowledging the potential disparity of  $\\mathbf{C}$  in the feature space between training and test datasets, we employ a MLP to map these features and enhance the network's ability to generalize spatial context. Subsequently, we use the concatenation operation to integrate the aforementioned embeddings and obtain the initial spatio-temporal embedding:",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\bar {\\mathbf {E}} = \\operatorname {C A T} [ \\mathbf {E}, \\mathbf {M}, \\bar {\\mathbf {C}} ]; \\quad \\bar {\\mathbf {C}} _ {r, f} = \\operatorname {M L P} \\left(\\mathbf {C} _ {r, f}\\right) \\tag {6}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "$\\bar{\\mathbf{E}}$  denotes the integrated embeddings, where we concatenate the original representation  $\\mathbf{E}$ , the temporal context embeddings  $\\mathbf{M}$ , and the transformed regional properties  $\\bar{\\mathbf{C}}$ . The transformation of  $\\mathbf{C}$  is achieved through the MLP function.",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "4.1.2. Spatio-Temporal Dependency Modeling",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Temporal Dependency Encoder. To capture dependencies across different time slots and preserve time-evolving data patterns, we introduce a lightweight gating mechanism. The formal operations of this mechanism are presented below:",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathbf {H} _ {r, f} = \\left(W _ {2} \\sigma \\left(W _ {1} \\overline {{\\mathbf {E}}} _ {r, f} + b _ {1}\\right) + b _ {2}\\right) + \\overline {{\\mathbf {E}}} _ {r, f} \\tag {7}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The trainable parameters of this gating mechanism are denoted by  $W_{1}, W_{2} \\in \\mathbb{R}^{d' \\times d'}$  and  $b_{1}, b_{2} \\in \\mathbb{R}^{d'}$ . The initial spatio-temporal embedding is represented by  $\\bar{\\mathbf{E}}_{r,f}$ , while the encoded temporal embedding is denoted by  $\\mathbf{H}_{r,f} \\in \\mathbb{R}^{d' \\times d'}$ . The embedding  $\\bar{\\mathbf{E}}_{r,f}$  contains valuable contextual information about temporal dynamics and regional characteristics, which is essential for the temporal dependency encoder. This enriched information enables our in-context learning to effectively identify variations in spatio-temporal patterns across different regions and time intervals, thereby facilitating precise modeling of temporal correlations.",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Spatial Dependency Encoder. Drawing inspiration from the effectiveness of graph neural networks in capturing spatial correlations between geographical locations (Wu et al., 2020b; Zheng et al., 2020), we leverage graph convolution-based message passing to encode inter-regional correlations. The adjacency matrix  $\\mathbf{A}$ , as defined in Eq 5, acts as the connectivity matrix within the graph network framework. The process of spatial encoding is formalized as follows:",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathbf {S} _ {f} = \\sigma (\\mathbf {A} \\cdot \\mathbf {H} _ {f} \\cdot W _ {3}) + \\mathbf {H} _ {f} \\tag {8}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Here,  $W_{3} \\in \\mathbb{R}^{d' \\times d'}$  represents the trainable parameters. To mitigate the potential over-smoothing phenomenon caused by multi-layer GNNs, we employ residual networks. By stacking multiple layers of spatio-temporal encoders, our model generates a representation  $\\mathbf{E}_{pro}$  that captures rich spatial and temporal dependencies across time and space.",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "4.2. Unified Distribution Mapping Mechanism",
+        "text_level": 1,
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "To bridge the distribution gap between the pre-training and diverse unseen data in downstream tasks, we enhance FlashST by incorporating a distribution mapping mechanism. The objective of this mechanism is to transform both the pre-training data and the downstream data into a shared distribution space. This alignment of data distributions enables a seamless transfer of knowledge, ensuring that the insights gained from the pre-training stage can be effectively applied to the downstream spatio-temporal context.",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "To achieve this goal, our FlashST involves standardizing the prompt embedding to ensure a consistent distribution across diverse downstream datasets. We draw inspiration from various works in contrastive learning (Wang & Isola, 2020; Chuang et al., 2020; Wang & Liu, 2021) and incorporate the infoNCE-based loss function to regularize the prompt network for representation generation. This loss function is designed to bring the representations of positive sample pairs closer together while pushing apart those of negative pairs. By leveraging self-supervised learning without the need for additional labeling, optimizing the infoNCE loss helps attain a more uniform embedding distribution. Empirical evidence suggests that through this loss alone, an almost entirely uniform distribution can be achieved (Yu et al., 2022). Building upon this, we employ the infoNCE loss to adjust the distribution of the learned spatio-temporal prompt embedding  $\\mathbf{E}_{pro} \\in \\mathbb{R}^{R \\times F \\times d'}$ , formalized as:",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathcal {L} _ {U n i} = \\sum_ {r = 1} ^ {R} \\sum_ {f = 1} ^ {F} - \\log \\sum_ {r ^ {\\prime}} \\exp \\left(\\cos \\left(\\mathbf {E} _ {p r o} ^ {r, f}, \\mathbf {E} _ {p r o} ^ {r ^ {\\prime}, f}\\right) / \\tau\\right) \\tag {9}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The cosine similarity function  $\\cos (\\cdot)$  is used to measure the similarity between embeddings, and the temperature coefficient  $\\tau$  is used to adjust the softmax scale. In our FlashST, we enhance the uniformity of prompt embeddings by increasing the separation among embeddings that correspond",
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+        "page_idx": 3
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+    {
+        "type": "header",
+        "text": "Title Suppressed Due to Excessive Size",
+        "bbox": [
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+    {
+        "type": "page_number",
+        "text": "4",
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "to different regions. This improvement allows downstream models to effectively utilize the provided prompts  $\\mathbf{E}_{pro}^{(U)}$  for rapid generalization across new data and tasks.",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4.3. Unifying Pre-training and Downstream Tasks",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Our spatio-temporal pretrain-prompt framework integrates the model pre-training process with downstream forecasting tasks through efficient prompt-tuning. In the pre-training phase, we train and optimize all parameters using dedicated pre-training datasets. Subsequently, in the prompt fine-tuning phase, we exclusively update the parameters of the prompt network by conducting a limited number of training epochs on unseen datasets. This approach enables the downstream model to effectively adapt to new data. Moreover, our framework is model-agnostic, allowing seamless integration with various existing spatio-temporal prediction baselines as the downstream model.",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "i) Pre-training Phase: Our goal is to forecast future trends using historical spatio-temporal records from the pre-training data. We achieve this with following process:",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\hat {\\mathbf {Y}} ^ {A} = g \\left(\\mathbf {E} _ {p r o} ^ {(U)}\\right); \\quad \\mathbf {E} _ {p r o} ^ {(U)} = f _ {P r o m p t} \\left(\\mathbf {X} ^ {A}\\right) \\tag {10}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Here, we have adjustable parameters for both  $f_{Prompt}(\\cdot)$  and  $g(\\cdot)$ . To update these parameters, we utilize the loss function  $\\mathcal{L}$  in conjunction with the Adam optimizer. The loss function  $\\mathcal{L}$  is a combination of the regression loss  $\\mathcal{L}_r$  and the uniformity loss  $\\mathcal{L}_{Uni}$ , with  $\\lambda$  as the balance coefficient between the two losses. The label is represented by  $Y \\in \\mathbb{R}^{R \\times T \\times F}$ . The regression loss  $\\mathcal{L}_r$  is defined as:",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathcal {L} _ {r} = \\frac {1}{R T F} \\sum_ {r = 1} ^ {R} \\sum_ {t = 1} ^ {T} \\sum_ {f = 1} ^ {F} | \\left(\\mathbf {Y} _ {r, t, f} - \\hat {\\mathbf {Y}} _ {r, t, f}\\right) | \\tag {11}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "This formulation calculates the average absolute difference between the predicted values  $\\hat{\\mathbf{Y}}_{r,t,f}$  and the actual labels  $\\mathbf{Y}_{r,t,f}$  across all regions  $R$ , time steps  $T$ , and features  $F$ .",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "ii) Prompt-Tuning Phase: We freeze the parameters of the downstream model and focus exclusively on fine-tuning the prompt network. This fine-tuning process is carried out on the test dataset to perform spatio-temporal prediction tasks, following the formulation described in Eq 10:",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\hat {\\mathbf {Y}} ^ {B} = \\tilde {g} \\left(f _ {\\text {P r o m p t}} \\left(\\mathbf {X} ^ {B}\\right)\\right) \\tag {12}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Here,  $\\tilde{g}$  represents the downstream model with frozen parameters, and  $B$  indicates the test datasets. During the prompt-tuning phase, we conduct a limited number of optimization epochs to generate high-quality spatio-temporal prompt representations for the downstream models.",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "5. Evaluation",
+        "text_level": 1,
+        "bbox": [
+            84,
+            859,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In this section, we analyze the performance of our model by addressing the following several research questions:",
+        "bbox": [
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+            886,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "table",
+        "img_path": "images/3e747514a2f7577f741660f0fe710550d9d9f06b67371c8e9d6ec9046e94af1e.jpg",
+        "table_caption": [
+            "Table 1. Statistics of pre-training dataset."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Dataset</td><td>Data Record</td><td># Region</td><td>Time Steps</td><td>Sample Date</td></tr><tr><td>PEMS03</td><td>traffic flow</td><td>358</td><td>26208</td><td>1/Sep/2018 - 30/Nov/2018</td></tr><tr><td>PEMS04</td><td>traffic flow</td><td>307</td><td>16992</td><td>1/Jan/2018 - 28/Feb/2018</td></tr><tr><td>PEMS07</td><td>traffic flow</td><td>883</td><td>26208</td><td>1/May/2017 - 30/Aug/2017</td></tr><tr><td>PEMS08</td><td>traffic flow</td><td>170</td><td>17856</td><td>1/Jul/2016 - 31/Aug/2016</td></tr></table>",
+        "bbox": [
+            501,
+            97,
+            883,
+            152
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "table",
+        "img_path": "images/d0d82ae1903627ba202acbbac18806e0aa3daec21ef4935d9adcf6d87e478494.jpg",
+        "table_caption": [
+            "Table 2. Statistics of downstream tasks dataset."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Dataset</td><td>Data Record</td><td># Region</td><td>Time Steps</td><td>Sample Date</td></tr><tr><td>PEMS07(M)</td><td>traffic speed</td><td>170</td><td>12672</td><td>1/Jul/2016 - 31/Aug/2016</td></tr><tr><td>CA-D5</td><td>traffic flow</td><td>211</td><td>16992</td><td>1/Jan/2017 - 28/Feb/2017</td></tr><tr><td>ChengDu-DIDI</td><td>traffic index</td><td>524</td><td>17280</td><td>1/Jan/2018 - 30/Apr/2018</td></tr><tr><td>NYC Citi Bike</td><td>bike orders</td><td>250</td><td>4368</td><td>1/Apr/2016 - 30/Jun/2016</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "- RQ1: Does FlashST effectively generalize pre-trained models to new spatio-temporal prediction data and tasks?",
+            "- RQ2: How does the efficiency of our model compare to that of end-to-end training and fine-tuning approaches?",
+            "- RQ3: What is the impact of the key components of our FlashST on the performance of downstream models?",
+            "- RQ4: How do parameters affect the model performance?"
+        ],
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "5.1. Experimental Setting",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Datasets. To evaluate how well the model can generalize across different urban spatio-temporal contexts, our experimental setup is designed as follows: (i) In the pre-training phase, we use four datasets (PEMS03, PEMS04, PEMS07, and PEMS08 (Song et al., 2020)) as our training sets. These datasets consist of records detailing traffic flow conditions across various streets and cities in California, USA. (ii) During the subsequent prompt-tuning phase, we concentrate on four distinct target datasets to fine-tune and evaluate our framework: PEMS07(M) (Yu et al., 2018), CA-D5 (Liu et al., 2023b), ChengDu-DIDI (Lu et al., 2022), and NYC Citi Bike (Ye et al., 2021). These datasets correspondingly represent traffic speeds in Los Angeles, traffic flow in California, traffic flow in Chengdu, and bicycle demand in New York City. Each target dataset is partitioned into training, validation, and test sets in a ratio of 6:2:2. Detailed statistics for each dataset can be found in Table 1 and Table 2.",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Experimental Description. We configure the model with 32 hidden units for  $d$ ,  $d_t$ , and  $d_r$ , while employing a spatial and temporal encoder with 2 layers. The pre-training phase involves alternating training for 300 epochs on the four pre-training datasets. Subsequently, we conduct fine-tuning for 20 epochs during the prompt-tuning phase. In both the baseline and full fine-tuning setups, the maximum number of training epochs is limited to 100. Furthermore, a batch size of 64 is employed for both phases and all baselines. For the dataset-specific parameters in the baselines (e.g., node embeddings), we initialize them randomly and make them trainable during the prompt fine-tuning phase. Additionally, the final regression layers of some baseline models are set to be trainable to ensure accurate numerical predictions.",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Evaluation Metrics. In order to assess the performance of the model in spatiotemporal prediction, we employ three widely-used evaluation metrics: Mean Absolute Error (MAE), Root Mean Squared Error (RMSE), and Mean",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "header",
+        "text": "Title Suppressed Due to Excessive Size",
+        "bbox": [
+            362,
+            56,
+            609,
+            71
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "page_number",
+        "text": "5",
+        "bbox": [
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+            922,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "table",
+        "img_path": "images/cc90fdfd45b97da7eccb04abf8aaefd2cc0df8dc6e098352c7fe3c4943aef85b.jpg",
+        "table_caption": [
+            "Table 3. Overall performance on PEMS07(M), CA-D5, ChengDu-DIDI and NYC Citi Bike datasets in terms of MAE, RMSE and MAPE."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Model</td><td rowspan=\"2\">Dataset Metrics</td><td colspan=\"3\">PEMS07(M)</td><td colspan=\"3\">CA-D5</td><td colspan=\"3\">ChengDu-DIDI</td><td colspan=\"3\">NYC Citi Bike</td></tr><tr><td>MAE</td><td>RMSE</td><td>MAPE</td><td>MAE</td><td>RMSE</td><td>MAPE</td><td>MAE</td><td>RMSE</td><td>MAPE</td><td>MAE</td><td>RMSE</td><td>MAPE</td></tr><tr><td>TGCN</td><td>5.05</td><td>8.56</td><td>13.68%</td><td>15.36</td><td>24.52</td><td>24.42%</td><td>3.18</td><td>4.72</td><td>14.48%</td><td>2.19</td><td>3.54</td><td>63.31%</td><td></td></tr><tr><td>STGCN</td><td>3.14</td><td>6.25</td><td>8.05%</td><td>15.02</td><td>23.73</td><td>27.84%</td><td>2.59</td><td>3.97</td><td>12.20%</td><td>2.04</td><td>3.21</td><td>52.06%</td><td></td></tr><tr><td>ASTGCN</td><td>3.13</td><td>6.19</td><td>8.09%</td><td>15.26</td><td>24.37</td><td>24.45%</td><td>2.56</td><td>3.83</td><td>11.58%</td><td>2.03</td><td>3.10</td><td>55.13%</td><td></td></tr><tr><td>GWN</td><td>2.67</td><td>5.40</td><td>6.62%</td><td>13.63</td><td>22.30</td><td>24.68%</td><td>2.35</td><td>3.56</td><td>10.72%</td><td>1.81</td><td>2.71</td><td>50.05%</td><td></td></tr><tr><td>STSGCN</td><td>2.93</td><td>5.87</td><td>7.52%</td><td>15.30</td><td>24.44</td><td>24.70%</td><td>2.86</td><td>4.29</td><td>13.02%</td><td>1.89</td><td>2.78</td><td>52.48%</td><td></td></tr><tr><td>AGCRN</td><td>2.89</td><td>5.69</td><td>7.32%</td><td>14.44</td><td>23.10</td><td>23.02%</td><td>2.87</td><td>4.27</td><td>13.05%</td><td>2.16</td><td>3.41</td><td>60.95%</td><td></td></tr><tr><td>MTGNN</td><td>2.70</td><td>5.50</td><td>6.81%</td><td>13.90</td><td>22.44</td><td>23.86%</td><td>2.33</td><td>3.56</td><td>10.63%</td><td>1.84</td><td>2.80</td><td>48.69%</td><td></td></tr><tr><td>STFGNN</td><td>3.14</td><td>6.21</td><td>7.81%</td><td>15.38</td><td>24.37</td><td>25.78%</td><td>2.83</td><td>4.24</td><td>13.05%</td><td>2.36</td><td>3.73</td><td>61.93%</td><td></td></tr><tr><td>STGODE</td><td>2.81</td><td>5.57</td><td>7.09%</td><td>14.13</td><td>22.77</td><td>23.41%</td><td>2.51</td><td>3.75</td><td>11.48%</td><td>1.94</td><td>2.83</td><td>56.25%</td><td></td></tr><tr><td>DMSTGCN</td><td>2.94</td><td>5.92</td><td>7.45%</td><td>13.88</td><td>22.81</td><td>21.97%</td><td>2.67</td><td>4.00</td><td>12.34%</td><td>2.09</td><td>3.32</td><td>60.00%</td><td></td></tr><tr><td>MSDR</td><td>2.79</td><td>5.53</td><td>7.07%</td><td>15.31</td><td>23.71</td><td>33.01%</td><td>2.56</td><td>3.85</td><td>11.94%</td><td>1.87</td><td>2.86</td><td>51.25%</td><td></td></tr><tr><td>STWA</td><td>2.74</td><td>5.52</td><td>6.97%</td><td>14.00</td><td>22.75</td><td>23.22%</td><td>2.44</td><td>3.68</td><td>11.33%</td><td>1.88</td><td>2.83</td><td>49.93%</td><td></td></tr><tr><td>PDFormer</td><td>2.68</td><td>5.46</td><td>6.66%</td><td>13.85</td><td>22.59</td><td>23.11%</td><td>2.43</td><td>3.70</td><td>11.17%</td><td>1.89</td><td>2.88</td><td>51.98%</td><td></td></tr><tr><td>Ours</td><td>2.59</td><td>5.25</td><td>6.54%</td><td>13.26</td><td>21.83</td><td>21.49%</td><td>2.31</td><td>3.52</td><td>10.51%</td><td>1.79</td><td>2.63</td><td>49.77%</td><td></td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Absolute Percentage Error (MAPE). These metrics quantify the disparities between the predicted data and the ground truth data. It is important to note that lower values of these metrics indicate better performance.",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Baseline Description. To evaluate the effectiveness of our FlashST, we selected 13 advanced spatio-temporal prediction models as baselines. These include RNNs-based models, attention-based models, GNNs-based models, and differential equation-based spatio-temporal prediction methods.",
+        "bbox": [
+            84,
+            414,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "RNN-based Spatio-Temporal Prediction Approaches:",
+        "text_level": 1,
+        "bbox": [
+            86,
+            497,
+            460,
+            512
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "- AGCRN (Bai et al., 2020): This method incorporates RNNs with learnable node embeddings to capture personalized spatio-temporal patterns of regions.",
+            "- MSDR (Liu et al., 2022): This model addresses the issue of long-term information forgetting in RNNs by introducing a variant of RNNs. It further combines this variant with GNNs to capture spatio-temporal correlations."
+        ],
+        "bbox": [
+            84,
+            529,
+            473,
+            646
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "GNN-based Spatio-Temporal Prediction Models:",
+        "text_level": 1,
+        "bbox": [
+            84,
+            662,
+            429,
+            678
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "- TGCN (Zhao et al., 2020): This model leverages a combination of RNNs and GCNs to separately model temporal dependencies and spatial correlations.",
+            "- STGCN (Yu et al., 2018): This method utilizes gated convolutional networks to encode temporal dependencies and GCNs to capture local spatial relationships.",
+            "- GWN (Wu et al., 2019): This approach incorporates a learnable graph structure to model spatial correlations, while TCNs are used to extract temporal features.",
+            "- STSGCN (Song et al., 2020): It synchronously models dynamic spatio-temporal correlations by constructing a graph network that allows interactions across time slices."
+        ],
+        "bbox": [
+            84,
+            694,
+            473,
+            905
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "- MTGNN (Wu et al., 2020b): This method captures temporal dependencies using time convolutional networks combined with skip connections, and employs learnable graph networks to model spatial correlations.",
+            "- STFGNN (Li & Zhu, 2021): This method proposes a data-driven spatio-temporal graph construction approach to capture dynamic spatio-temporal correlations.",
+            "- DMSTGCN (Han et al., 2021): This method models dynamic spatial correlations by constructing a learnable dynamic graph network."
+        ],
+        "bbox": [
+            496,
+            345,
+            885,
+            521
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Attention-based Spatio-Temporal Models:",
+        "text_level": 1,
+        "bbox": [
+            496,
+            540,
+            794,
+            555
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "- ASTGCN (Guo et al., 2019): This method leverages attention mechanisms to effectively capture the periodic spatio-temporal correlations inherent in the data.",
+            "- STWA(Cirstea et al., 2022): This approach goes beyond standard attention mechanisms by incorporating specialized node features and time-dynamic parameters.",
+            "- **PDFformer** (Jiang et al., 2023): This model leverages the power of the transformer architecture to effectively encode the intricate spatio-temporal dependencies present in the data. Furthermore, it introduces a novel time-delayed spatial correlation modeling technique."
+        ],
+        "bbox": [
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+            573,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "ODE-based Spatio-Temporal Prediction Methods:",
+        "text_level": 1,
+        "bbox": [
+            496,
+            782,
+            849,
+            797
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "- STGODE (Fang et al., 2021): This method enriches graph networks with ordinary differential equations (ODEs) to extract continuous spatial correlation information. Time convolutional networks are employed to learn time-dependent relationships within the data.",
+        "bbox": [
+            496,
+            816,
+            885,
+            892
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "header",
+        "text": "Title Suppressed Due to Excessive Size",
+        "bbox": [
+            362,
+            56,
+            609,
+            71
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "page_number",
+        "text": "6",
+        "bbox": [
+            480,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/2f32ef6cb10c29dabf96312b63f47e55f65490ca681394c7185d1357ab9cdb2c.jpg",
+        "table_caption": [
+            "Table 4. Model-agnostic experiments on PEMs07(M), CA-D5, ChengDu-DIDI and NYC Citi Bike datasets."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Model</td><td>Dataset</td><td colspan=\"3\">PEMS07(M)</td><td colspan=\"3\">CA-D5</td><td colspan=\"3\">ChengDu-DIDI</td><td colspan=\"3\">NYC Citi Bike</td></tr><tr><td>Metrics</td><td>MAE</td><td>RMSE</td><td>MAPE</td><td>MAE</td><td>RMSE</td><td>MAPE</td><td>MAE</td><td>RMSE</td><td>MAPE</td><td>MAE</td><td>RMSE</td><td>MAPE</td></tr><tr><td>STGCN</td><td>3.14</td><td>6.25</td><td>8.05%</td><td>15.02</td><td>23.73</td><td>27.84%</td><td>2.59</td><td>3.97</td><td>12.20%</td><td>2.04</td><td>3.21</td><td>52.06%</td><td></td></tr><tr><td>w/o Finetune</td><td>8.07</td><td>11.77</td><td>23.02%</td><td>31.20</td><td>41.82</td><td>68.75%</td><td>4.82</td><td>6.53</td><td>22.87%</td><td>4.10</td><td>5.60</td><td>148.20%</td><td></td></tr><tr><td>w/ Finetune</td><td>3.18</td><td>6.25</td><td>8.04%</td><td>15.15</td><td>24.18</td><td>23.30%</td><td>2.58</td><td>3.93</td><td>12.01%</td><td>1.99</td><td>3.11</td><td>50.38%</td><td></td></tr><tr><td>Ours</td><td>2.68</td><td>5.37</td><td>6.80%</td><td>13.47</td><td>22.01</td><td>22.62%</td><td>2.37</td><td>3.60</td><td>10.92%</td><td>1.80</td><td>2.67</td><td>50.06%</td><td></td></tr><tr><td>GWN</td><td>2.67</td><td>5.40</td><td>6.62%</td><td>13.63</td><td>22.30</td><td>24.68%</td><td>2.35</td><td>3.56</td><td>10.72%</td><td>1.81</td><td>2.71</td><td>50.05%</td><td></td></tr><tr><td>w/o Finetune</td><td>7.41</td><td>12.78</td><td>24.30%</td><td>27.28</td><td>41.46</td><td>48.26%</td><td>4.89</td><td>6.99</td><td>23.42%</td><td>3.39</td><td>4.99</td><td>112.57%</td><td></td></tr><tr><td>w/ Finetune</td><td>2.69</td><td>5.38</td><td>6.74%</td><td>13.61</td><td>22.44</td><td>22.82%</td><td>2.35</td><td>3.55</td><td>10.72%</td><td>1.85</td><td>2.79</td><td>51.88%</td><td></td></tr><tr><td>Ours</td><td>2.67</td><td>5.36</td><td>6.79%</td><td>13.19</td><td>21.68</td><td>21.21%</td><td>2.37</td><td>3.58</td><td>10.77%</td><td>1.80</td><td>2.66</td><td>51.16%</td><td></td></tr><tr><td>MTGNN</td><td>2.70</td><td>5.50</td><td>6.81%</td><td>13.90</td><td>22.44</td><td>23.86%</td><td>2.33</td><td>3.56</td><td>10.63%</td><td>1.84</td><td>2.80</td><td>48.69%</td><td></td></tr><tr><td>w/o Finetune</td><td>5.91</td><td>9.21</td><td>14.56%</td><td>50.07</td><td>65.35</td><td>77.26%</td><td>4.45</td><td>6.27</td><td>20.06%</td><td>3.32</td><td>4.74</td><td>118.01%</td><td></td></tr><tr><td>w/ Finetune</td><td>2.62</td><td>5.33</td><td>6.56%</td><td>13.46</td><td>22.02</td><td>22.10%</td><td>2.33</td><td>3.54</td><td>10.52%</td><td>1.81</td><td>2.74</td><td>49.35%</td><td></td></tr><tr><td>Ours</td><td>2.59</td><td>5.25</td><td>6.54%</td><td>13.26</td><td>21.83</td><td>21.49%</td><td>2.31</td><td>3.52</td><td>10.51%</td><td>1.79</td><td>2.63</td><td>49.77%</td><td></td></tr><tr><td>PDForner</td><td>2.68</td><td>5.46</td><td>6.66%</td><td>13.85</td><td>22.59</td><td>23.11%</td><td>2.43</td><td>3.70</td><td>11.17%</td><td>1.89</td><td>2.88</td><td>51.98%</td><td></td></tr><tr><td>w/o Finetune</td><td>5.57</td><td>10.03</td><td>19.16%</td><td>20.45</td><td>31.56</td><td>30.21%</td><td>3.60</td><td>5.20</td><td>16.93%</td><td>3.19</td><td>4.76</td><td>111.40%</td><td></td></tr><tr><td>w/ Finetune</td><td>2.71</td><td>5.44</td><td>6.80%</td><td>13.77</td><td>22.31</td><td>24.90%</td><td>2.43</td><td>3.69</td><td>11.15%</td><td>1.83</td><td>2.78</td><td>48.56%</td><td></td></tr><tr><td>Ours</td><td>2.68</td><td>5.31</td><td>6.79%</td><td>13.55</td><td>22.08</td><td>21.98%</td><td>2.41</td><td>3.64</td><td>11.05%</td><td>1.81</td><td>2.68</td><td>52.06%</td><td></td></tr></table>",
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+    {
+        "type": "text",
+        "text": "5.2. Overall Performance (RQ1)",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "In this section, we investigate the effectiveness of FlashST in enhancing the generalization capabilities of downstream models across new spatio-temporal contexts. As shown in Table 3, the baseline models were exclusively trained and tested on the target dataset to evaluate their performance.",
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+    {
+        "type": "text",
+        "text": "Superiority over End-to-End Prediction Models. Upon analyzing the results presented in Table 3, it becomes evident that our approach exhibits a significant advantage over the end-to-end spatio-temporal models in diverse urban data prediction scenarios. These findings provide compelling evidence of the effectiveness of our FlashST method in accurately capturing the intricate spatio-temporal invariant patterns present in urban data. Our new in-context learning paradigm excels in transferring this acquired knowledge to adapt the pre-trained model to new downstream tasks. By effectively handling the distribution gap, our method bridges the semantic gap between the pre-trained model and the specific prediction scenarios encountered in practice.",
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+    {
+        "type": "text",
+        "text": "Model-Agnostic Advantage. The model-agnostic advantage is a notable strength of our approach. We have designed our method to seamlessly integrate with various spatio-temporal backbone encoders, providing flexibility and avoiding the constraints of a specific model choice. This adaptability is demonstrated in Table 4, where our FlashST approach showcases easy adaptation with four state-of-the-art spatio-temporal models (i.e., STGCN, GWN, MTGNN, PDFormer). The evaluation results highlight the versatility of our FlashST approach, showcasing its exceptional performance enhancement when combined with these cutting-edge spatio-temporal models. The successful integration of our approach with state-of-the-art models further reinforces its adaptability and capability to improve prediction accuracy in diverse urban data scenarios.",
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+    {
+        "type": "text",
+        "text": "Comparison with Model Fine-Tuning. To further demonstrate the effectiveness of our spatio-temporal in-context framework, we compare our prompt-tuning approach with the full fine-tuning of the model parameters. The \"w/o Finetune\" approach refers to direct prediction on the target dataset without any fine-tuning after pre-training. On the other hand, \"w/ Finetune\" indicates the utilization of full-parameter fine-tuning to adapt the models to the target data after pre-training. However, it is worth noting that the observed improvements with direct fine-tuning, compared to end-to-end prediction outcomes, suggest that pre-training may not have provided a significant advantage. In the absence of effective alignment between the pre-trained model and downstream tasks, noise can be introduced, leading to misleading fine-tuning and suboptimal performance.",
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+    {
+        "type": "text",
+        "text": "5.3. Assessment of Model Efficiency (RQ2)",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Training Time. In this section, we focus on evaluating the efficiency of our proposed method. We measured the training time for three different scenarios: end-to-end training, full fine-tuning, and our FlashST approach. The results are presented in Table 5. For end-to-end training and full fine-tuning, we followed the settings of existing baselines, configuring the training epochs to 100. We also set an early stopping criterion of 25, which halts the training process if no decrease in validation loss is observed. Regarding our FlashST approach, we limited the number of epochs for prompt-tuning to 20. This constraint was implemented to promote swift adaptation of the downstream models to the new datasets. The results suggest that the same baseline model showcases comparable efficiency in both end-to-end training and full-parameter fine-tuning. The disparity in training time between these two settings primarily arises from variations in convergence speed caused by different initialization parameters. One of the significant contribu",
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+    {
+        "type": "header",
+        "text": "Title Suppressed Due to Excessive Size",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "page_number",
+        "text": "7",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/c422d628cf15114e7038736686bcec2fae887fde3126549fbc9837d7c343bce8.jpg",
+        "table_caption": [
+            "Table 5. Computational time cost investigation (seconds)."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>time(s)</td><td>Model</td><td>time(s)</td></tr><tr><td>ASTGCN</td><td>1310</td><td>TGCN</td><td>927</td></tr><tr><td>AGCRN</td><td>689</td><td>STSGCN</td><td>2253</td></tr><tr><td>DMSTGCN</td><td>1330</td><td>STFGNN</td><td>990</td></tr><tr><td>MSDR</td><td>5190</td><td>STWA</td><td>5143</td></tr><tr><td>STGCN</td><td>411</td><td>GWN</td><td>1042</td></tr><tr><td>STGCN Finetune</td><td>445</td><td>GWN Finetune</td><td>1185</td></tr><tr><td>STGCN w/ Ours</td><td>190</td><td>GWN w/ Ours</td><td>222</td></tr><tr><td>MTGNN</td><td>962</td><td>PDForner</td><td>7524</td></tr><tr><td>MTGNN Finetune</td><td>1095</td><td>PDForner Finetune</td><td>6466</td></tr><tr><td>MTGNN w/ Ours</td><td>230</td><td>PDForner w/ Ours</td><td>1220</td></tr></table>",
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+    {
+        "type": "image",
+        "img_path": "images/c90212b93992b54d893ad9a9d2108a304e9c21f3a764331a685d62e54ea7c603.jpg",
+        "image_caption": [
+            "Figure 3. The convergence efficiency of FlashST."
+        ],
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+    {
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+    {
+        "type": "text",
+        "text": "tions of our FlashST framework is the notable increase in computational efficiency. It reduces the training time of baseline models by  $20\\%$  to  $80\\%$ . This substantial enhancement significantly improves their efficiency in adapting to new spatio-temporal data.",
+        "bbox": [
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+    {
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+        "text": "Faster Convergence Rate. To further validate the ability of our FlashST framework to swiftly adapt to new spatiotemporal scenarios, we conducted an investigation into its convergence speed on different datasets. We specifically focused on the decreasing trend of the validation error when using the PEMs07(M) and CA-D5 datasets, with MTGNN adopted as the downstream model. The results are visualized in Figure 3. The findings clearly demonstrate that with the integration of our FlashST approach, the downstream model achieves convergence within a few tuning epochs. In contrast, the end-to-end training and fine-tuning paradigms require a greater number of training rounds to fit the new data. This phenomenon can be attributed to the effectiveness of our proposed spatio-temporal prompt network and data mapping strategy. These components enable the model to leverage the spatio-temporal characteristics of the new data in conjunction with the pre-trained knowledge, facilitating rapid adaptation to diverse spatio-temporal scenes.",
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+    {
+        "type": "text",
+        "text": "5.4. Ablation Study (RQ3)",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "In this section, we conduct an ablation study to investigate the contributions of each key module in our FlashST framework. The corresponding experimental results are depicted in Figure 4. We utilize MTGNN as the downstream baseline and assess the performance of each component on the",
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+        "type": "text",
+        "text": "PEMS07(M) and CA-D5 data. Before the prompt-tuning step, we conduct re-pre-training to adapt to each variant.",
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+    {
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+        "text": "- Impact of Spatio-Temporal Context Distillation. To assess the importance of temporal and spatial context distillation, we conducted experiments by removing the distillation with temporal context (-TC) and spatial context (-SC) separately. The results clearly demonstrate a substantial performance drop across most metrics when the distilled temporal and spatial context are removed. This highlights the critical importance of preserving both temporal and spatial context during the in-context learning process. Effective encoding of the temporal information and the incorporation of the spatial information are vital for capturing dynamic periodic temporal patterns and region-specific properties, thereby enhancing the model's ability to recognize spatio-temporal invariant patterns and improve its data understanding.",
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+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "- Impact of Spatio-Temporal Dependency Modeling. In our analysis, we individually removed the temporal encoder (-TE) and spatial encoder (-SE) to investigate their contributions. Our analysis revealed that spatio-temporal dependency encoding plays a vital role in effectively integrating complex relationships among different time slots and locations during the in-context learning process. The inclusion of both temporal and spatial dependency encoders enables the model to comprehend and leverage the intricate interactions between time and space. This proficiency facilitates a swifter adaptation of downstream models to new spatio-temporal scenarios.",
+            "- Impact of Unified Distribution Mapping Mechanism. In our variants, we assess the utility of the unified distribution mapping strategy through two aspects. i) -Uni. We did not explore the unified distribution mapping strategy. However, the decline in performance indicates its positive impact on the model. By mapping the diverse spatiotemporal data embeddings into a uniform distribution, our FlashST effectively mitigates the influence of distributional disparities between the pre-training data and unseen spatio-temporal data. ii) r/BN. The unified distribution mapping strategy has been replaced with batch normalization. Batch normalization standardizes the data based on local statistical properties of mini-batches, mitigating the issue of internal covariate shift during neural network training and thereby enhancing convergence efficiency. However, due to the lack of an established connection between the pretrained data and the downstream task data, it is challenging for the downstream model to effectively transfer knowledge from the pretraining process. The unified distribution mapping strategy ensures that the model can effectively leverage the knowledge acquired during the pre-training. By aligning the distributions of different data sources, the model can better adapt to new spatiotemporal scenarios and make more accurate predictions."
+        ],
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+    },
+    {
+        "type": "header",
+        "text": "Title Suppressed Due to Excessive Size",
+        "bbox": [
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+    },
+    {
+        "type": "page_number",
+        "text": "8",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/b179fa4c5df20fcdccf3acb7141055e3c3b7e8c4348914ab2eb9562a3f94a030.jpg",
+        "image_caption": [
+            "(b) MAE, RMSE and MAPE on PEMS07(M) dataset."
+        ],
+        "image_footnote": [],
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+    {
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+        "img_path": "images/6184d5046e1cd0f84dbcf889766a6f99eb1b64a10a2891fe2804cd2bda196307.jpg",
+        "image_caption": [
+            "(c) MAE, RMSE and MAPE on CA-D5 dataset.",
+            "Figure 4. Ablation study of FlashST."
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "5.5. Hyperparameter Analysis (RQ4)",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In this section, we investigate the impact of different hyperparameter settings, specifically across various temperature coefficients and loss weight coefficients, on the model's performance. Our findings reveal that the model achieves optimal performance with a parameter configuration of  $\\tau = 0.3$  and  $\\lambda = 1.0$ , as shown in Figure 6. Notably, variations in these parameters minimally influence the final result, highlighting the model's effective adaptability to different parameters settings. It learns efficient representations that distinguish embedded features in various regions, even with differences in feature scales. Additionally, the model's performance remains unaffected by an increase in the uniformity loss. This indicates that the key component of our distribution mapping strategy does not interfere with the predictive loss. This further supports the viability of our strategy and facilitates the downstream model's rapid generalization to novel spatio-temporal contexts.",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "5.6. Case Study (RQ5)",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "To assess the effectiveness of our proposed uniform distribution mapping approach in transforming various data representations into uniform distributions, we conducted visualizations of prompt embeddings with and without the application of our distribution mapping mechanism. These visualizations were performed using two variants of the downstream model MTGNN and evaluated on the test sets of the PEMS07(M) and CA-D5 datasets. Given that prompt embeddings are high-dimensional vectors, we initially applied the PCA technique to reduce the dimensionality of each embedding sample to 2-D (Wold et al., 1987). Subsequently, we projected the reduced embeddings onto the unit circle using the L2 norm, as depicted in Figure 6. The visualization results provide compelling evidence that our uniform distribution mapping strategy effectively transforms the prompt embeddings into an approximate uniform distri",
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+    {
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+    {
+        "type": "image",
+        "img_path": "images/3f905c64a01da43cc54d69c9805e2274f1e917de41df11a71a0164e0b38bf2ee.jpg",
+        "image_caption": [
+            "Figure 5. Hyperparameter study of  $\\tau$  and  $\\lambda$ .",
+            "Figure 6. Distribution of the prompt embedding."
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "image",
+        "img_path": "images/35f7f864ef170ab82fa1e41ae7cd1f8cfcd04dcaafff6f39019e8878caba13c7.jpg",
+        "image_caption": [],
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+    {
+        "type": "text",
+        "text": "bution. In contrast, the variants that lack this strategy fail to achieve such desirable distribution properties. By converting data from new spatio-temporal contexts into a consistent distribution, our FlashST gains the ability to utilize pre-trained knowledge and rapidly adapt to new datasets, facilitating its performance on various spatio-temporal tasks.",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "6. Conclusion",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "This paper presents FlashST, a new and efficient method for adapting spatio-temporal predictive models to various downstream tasks that involve previously unseen data. Our in-context learning framework utilizes a spatio-temporal prompt network, which consists of a context distillation mechanism and a dependency modeling scheme. This framework effectively adapts to different spatio-temporal scenarios by capturing contextual signals and modeling complex relationships across time and locations. To address the distribution gap, we enhance FlashST by incorporating a distribution mapping mechanism that aligns the data distributions of pre-training and downstream data, facilitating effective knowledge transfer in spatio-temporal forecasting. Extensive experiments demonstrate the effectiveness and generalization capabilities of our FlashST in diverse downstream spatio-temporal forecasting scenarios. For future work, an interesting direction would be to explore the potential of incorporating large language models as knowledge guidance within our FlashST framework.",
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+    {
+        "type": "text",
+        "text": "Impact Statement",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "This paper presents work whose goal is to advance the field of Machine Learning. There are many potential societal consequences of our work, none which we feel must be specifically highlighted here.",
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+    {
+        "type": "text",
+        "text": "Acknowledgments",
+        "text_level": 1,
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+        "type": "text",
+        "text": "This work is partially supported by the Guangzhou Municipal Science and Technology Plan Project - Key Research and Development Program (Grant No. 2024B01W0007).",
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+    {
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+]

2024/FlashST_ A Simple and Universal Prompt-Tuning Framework for Traffic Prediction/1591a92e-d519-49e7-ace8-85352ae1ed50_origin.pdf

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2024/FlashST_ A Simple and Universal Prompt-Tuning Framework for Traffic Prediction/full.md

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+# FlashST: A Simple and Universal Prompt-Tuning Framework for Traffic Prediction
+
+Zhonghang Li  $^{1,2}$  Lianghao Xia  $^{2}$  Yong Xu  $^{1,3}$  Chao Huang*
+
+# Abstract
+
+The objective of traffic prediction is to accurately forecast and analyze the dynamics of transportation patterns, considering both space and time. However, the presence of distribution shift poses a significant challenge in this field, as existing models struggle to generalize well when faced with test data that significantly differs from the training distribution. To tackle this issue, this paper introduces a simple and universal spatio-temporal prompt-tuning framework-FlashST, which adapts pre-trained models to the specific characteristics of diverse downstream datasets, improving generalization in diverse traffic prediction scenarios. Specifically, the FlashST framework employs a lightweight spatio-temporal prompt network for in-context learning, capturing spatio-temporal invariant knowledge and facilitating effective adaptation to diverse scenarios. Additionally, we incorporate a distribution mapping mechanism to align the data distributions of pre-training and downstream data, facilitating effective knowledge transfer in spatio-temporal forecasting. Empirical evaluations demonstrate the effectiveness of our FlashST across different spatio-temporal prediction tasks using diverse urban datasets. Code is available at https://github.com/HKUDS/FlashST.
+
+# 1. Introduction
+
+Accurate traffic forecasting is a crucial objective in the domain of urban computing. It aims to precisely predict and analyze the dynamic transportation patterns within cities. The key goals of traffic prediction include supporting urban planning, enabling real-time monitoring and management of traffic, and contributing to the development of smart city
+
+applications (Guo et al., 2019; Han et al., 2021; Li et al., 2024). The primary challenge lies in effectively modeling the intricate spatial and temporal correlations among different time intervals and geographical locations.
+
+Neural network techniques have become a prominent approach for spatio-temporal prediction tasks, thanks to their remarkable capabilities in feature representation. Initially, researchers leveraged Recurrent Neural Networks (RNNs) (Yu et al., 2017) to capture temporal dependencies, and Convolutional Neural Networks (CNNs) (Zhang et al., 2017; Yao et al., 2018) to model spatial correlations. Subsequently, Graph Neural Networks (GNNs) have emerged as effective models for capturing complex spatial dependencies, utilizing message passing mechanisms to learn the relationships among spatial units such as regions and road segments. In constructing the graph adjacency matrices, researchers have explored various factors. Some have considered static geographical distance (Li et al., 2018; Yu et al., 2018), while others have incorporated time-aware region correlations (Han et al., 2021; Li et al., 2022). Additionally, there have been efforts to learn the region-wise relevance in a data-driven manner (Wu et al., 2019; 2020a).
+
+Although the aforementioned methods have shown effectiveness, most current spatio-temporal prediction models struggle to generalize effectively when confronted with distribution shifts across diverse downstream datasets and tasks. In their methods, the assumption of an inconsistent distribution between training and testing data becomes a hindrance to accurate predictions in real-life urban scenarios with uncontrolled shifts from unseen spatio-temporal data. As illustrated in Figure 1, directly applying parameters learned from training on dataset A to test on dataset B can lead to suboptimal performance due to significant variations in spatio-temporal characteristics across different data distributions. Therefore, there is a need to enhance the generalization ability of spatio-temporal forecasting models by efficiently adapting them to handle such distribution shifts.
+
+While there are potential benefits to enabling model adaptation for spatio-temporal prediction methods, several key challenges remain to be addressed. Firstly, C1: efficiently distilling specific and complex spatio-temporal contextual information from the downstream tasks is essential. How-
+
+![](images/db884f595d7f7fcd4b850a8936bc81034dcf9ab370fe1f36937aa5473fc80490.jpg)  
+Figure 1. Motivations behind FlashST. The left figure illustrates the diverse data distributions across various ST datasets, while the right figure demonstrates that the end-to-end model's parameters are overfit to training set A and fail to generalize to test set B.
+
+ever, equipping pre-trained models with the ability to understand and incorporate the spatial and temporal characteristics of new domain data that is only accessible during testing is a formidable challenge. Secondly, C2: there is often a significant distribution gap between the training and test datasets, especially when they are collected from different spatio-temporal scenarios and domains. Consequently, it is essential to enhance the model adaptation framework by enabling it to effectively bridge the distribution gap and capture spatio-temporal invariants. This enhancement will facilitate the successful knowledge transfer from the pretrained phase to the downstream prediction tasks.
+
+Contributions. To overcome the challenges mentioned above, we introduce a lightweight and innovative prompting framework (FlashST) that aims to achieve spatiotemporal in-context learning, enabling efficient and effective model adaptation across different spatio-temporal prediction tasks. Within our framework, we employ a spatio-temporal prompt network to enable in-context learning. To tackle challenge C1, we propose a context distillation mechanism that captures contextual signals from unseen data, facilitating adaptation to diverse spatio-temporal scenarios. Furthermore, a dependency modeling scheme is introduced to capture and understand the relationships between time and locations, enabling effective analysis of inter-dependencies among spatio-temporal elements. To overcome challenge C2, we enhance our FlashST framework by incorporating a unified distribution mapping mechanism, which bridges the distribution gap between pre-trained and downstream tasks. This mechanism facilitates the effective transfer of knowledge from the pre-trained insights to downstream spatio-temporal forecasting by aligning the data distributions through the regularization of prompt embeddings.
+
+We conduct extensive experiments on four distinct types of spatio-temporal data tasks to evaluate the effectiveness of our proposed framework. The results demonstrate that our framework substantially improves the generalization capabilities in downstream prediction tasks across different spatio-temporal datasets. The noteworthy enhancement in predictive performance, achieved with model efficiency, underscores the efficacy of our FlashST framework.
+
+# 2. Related Work
+
+Deep Spatio-Temporal Learning is a thriving research field that aims to model and understand the intricate spatiotemporal dynamics in real-world urban data. Various approaches have been proposed, including integrating recurrent neural networks (RNNs) (Yao et al., 2018; Zhao et al., 2020) or Transformers (Wang et al., 2020; Xu et al., 2020) to capture long-term patterns and short-term fluctuations. Attention mechanisms have been widely used to dynamically weigh spatial and temporal features (Zheng et al., 2020; Zhang et al., 2021). MLPs are also utilized as temporal encoders to capture temporal correlations (Shao et al., 2022a). Another important direction involves leveraging Graph Neural Networks (GNNs) (Yu et al., 2018; Li et al., 2018; Yu et al., 2018; Ye et al., 2021) to simultaneously capture spatial dependencies and temporal evolutions. These models employ graph structures to represent relationships between spatial entities, facilitating information propagation. However, most existing deep spatio-temporal models adopt an end-to-end approach, limiting their generalization across diverse urban forecasting tasks and domains.
+
+Pre-Training with Spatio-Temporal Data. There has been a recent surge of interest in using pre-training for acquiring comprehensive representations from spatio-temporal data through self-supervised learning. This approach aims to learn generalizable representations that capture the underlying patterns and dynamics. Two dominant research lines have emerged: i) integrating contrastive learning into the pre-training stage for spatio-temporal data (Zhang et al., 2023a), and ii) utilizing masked autoencoders (Shao et al., 2022b; Li et al., 2023) to reconstruct the masked input data to learn the spatio-temporal relationships.
+
+Prompt-Tuning is a technique that optimizes prompts or instructions during inference to fine-tune the model, resulting in more accurate and context-specific predictions (Shrivastava et al., 2023; Liu et al., 2023a). It has gained recognition as a promising approach to improve the performance and adaptability of pre-trained models in various domains. For example, in prompt-based language generation, it allows precise control over the generated output, aligning it with specific requirements or desired styles (Brown et al., 2020; Schick & Schütze, 2020; Zhang et al., 2023b). Similarly, in vision prompt learning, specific instructions guide the model's attention towards relevant visual features or concepts (Zhou et al., 2022b;a; Sohn et al., 2023). This study explores spatio-temporal prompt-tuning to enhance prediction models in urban computing.
+
+# 3. Preliminaries
+
+Representation of Spatial-Temporal Data: The spatial-temporal information is captured and encoded using a three-
+
+way tensor  $\mathbf{X}$ , where  $\mathbf{X} \in \mathbb{R}^{R \times T \times F}$ . In this representation,  $R$  represents the number of regions,  $T$  denotes the time slots, and  $F$  indicates the number of features. Each entry  $\mathbf{X}_{r,t,f}$  of the tensor corresponds to the  $f$ -th feature of the  $r$ -th region at the  $t$ -th time slot. For example, in the context of traffic flow prediction, the tensor  $\mathbf{X}$  captures traffic volume data, which quantifies the number of vehicles passing through a specific region within a fixed time interval (e.g., 5 minutes).
+
+Spatio-Temporal Prediction. In the context of spatiotemporal learning, a common scenario revolves around predicting future urban spatio-temporal conditions using historical records. This scenario can be described as follows:
+
+$$
+\mathbf {X} _ {t _ {K + 1}: t _ {K + P}} ^ {A} = g \left(\mathbf {X} _ {t _ {K - H + 1}: t _ {K}} ^ {A}\right) \tag {1}
+$$
+
+In this scenario,  $\mathbf{X}^A$  represents the spatio-temporal data from dataset A. The function  $g(\cdot)$  corresponds to the spatiotemporal predictive function, which could be methods like spatio-temporal transformers or spatio-temporal graph neural networks (GNNs). These predictive methods generate forecasts for the next  $P$  time slots by leveraging the historical spatio-temporal data from the previous  $H$  time slots.
+
+$$
+\mathbf {X} _ {t _ {K + 1}: t _ {K + P}} ^ {B} = \tilde {g} \left(f \left(\mathbf {X} _ {t _ {K - H + 1}: t _ {K}} ^ {B}\right)\right) \tag {2}
+$$
+
+Furthermore, we introduce  $\tilde{g} (\cdot)$  as the spatio-temporal prediction model with fixed parameters, and  $\mathbf{X}^B$  represents an unseen spatio-temporal dataset. The goal of this work is to enable swift adaptation of the  $\tilde{g} (\cdot)$  model to a new dataset while ensuring parameter efficiency. To achieve this, we propose incorporating a lightweight and effective prompt-tuning paradigm into the pre-trained  $\tilde{g} (\cdot)$  model.
+
+# 4. Methodology
+
+This section provides an in-depth explanation of the technical details of the proposed FlashST framework. We commence by outlining the foundational paradigm and design principles that guide the development of the FlashST framework. Subsequently, we present a comprehensive overview of the framework's main components: the Spatio-Temporal In-Context Learning and the Unified Distribution Mapping Mechanism. To provide a visual representation of the overall model architecture, please refer to Figure 2.
+
+# 4.1. Spatio-Temporal In-Context Learning
+
+Our in-context learning framework is implemented through a spatio-temporal prompt network, which comprises two primary components: (i) Spatio-temporal context distillation mechanism efficiently captures contextual signals that are aware of both time and location from the unseen data. By doing so, it enables the model to learn from the specific context of the data, facilitating effective adaptation to various spatio-temporal scenarios. (ii) Spatio-temporal dependency
+
+modeling scheme incorporates complex relationships across time and locations into the in-context spatio-temporal network. By capturing and modeling these dependencies, the network can effectively understand the interdependencies and interactions between different spatio-temporal elements.
+
+# 4.1.1. Spatio-Temporal Context Distillation
+
+Spatio-Temporal Data Projection. To initialize the representation of the spatio-temporal data, we employ a projection layer with two steps: normalization using the Z-Score function and augmentation through linear transformation:
+
+$$
+\mathbf {E} _ {r, f} = \bar {\mathbf {X}} _ {r, f} \mathbf {e} _ {0}; \quad \bar {\mathbf {X}} _ {r, t, f} = \mathbf {Z} \left(\mathbf {X} _ {r, t, f}\right) = \frac {\mathbf {X} _ {r , t , f} - \mu}{\sigma} \tag {3}
+$$
+
+$\mathbf{E}_{r,f} \in \mathbb{R}^d$  represents the representation for the  $r$ -th region of the  $f$ -th feature, where  $d$  denotes the number of hidden units.  $\bar{\mathbf{X}}_{r,t,f} \in \mathbb{R}$  corresponds to the normalized spatiotemporal data feature obtained through the Z-Score  $(Z(\cdot))$  function. The parameters  $\mu$  and  $\sigma$  represent the mean and standard deviation of the raw ST matrix  $\mathbf{X}$ . The parametric vector  $\mathbf{e}_0 \in \mathbb{R}^{T \times d}$  denotes the unit embedding vector for the normalized spatio-temporal feature.
+
+Temporal Context Incorporation. In order to capture dynamic and periodic spatio-temporal patterns from diverse urban data, we introduce the time-aware context into our prompt network. This context is based on multi-resolution temporal features, specifically the hour of the day  $(\mathbf{z}^{(d)} \in \mathbb{R}^T)$  and the day of the week  $(\mathbf{z}^{(w)} \in \mathbb{R}^T)$ . To effectively extract the temporal contextual signals, we employ a process of transformation and concatenation (denoted as  $\mathrm{CAT}[,]$ ), which can be summarized as follows:
+
+$$
+\mathbf {M} _ {t} = \operatorname {C A T} \left[ \mathbf {z} _ {t} ^ {(d)} \cdot \mathbf {e} _ {1}, \mathbf {z} _ {t} ^ {(w)} \cdot \mathbf {e} _ {2} \right] \tag {4}
+$$
+
+Spatial Context Incorporation. To enhance the prompt network with geographical contextual information related to regional properties, we incorporate the citywide road network structure as an encoding feature that reflects spatial context. This process starts by formulating the normalized Laplacian matrix, which is defined as follows:
+
+$$
+\triangle = \mathbf {I} - \mathbf {D} ^ {- 1 / 2} \mathbf {A D} ^ {- 1 / 2} \tag {5}
+$$
+
+Here,  $\mathbf{I}$ ,  $\mathbf{D}$ , and  $\mathbf{A}$  denote the identity matrix, the degree matrix, and the adjacency matrix, respectively. The adjacency matrix is computed by considering the distances between areas and the road structure. Since Laplacian eigenvectors effectively preserve global graph structure information in Euclidean space, we perform eigenvalue decomposition to obtain  $\triangle = \mathbf{U} \boldsymbol{\Lambda} \mathbf{U}^{\top}$ . After extracting the eigenvalue matrix  $\boldsymbol{\Lambda}$  and the corresponding eigenvector matrix  $\mathbf{U}$ , we derive the structure-aware node properties  $\mathbf{C} \in \mathbb{R}^{R \times d_r}$  by projecting  $\mathbf{U}$  to obtain the  $d_r$  smallest nontrivial eigenvectors.
+
+![](images/effd40696667807a4483bb063bad61144fabe5d2457046964e71dbf7b064d2c9.jpg)  
+Figure 2. Our proposed FlashST framework adopts an architecture that integrates spatio-temporal in-context learning and a unified distribution mapping mechanism, offering an efficient and effective approach for spatio-temporal prompt-tuning across diverse scenarios.
+
+Acknowledging the potential disparity of  $\mathbf{C}$  in the feature space between training and test datasets, we employ a MLP to map these features and enhance the network's ability to generalize spatial context. Subsequently, we use the concatenation operation to integrate the aforementioned embeddings and obtain the initial spatio-temporal embedding:
+
+$$
+\bar {\mathbf {E}} = \operatorname {C A T} [ \mathbf {E}, \mathbf {M}, \bar {\mathbf {C}} ]; \quad \bar {\mathbf {C}} _ {r, f} = \operatorname {M L P} \left(\mathbf {C} _ {r, f}\right) \tag {6}
+$$
+
+$\bar{\mathbf{E}}$  denotes the integrated embeddings, where we concatenate the original representation  $\mathbf{E}$ , the temporal context embeddings  $\mathbf{M}$ , and the transformed regional properties  $\bar{\mathbf{C}}$ . The transformation of  $\mathbf{C}$  is achieved through the MLP function.
+
+# 4.1.2. Spatio-Temporal Dependency Modeling
+
+Temporal Dependency Encoder. To capture dependencies across different time slots and preserve time-evolving data patterns, we introduce a lightweight gating mechanism. The formal operations of this mechanism are presented below:
+
+$$
+\mathbf {H} _ {r, f} = \left(W _ {2} \sigma \left(W _ {1} \overline {{\mathbf {E}}} _ {r, f} + b _ {1}\right) + b _ {2}\right) + \overline {{\mathbf {E}}} _ {r, f} \tag {7}
+$$
+
+The trainable parameters of this gating mechanism are denoted by  $W_{1}, W_{2} \in \mathbb{R}^{d' \times d'}$  and  $b_{1}, b_{2} \in \mathbb{R}^{d'}$ . The initial spatio-temporal embedding is represented by  $\bar{\mathbf{E}}_{r,f}$ , while the encoded temporal embedding is denoted by  $\mathbf{H}_{r,f} \in \mathbb{R}^{d' \times d'}$ . The embedding  $\bar{\mathbf{E}}_{r,f}$  contains valuable contextual information about temporal dynamics and regional characteristics, which is essential for the temporal dependency encoder. This enriched information enables our in-context learning to effectively identify variations in spatio-temporal patterns across different regions and time intervals, thereby facilitating precise modeling of temporal correlations.
+
+Spatial Dependency Encoder. Drawing inspiration from the effectiveness of graph neural networks in capturing spatial correlations between geographical locations (Wu et al., 2020b; Zheng et al., 2020), we leverage graph convolution-based message passing to encode inter-regional correlations. The adjacency matrix  $\mathbf{A}$ , as defined in Eq 5, acts as the connectivity matrix within the graph network framework. The process of spatial encoding is formalized as follows:
+
+$$
+\mathbf {S} _ {f} = \sigma (\mathbf {A} \cdot \mathbf {H} _ {f} \cdot W _ {3}) + \mathbf {H} _ {f} \tag {8}
+$$
+
+Here,  $W_{3} \in \mathbb{R}^{d' \times d'}$  represents the trainable parameters. To mitigate the potential over-smoothing phenomenon caused by multi-layer GNNs, we employ residual networks. By stacking multiple layers of spatio-temporal encoders, our model generates a representation  $\mathbf{E}_{pro}$  that captures rich spatial and temporal dependencies across time and space.
+
+# 4.2. Unified Distribution Mapping Mechanism
+
+To bridge the distribution gap between the pre-training and diverse unseen data in downstream tasks, we enhance FlashST by incorporating a distribution mapping mechanism. The objective of this mechanism is to transform both the pre-training data and the downstream data into a shared distribution space. This alignment of data distributions enables a seamless transfer of knowledge, ensuring that the insights gained from the pre-training stage can be effectively applied to the downstream spatio-temporal context.
+
+To achieve this goal, our FlashST involves standardizing the prompt embedding to ensure a consistent distribution across diverse downstream datasets. We draw inspiration from various works in contrastive learning (Wang & Isola, 2020; Chuang et al., 2020; Wang & Liu, 2021) and incorporate the infoNCE-based loss function to regularize the prompt network for representation generation. This loss function is designed to bring the representations of positive sample pairs closer together while pushing apart those of negative pairs. By leveraging self-supervised learning without the need for additional labeling, optimizing the infoNCE loss helps attain a more uniform embedding distribution. Empirical evidence suggests that through this loss alone, an almost entirely uniform distribution can be achieved (Yu et al., 2022). Building upon this, we employ the infoNCE loss to adjust the distribution of the learned spatio-temporal prompt embedding  $\mathbf{E}_{pro} \in \mathbb{R}^{R \times F \times d'}$ , formalized as:
+
+$$
+\mathcal {L} _ {U n i} = \sum_ {r = 1} ^ {R} \sum_ {f = 1} ^ {F} - \log \sum_ {r ^ {\prime}} \exp \left(\cos \left(\mathbf {E} _ {p r o} ^ {r, f}, \mathbf {E} _ {p r o} ^ {r ^ {\prime}, f}\right) / \tau\right) \tag {9}
+$$
+
+The cosine similarity function  $\cos (\cdot)$  is used to measure the similarity between embeddings, and the temperature coefficient  $\tau$  is used to adjust the softmax scale. In our FlashST, we enhance the uniformity of prompt embeddings by increasing the separation among embeddings that correspond
+
+to different regions. This improvement allows downstream models to effectively utilize the provided prompts  $\mathbf{E}_{pro}^{(U)}$  for rapid generalization across new data and tasks.
+
+# 4.3. Unifying Pre-training and Downstream Tasks
+
+Our spatio-temporal pretrain-prompt framework integrates the model pre-training process with downstream forecasting tasks through efficient prompt-tuning. In the pre-training phase, we train and optimize all parameters using dedicated pre-training datasets. Subsequently, in the prompt fine-tuning phase, we exclusively update the parameters of the prompt network by conducting a limited number of training epochs on unseen datasets. This approach enables the downstream model to effectively adapt to new data. Moreover, our framework is model-agnostic, allowing seamless integration with various existing spatio-temporal prediction baselines as the downstream model.
+
+i) Pre-training Phase: Our goal is to forecast future trends using historical spatio-temporal records from the pre-training data. We achieve this with following process:
+
+$$
+\hat {\mathbf {Y}} ^ {A} = g \left(\mathbf {E} _ {p r o} ^ {(U)}\right); \quad \mathbf {E} _ {p r o} ^ {(U)} = f _ {P r o m p t} \left(\mathbf {X} ^ {A}\right) \tag {10}
+$$
+
+Here, we have adjustable parameters for both  $f_{Prompt}(\cdot)$  and  $g(\cdot)$ . To update these parameters, we utilize the loss function  $\mathcal{L}$  in conjunction with the Adam optimizer. The loss function  $\mathcal{L}$  is a combination of the regression loss  $\mathcal{L}_r$  and the uniformity loss  $\mathcal{L}_{Uni}$ , with  $\lambda$  as the balance coefficient between the two losses. The label is represented by  $Y \in \mathbb{R}^{R \times T \times F}$ . The regression loss  $\mathcal{L}_r$  is defined as:
+
+$$
+\mathcal {L} _ {r} = \frac {1}{R T F} \sum_ {r = 1} ^ {R} \sum_ {t = 1} ^ {T} \sum_ {f = 1} ^ {F} | \left(\mathbf {Y} _ {r, t, f} - \hat {\mathbf {Y}} _ {r, t, f}\right) | \tag {11}
+$$
+
+This formulation calculates the average absolute difference between the predicted values  $\hat{\mathbf{Y}}_{r,t,f}$  and the actual labels  $\mathbf{Y}_{r,t,f}$  across all regions  $R$ , time steps  $T$ , and features  $F$ .
+
+ii) Prompt-Tuning Phase: We freeze the parameters of the downstream model and focus exclusively on fine-tuning the prompt network. This fine-tuning process is carried out on the test dataset to perform spatio-temporal prediction tasks, following the formulation described in Eq 10:
+
+$$
+\hat {\mathbf {Y}} ^ {B} = \tilde {g} \left(f _ {\text {P r o m p t}} \left(\mathbf {X} ^ {B}\right)\right) \tag {12}
+$$
+
+Here,  $\tilde{g}$  represents the downstream model with frozen parameters, and  $B$  indicates the test datasets. During the prompt-tuning phase, we conduct a limited number of optimization epochs to generate high-quality spatio-temporal prompt representations for the downstream models.
+
+# 5. Evaluation
+
+In this section, we analyze the performance of our model by addressing the following several research questions:
+
+Table 1. Statistics of pre-training dataset.  
+
+<table><tr><td>Dataset</td><td>Data Record</td><td># Region</td><td>Time Steps</td><td>Sample Date</td></tr><tr><td>PEMS03</td><td>traffic flow</td><td>358</td><td>26208</td><td>1/Sep/2018 - 30/Nov/2018</td></tr><tr><td>PEMS04</td><td>traffic flow</td><td>307</td><td>16992</td><td>1/Jan/2018 - 28/Feb/2018</td></tr><tr><td>PEMS07</td><td>traffic flow</td><td>883</td><td>26208</td><td>1/May/2017 - 30/Aug/2017</td></tr><tr><td>PEMS08</td><td>traffic flow</td><td>170</td><td>17856</td><td>1/Jul/2016 - 31/Aug/2016</td></tr></table>
+
+Table 2. Statistics of downstream tasks dataset.  
+
+<table><tr><td>Dataset</td><td>Data Record</td><td># Region</td><td>Time Steps</td><td>Sample Date</td></tr><tr><td>PEMS07(M)</td><td>traffic speed</td><td>170</td><td>12672</td><td>1/Jul/2016 - 31/Aug/2016</td></tr><tr><td>CA-D5</td><td>traffic flow</td><td>211</td><td>16992</td><td>1/Jan/2017 - 28/Feb/2017</td></tr><tr><td>ChengDu-DIDI</td><td>traffic index</td><td>524</td><td>17280</td><td>1/Jan/2018 - 30/Apr/2018</td></tr><tr><td>NYC Citi Bike</td><td>bike orders</td><td>250</td><td>4368</td><td>1/Apr/2016 - 30/Jun/2016</td></tr></table>
+
+- RQ1: Does FlashST effectively generalize pre-trained models to new spatio-temporal prediction data and tasks?  
+- RQ2: How does the efficiency of our model compare to that of end-to-end training and fine-tuning approaches?  
+- RQ3: What is the impact of the key components of our FlashST on the performance of downstream models?  
+- RQ4: How do parameters affect the model performance?
+
+# 5.1. Experimental Setting
+
+Datasets. To evaluate how well the model can generalize across different urban spatio-temporal contexts, our experimental setup is designed as follows: (i) In the pre-training phase, we use four datasets (PEMS03, PEMS04, PEMS07, and PEMS08 (Song et al., 2020)) as our training sets. These datasets consist of records detailing traffic flow conditions across various streets and cities in California, USA. (ii) During the subsequent prompt-tuning phase, we concentrate on four distinct target datasets to fine-tune and evaluate our framework: PEMS07(M) (Yu et al., 2018), CA-D5 (Liu et al., 2023b), ChengDu-DIDI (Lu et al., 2022), and NYC Citi Bike (Ye et al., 2021). These datasets correspondingly represent traffic speeds in Los Angeles, traffic flow in California, traffic flow in Chengdu, and bicycle demand in New York City. Each target dataset is partitioned into training, validation, and test sets in a ratio of 6:2:2. Detailed statistics for each dataset can be found in Table 1 and Table 2.
+
+Experimental Description. We configure the model with 32 hidden units for  $d$ ,  $d_t$ , and  $d_r$ , while employing a spatial and temporal encoder with 2 layers. The pre-training phase involves alternating training for 300 epochs on the four pre-training datasets. Subsequently, we conduct fine-tuning for 20 epochs during the prompt-tuning phase. In both the baseline and full fine-tuning setups, the maximum number of training epochs is limited to 100. Furthermore, a batch size of 64 is employed for both phases and all baselines. For the dataset-specific parameters in the baselines (e.g., node embeddings), we initialize them randomly and make them trainable during the prompt fine-tuning phase. Additionally, the final regression layers of some baseline models are set to be trainable to ensure accurate numerical predictions.
+
+Evaluation Metrics. In order to assess the performance of the model in spatiotemporal prediction, we employ three widely-used evaluation metrics: Mean Absolute Error (MAE), Root Mean Squared Error (RMSE), and Mean
+
+Table 3. Overall performance on PEMS07(M), CA-D5, ChengDu-DIDI and NYC Citi Bike datasets in terms of MAE, RMSE and MAPE.  
+
+<table><tr><td rowspan="2">Model</td><td rowspan="2">Dataset Metrics</td><td colspan="3">PEMS07(M)</td><td colspan="3">CA-D5</td><td colspan="3">ChengDu-DIDI</td><td colspan="3">NYC Citi Bike</td></tr><tr><td>MAE</td><td>RMSE</td><td>MAPE</td><td>MAE</td><td>RMSE</td><td>MAPE</td><td>MAE</td><td>RMSE</td><td>MAPE</td><td>MAE</td><td>RMSE</td><td>MAPE</td></tr><tr><td>TGCN</td><td>5.05</td><td>8.56</td><td>13.68%</td><td>15.36</td><td>24.52</td><td>24.42%</td><td>3.18</td><td>4.72</td><td>14.48%</td><td>2.19</td><td>3.54</td><td>63.31%</td><td></td></tr><tr><td>STGCN</td><td>3.14</td><td>6.25</td><td>8.05%</td><td>15.02</td><td>23.73</td><td>27.84%</td><td>2.59</td><td>3.97</td><td>12.20%</td><td>2.04</td><td>3.21</td><td>52.06%</td><td></td></tr><tr><td>ASTGCN</td><td>3.13</td><td>6.19</td><td>8.09%</td><td>15.26</td><td>24.37</td><td>24.45%</td><td>2.56</td><td>3.83</td><td>11.58%</td><td>2.03</td><td>3.10</td><td>55.13%</td><td></td></tr><tr><td>GWN</td><td>2.67</td><td>5.40</td><td>6.62%</td><td>13.63</td><td>22.30</td><td>24.68%</td><td>2.35</td><td>3.56</td><td>10.72%</td><td>1.81</td><td>2.71</td><td>50.05%</td><td></td></tr><tr><td>STSGCN</td><td>2.93</td><td>5.87</td><td>7.52%</td><td>15.30</td><td>24.44</td><td>24.70%</td><td>2.86</td><td>4.29</td><td>13.02%</td><td>1.89</td><td>2.78</td><td>52.48%</td><td></td></tr><tr><td>AGCRN</td><td>2.89</td><td>5.69</td><td>7.32%</td><td>14.44</td><td>23.10</td><td>23.02%</td><td>2.87</td><td>4.27</td><td>13.05%</td><td>2.16</td><td>3.41</td><td>60.95%</td><td></td></tr><tr><td>MTGNN</td><td>2.70</td><td>5.50</td><td>6.81%</td><td>13.90</td><td>22.44</td><td>23.86%</td><td>2.33</td><td>3.56</td><td>10.63%</td><td>1.84</td><td>2.80</td><td>48.69%</td><td></td></tr><tr><td>STFGNN</td><td>3.14</td><td>6.21</td><td>7.81%</td><td>15.38</td><td>24.37</td><td>25.78%</td><td>2.83</td><td>4.24</td><td>13.05%</td><td>2.36</td><td>3.73</td><td>61.93%</td><td></td></tr><tr><td>STGODE</td><td>2.81</td><td>5.57</td><td>7.09%</td><td>14.13</td><td>22.77</td><td>23.41%</td><td>2.51</td><td>3.75</td><td>11.48%</td><td>1.94</td><td>2.83</td><td>56.25%</td><td></td></tr><tr><td>DMSTGCN</td><td>2.94</td><td>5.92</td><td>7.45%</td><td>13.88</td><td>22.81</td><td>21.97%</td><td>2.67</td><td>4.00</td><td>12.34%</td><td>2.09</td><td>3.32</td><td>60.00%</td><td></td></tr><tr><td>MSDR</td><td>2.79</td><td>5.53</td><td>7.07%</td><td>15.31</td><td>23.71</td><td>33.01%</td><td>2.56</td><td>3.85</td><td>11.94%</td><td>1.87</td><td>2.86</td><td>51.25%</td><td></td></tr><tr><td>STWA</td><td>2.74</td><td>5.52</td><td>6.97%</td><td>14.00</td><td>22.75</td><td>23.22%</td><td>2.44</td><td>3.68</td><td>11.33%</td><td>1.88</td><td>2.83</td><td>49.93%</td><td></td></tr><tr><td>PDFormer</td><td>2.68</td><td>5.46</td><td>6.66%</td><td>13.85</td><td>22.59</td><td>23.11%</td><td>2.43</td><td>3.70</td><td>11.17%</td><td>1.89</td><td>2.88</td><td>51.98%</td><td></td></tr><tr><td>Ours</td><td>2.59</td><td>5.25</td><td>6.54%</td><td>13.26</td><td>21.83</td><td>21.49%</td><td>2.31</td><td>3.52</td><td>10.51%</td><td>1.79</td><td>2.63</td><td>49.77%</td><td></td></tr></table>
+
+Absolute Percentage Error (MAPE). These metrics quantify the disparities between the predicted data and the ground truth data. It is important to note that lower values of these metrics indicate better performance.
+
+Baseline Description. To evaluate the effectiveness of our FlashST, we selected 13 advanced spatio-temporal prediction models as baselines. These include RNNs-based models, attention-based models, GNNs-based models, and differential equation-based spatio-temporal prediction methods.
+
+# RNN-based Spatio-Temporal Prediction Approaches:
+
+- AGCRN (Bai et al., 2020): This method incorporates RNNs with learnable node embeddings to capture personalized spatio-temporal patterns of regions.  
+- MSDR (Liu et al., 2022): This model addresses the issue of long-term information forgetting in RNNs by introducing a variant of RNNs. It further combines this variant with GNNs to capture spatio-temporal correlations.
+
+# GNN-based Spatio-Temporal Prediction Models:
+
+- TGCN (Zhao et al., 2020): This model leverages a combination of RNNs and GCNs to separately model temporal dependencies and spatial correlations.  
+- STGCN (Yu et al., 2018): This method utilizes gated convolutional networks to encode temporal dependencies and GCNs to capture local spatial relationships.  
+- GWN (Wu et al., 2019): This approach incorporates a learnable graph structure to model spatial correlations, while TCNs are used to extract temporal features.  
+- STSGCN (Song et al., 2020): It synchronously models dynamic spatio-temporal correlations by constructing a graph network that allows interactions across time slices.
+
+- MTGNN (Wu et al., 2020b): This method captures temporal dependencies using time convolutional networks combined with skip connections, and employs learnable graph networks to model spatial correlations.  
+- STFGNN (Li & Zhu, 2021): This method proposes a data-driven spatio-temporal graph construction approach to capture dynamic spatio-temporal correlations.  
+- DMSTGCN (Han et al., 2021): This method models dynamic spatial correlations by constructing a learnable dynamic graph network.
+
+# Attention-based Spatio-Temporal Models:
+
+- ASTGCN (Guo et al., 2019): This method leverages attention mechanisms to effectively capture the periodic spatio-temporal correlations inherent in the data.  
+- STWA(Cirstea et al., 2022): This approach goes beyond standard attention mechanisms by incorporating specialized node features and time-dynamic parameters.  
+- **PDFformer** (Jiang et al., 2023): This model leverages the power of the transformer architecture to effectively encode the intricate spatio-temporal dependencies present in the data. Furthermore, it introduces a novel time-delayed spatial correlation modeling technique.
+
+# ODE-based Spatio-Temporal Prediction Methods:
+
+- STGODE (Fang et al., 2021): This method enriches graph networks with ordinary differential equations (ODEs) to extract continuous spatial correlation information. Time convolutional networks are employed to learn time-dependent relationships within the data.
+
+Table 4. Model-agnostic experiments on PEMs07(M), CA-D5, ChengDu-DIDI and NYC Citi Bike datasets.  
+
+<table><tr><td rowspan="2">Model</td><td>Dataset</td><td colspan="3">PEMS07(M)</td><td colspan="3">CA-D5</td><td colspan="3">ChengDu-DIDI</td><td colspan="3">NYC Citi Bike</td></tr><tr><td>Metrics</td><td>MAE</td><td>RMSE</td><td>MAPE</td><td>MAE</td><td>RMSE</td><td>MAPE</td><td>MAE</td><td>RMSE</td><td>MAPE</td><td>MAE</td><td>RMSE</td><td>MAPE</td></tr><tr><td>STGCN</td><td>3.14</td><td>6.25</td><td>8.05%</td><td>15.02</td><td>23.73</td><td>27.84%</td><td>2.59</td><td>3.97</td><td>12.20%</td><td>2.04</td><td>3.21</td><td>52.06%</td><td></td></tr><tr><td>w/o Finetune</td><td>8.07</td><td>11.77</td><td>23.02%</td><td>31.20</td><td>41.82</td><td>68.75%</td><td>4.82</td><td>6.53</td><td>22.87%</td><td>4.10</td><td>5.60</td><td>148.20%</td><td></td></tr><tr><td>w/ Finetune</td><td>3.18</td><td>6.25</td><td>8.04%</td><td>15.15</td><td>24.18</td><td>23.30%</td><td>2.58</td><td>3.93</td><td>12.01%</td><td>1.99</td><td>3.11</td><td>50.38%</td><td></td></tr><tr><td>Ours</td><td>2.68</td><td>5.37</td><td>6.80%</td><td>13.47</td><td>22.01</td><td>22.62%</td><td>2.37</td><td>3.60</td><td>10.92%</td><td>1.80</td><td>2.67</td><td>50.06%</td><td></td></tr><tr><td>GWN</td><td>2.67</td><td>5.40</td><td>6.62%</td><td>13.63</td><td>22.30</td><td>24.68%</td><td>2.35</td><td>3.56</td><td>10.72%</td><td>1.81</td><td>2.71</td><td>50.05%</td><td></td></tr><tr><td>w/o Finetune</td><td>7.41</td><td>12.78</td><td>24.30%</td><td>27.28</td><td>41.46</td><td>48.26%</td><td>4.89</td><td>6.99</td><td>23.42%</td><td>3.39</td><td>4.99</td><td>112.57%</td><td></td></tr><tr><td>w/ Finetune</td><td>2.69</td><td>5.38</td><td>6.74%</td><td>13.61</td><td>22.44</td><td>22.82%</td><td>2.35</td><td>3.55</td><td>10.72%</td><td>1.85</td><td>2.79</td><td>51.88%</td><td></td></tr><tr><td>Ours</td><td>2.67</td><td>5.36</td><td>6.79%</td><td>13.19</td><td>21.68</td><td>21.21%</td><td>2.37</td><td>3.58</td><td>10.77%</td><td>1.80</td><td>2.66</td><td>51.16%</td><td></td></tr><tr><td>MTGNN</td><td>2.70</td><td>5.50</td><td>6.81%</td><td>13.90</td><td>22.44</td><td>23.86%</td><td>2.33</td><td>3.56</td><td>10.63%</td><td>1.84</td><td>2.80</td><td>48.69%</td><td></td></tr><tr><td>w/o Finetune</td><td>5.91</td><td>9.21</td><td>14.56%</td><td>50.07</td><td>65.35</td><td>77.26%</td><td>4.45</td><td>6.27</td><td>20.06%</td><td>3.32</td><td>4.74</td><td>118.01%</td><td></td></tr><tr><td>w/ Finetune</td><td>2.62</td><td>5.33</td><td>6.56%</td><td>13.46</td><td>22.02</td><td>22.10%</td><td>2.33</td><td>3.54</td><td>10.52%</td><td>1.81</td><td>2.74</td><td>49.35%</td><td></td></tr><tr><td>Ours</td><td>2.59</td><td>5.25</td><td>6.54%</td><td>13.26</td><td>21.83</td><td>21.49%</td><td>2.31</td><td>3.52</td><td>10.51%</td><td>1.79</td><td>2.63</td><td>49.77%</td><td></td></tr><tr><td>PDForner</td><td>2.68</td><td>5.46</td><td>6.66%</td><td>13.85</td><td>22.59</td><td>23.11%</td><td>2.43</td><td>3.70</td><td>11.17%</td><td>1.89</td><td>2.88</td><td>51.98%</td><td></td></tr><tr><td>w/o Finetune</td><td>5.57</td><td>10.03</td><td>19.16%</td><td>20.45</td><td>31.56</td><td>30.21%</td><td>3.60</td><td>5.20</td><td>16.93%</td><td>3.19</td><td>4.76</td><td>111.40%</td><td></td></tr><tr><td>w/ Finetune</td><td>2.71</td><td>5.44</td><td>6.80%</td><td>13.77</td><td>22.31</td><td>24.90%</td><td>2.43</td><td>3.69</td><td>11.15%</td><td>1.83</td><td>2.78</td><td>48.56%</td><td></td></tr><tr><td>Ours</td><td>2.68</td><td>5.31</td><td>6.79%</td><td>13.55</td><td>22.08</td><td>21.98%</td><td>2.41</td><td>3.64</td><td>11.05%</td><td>1.81</td><td>2.68</td><td>52.06%</td><td></td></tr></table>
+
+# 5.2. Overall Performance (RQ1)
+
+In this section, we investigate the effectiveness of FlashST in enhancing the generalization capabilities of downstream models across new spatio-temporal contexts. As shown in Table 3, the baseline models were exclusively trained and tested on the target dataset to evaluate their performance.
+
+Superiority over End-to-End Prediction Models. Upon analyzing the results presented in Table 3, it becomes evident that our approach exhibits a significant advantage over the end-to-end spatio-temporal models in diverse urban data prediction scenarios. These findings provide compelling evidence of the effectiveness of our FlashST method in accurately capturing the intricate spatio-temporal invariant patterns present in urban data. Our new in-context learning paradigm excels in transferring this acquired knowledge to adapt the pre-trained model to new downstream tasks. By effectively handling the distribution gap, our method bridges the semantic gap between the pre-trained model and the specific prediction scenarios encountered in practice.
+
+Model-Agnostic Advantage. The model-agnostic advantage is a notable strength of our approach. We have designed our method to seamlessly integrate with various spatio-temporal backbone encoders, providing flexibility and avoiding the constraints of a specific model choice. This adaptability is demonstrated in Table 4, where our FlashST approach showcases easy adaptation with four state-of-the-art spatio-temporal models (i.e., STGCN, GWN, MTGNN, PDFormer). The evaluation results highlight the versatility of our FlashST approach, showcasing its exceptional performance enhancement when combined with these cutting-edge spatio-temporal models. The successful integration of our approach with state-of-the-art models further reinforces its adaptability and capability to improve prediction accuracy in diverse urban data scenarios.
+
+Comparison with Model Fine-Tuning. To further demonstrate the effectiveness of our spatio-temporal in-context framework, we compare our prompt-tuning approach with the full fine-tuning of the model parameters. The "w/o Finetune" approach refers to direct prediction on the target dataset without any fine-tuning after pre-training. On the other hand, "w/ Finetune" indicates the utilization of full-parameter fine-tuning to adapt the models to the target data after pre-training. However, it is worth noting that the observed improvements with direct fine-tuning, compared to end-to-end prediction outcomes, suggest that pre-training may not have provided a significant advantage. In the absence of effective alignment between the pre-trained model and downstream tasks, noise can be introduced, leading to misleading fine-tuning and suboptimal performance.
+
+# 5.3. Assessment of Model Efficiency (RQ2)
+
+Training Time. In this section, we focus on evaluating the efficiency of our proposed method. We measured the training time for three different scenarios: end-to-end training, full fine-tuning, and our FlashST approach. The results are presented in Table 5. For end-to-end training and full fine-tuning, we followed the settings of existing baselines, configuring the training epochs to 100. We also set an early stopping criterion of 25, which halts the training process if no decrease in validation loss is observed. Regarding our FlashST approach, we limited the number of epochs for prompt-tuning to 20. This constraint was implemented to promote swift adaptation of the downstream models to the new datasets. The results suggest that the same baseline model showcases comparable efficiency in both end-to-end training and full-parameter fine-tuning. The disparity in training time between these two settings primarily arises from variations in convergence speed caused by different initialization parameters. One of the significant contribu
+
+Table 5. Computational time cost investigation (seconds).  
+
+<table><tr><td>Model</td><td>time(s)</td><td>Model</td><td>time(s)</td></tr><tr><td>ASTGCN</td><td>1310</td><td>TGCN</td><td>927</td></tr><tr><td>AGCRN</td><td>689</td><td>STSGCN</td><td>2253</td></tr><tr><td>DMSTGCN</td><td>1330</td><td>STFGNN</td><td>990</td></tr><tr><td>MSDR</td><td>5190</td><td>STWA</td><td>5143</td></tr><tr><td>STGCN</td><td>411</td><td>GWN</td><td>1042</td></tr><tr><td>STGCN Finetune</td><td>445</td><td>GWN Finetune</td><td>1185</td></tr><tr><td>STGCN w/ Ours</td><td>190</td><td>GWN w/ Ours</td><td>222</td></tr><tr><td>MTGNN</td><td>962</td><td>PDForner</td><td>7524</td></tr><tr><td>MTGNN Finetune</td><td>1095</td><td>PDForner Finetune</td><td>6466</td></tr><tr><td>MTGNN w/ Ours</td><td>230</td><td>PDForner w/ Ours</td><td>1220</td></tr></table>
+
+![](images/c90212b93992b54d893ad9a9d2108a304e9c21f3a764331a685d62e54ea7c603.jpg)  
+Figure 3. The convergence efficiency of FlashST.
+
+![](images/8f3e49eac5e7b2f697ebf449d92beaff68aa6c7f125100b187e72406d5fc2311.jpg)
+
+tions of our FlashST framework is the notable increase in computational efficiency. It reduces the training time of baseline models by  $20\%$  to  $80\%$ . This substantial enhancement significantly improves their efficiency in adapting to new spatio-temporal data.
+
+Faster Convergence Rate. To further validate the ability of our FlashST framework to swiftly adapt to new spatiotemporal scenarios, we conducted an investigation into its convergence speed on different datasets. We specifically focused on the decreasing trend of the validation error when using the PEMs07(M) and CA-D5 datasets, with MTGNN adopted as the downstream model. The results are visualized in Figure 3. The findings clearly demonstrate that with the integration of our FlashST approach, the downstream model achieves convergence within a few tuning epochs. In contrast, the end-to-end training and fine-tuning paradigms require a greater number of training rounds to fit the new data. This phenomenon can be attributed to the effectiveness of our proposed spatio-temporal prompt network and data mapping strategy. These components enable the model to leverage the spatio-temporal characteristics of the new data in conjunction with the pre-trained knowledge, facilitating rapid adaptation to diverse spatio-temporal scenes.
+
+# 5.4. Ablation Study (RQ3)
+
+In this section, we conduct an ablation study to investigate the contributions of each key module in our FlashST framework. The corresponding experimental results are depicted in Figure 4. We utilize MTGNN as the downstream baseline and assess the performance of each component on the
+
+PEMS07(M) and CA-D5 data. Before the prompt-tuning step, we conduct re-pre-training to adapt to each variant.
+
+- Impact of Spatio-Temporal Context Distillation. To assess the importance of temporal and spatial context distillation, we conducted experiments by removing the distillation with temporal context (-TC) and spatial context (-SC) separately. The results clearly demonstrate a substantial performance drop across most metrics when the distilled temporal and spatial context are removed. This highlights the critical importance of preserving both temporal and spatial context during the in-context learning process. Effective encoding of the temporal information and the incorporation of the spatial information are vital for capturing dynamic periodic temporal patterns and region-specific properties, thereby enhancing the model's ability to recognize spatio-temporal invariant patterns and improve its data understanding.
+
+- Impact of Spatio-Temporal Dependency Modeling. In our analysis, we individually removed the temporal encoder (-TE) and spatial encoder (-SE) to investigate their contributions. Our analysis revealed that spatio-temporal dependency encoding plays a vital role in effectively integrating complex relationships among different time slots and locations during the in-context learning process. The inclusion of both temporal and spatial dependency encoders enables the model to comprehend and leverage the intricate interactions between time and space. This proficiency facilitates a swifter adaptation of downstream models to new spatio-temporal scenarios.  
+- Impact of Unified Distribution Mapping Mechanism. In our variants, we assess the utility of the unified distribution mapping strategy through two aspects. i) -Uni. We did not explore the unified distribution mapping strategy. However, the decline in performance indicates its positive impact on the model. By mapping the diverse spatiotemporal data embeddings into a uniform distribution, our FlashST effectively mitigates the influence of distributional disparities between the pre-training data and unseen spatio-temporal data. ii) r/BN. The unified distribution mapping strategy has been replaced with batch normalization. Batch normalization standardizes the data based on local statistical properties of mini-batches, mitigating the issue of internal covariate shift during neural network training and thereby enhancing convergence efficiency. However, due to the lack of an established connection between the pretrained data and the downstream task data, it is challenging for the downstream model to effectively transfer knowledge from the pretraining process. The unified distribution mapping strategy ensures that the model can effectively leverage the knowledge acquired during the pre-training. By aligning the distributions of different data sources, the model can better adapt to new spatiotemporal scenarios and make more accurate predictions.
+
+![](images/b179fa4c5df20fcdccf3acb7141055e3c3b7e8c4348914ab2eb9562a3f94a030.jpg)  
+(b) MAE, RMSE and MAPE on PEMS07(M) dataset.
+
+![](images/6184d5046e1cd0f84dbcf889766a6f99eb1b64a10a2891fe2804cd2bda196307.jpg)  
+(c) MAE, RMSE and MAPE on CA-D5 dataset.  
+Figure 4. Ablation study of FlashST.
+
+# 5.5. Hyperparameter Analysis (RQ4)
+
+In this section, we investigate the impact of different hyperparameter settings, specifically across various temperature coefficients and loss weight coefficients, on the model's performance. Our findings reveal that the model achieves optimal performance with a parameter configuration of  $\tau = 0.3$  and  $\lambda = 1.0$ , as shown in Figure 6. Notably, variations in these parameters minimally influence the final result, highlighting the model's effective adaptability to different parameters settings. It learns efficient representations that distinguish embedded features in various regions, even with differences in feature scales. Additionally, the model's performance remains unaffected by an increase in the uniformity loss. This indicates that the key component of our distribution mapping strategy does not interfere with the predictive loss. This further supports the viability of our strategy and facilitates the downstream model's rapid generalization to novel spatio-temporal contexts.
+
+# 5.6. Case Study (RQ5)
+
+To assess the effectiveness of our proposed uniform distribution mapping approach in transforming various data representations into uniform distributions, we conducted visualizations of prompt embeddings with and without the application of our distribution mapping mechanism. These visualizations were performed using two variants of the downstream model MTGNN and evaluated on the test sets of the PEMS07(M) and CA-D5 datasets. Given that prompt embeddings are high-dimensional vectors, we initially applied the PCA technique to reduce the dimensionality of each embedding sample to 2-D (Wold et al., 1987). Subsequently, we projected the reduced embeddings onto the unit circle using the L2 norm, as depicted in Figure 6. The visualization results provide compelling evidence that our uniform distribution mapping strategy effectively transforms the prompt embeddings into an approximate uniform distri
+
+![](images/85e6cb98b5d546aec2afa421ed92f23ba51e5b5debaf3e25b243c9bc4cadccfe.jpg)
+
+![](images/c94f7e758afc1ed725444e72fa00da3172e56eb309933480f32bbca814c81bdf.jpg)
+
+![](images/3f905c64a01da43cc54d69c9805e2274f1e917de41df11a71a0164e0b38bf2ee.jpg)  
+Figure 5. Hyperparameter study of  $\tau$  and  $\lambda$ .  
+Figure 6. Distribution of the prompt embedding.
+
+![](images/35f7f864ef170ab82fa1e41ae7cd1f8cfcd04dcaafff6f39019e8878caba13c7.jpg)
+
+bution. In contrast, the variants that lack this strategy fail to achieve such desirable distribution properties. By converting data from new spatio-temporal contexts into a consistent distribution, our FlashST gains the ability to utilize pre-trained knowledge and rapidly adapt to new datasets, facilitating its performance on various spatio-temporal tasks.
+
+# 6. Conclusion
+
+This paper presents FlashST, a new and efficient method for adapting spatio-temporal predictive models to various downstream tasks that involve previously unseen data. Our in-context learning framework utilizes a spatio-temporal prompt network, which consists of a context distillation mechanism and a dependency modeling scheme. This framework effectively adapts to different spatio-temporal scenarios by capturing contextual signals and modeling complex relationships across time and locations. To address the distribution gap, we enhance FlashST by incorporating a distribution mapping mechanism that aligns the data distributions of pre-training and downstream data, facilitating effective knowledge transfer in spatio-temporal forecasting. Extensive experiments demonstrate the effectiveness and generalization capabilities of our FlashST in diverse downstream spatio-temporal forecasting scenarios. For future work, an interesting direction would be to explore the potential of incorporating large language models as knowledge guidance within our FlashST framework.
+
+# Impact Statement
+
+This paper presents work whose goal is to advance the field of Machine Learning. There are many potential societal consequences of our work, none which we feel must be specifically highlighted here.
+
+# Acknowledgments
+
+This work is partially supported by the Guangzhou Municipal Science and Technology Plan Project - Key Research and Development Program (Grant No. 2024B01W0007).
+
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+        "type": "text",
+        "text": "Flexible Residual Binarization for Image Super-Resolution",
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+        "text": "Yulun Zhang *1 Haotong Qin *2 Zixiang Zhao 2 Xianglong Liu 3 Martin Danelljan 2 Fisher Yu 2",
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+        "text": "Binarized image super-resolution (SR) has attracted much research attention due to its potential to drastically reduce parameters and operations. However, most binary SR works binarize network weights directly, which hinders high-frequency information extraction. Furthermore, as a pixel-wise reconstruction task, binarization often results in heavy representation content distortion. To address these issues, we propose a flexible residual binarization (FRB) method for image SR. We first propose a second-order residual binarization (SRB), to counter the information loss caused by binarization. In addition to the primary weight binarization, we also binarize the reconstruction error, which is added as a residual term in the prediction. Furthermore, to narrow the representation content gap between the binarized and full-precision networks, we propose Distillation-guided Binarization Training (DBT). We uniformly align the contents of different bit widths by constructing a normalized attention form. Finally, we generalize our method by applying our FRB to binarize convolution and Transformer-based SR networks, resulting in two binary baselines: FRBC and FRBT. We conduct extensive experiments and comparisons with recent leading binarization methods. Our proposed baselines, FRBC and FRBT, achieve superior performance both quantitatively and visually.",
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+    {
+        "type": "text",
+        "text": "1. Introduction",
+        "text_level": 1,
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+        "text": "Given a full-precision low-resolution (LR) input, single image super-resolution (SR) aims to obtain its high-resolution (HR) counterpart by reconstructing more details. Essentially, image SR is ill-posed, as there exist multiple HR candidi",
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+        "text": "*Equal contribution  $^{1}\\mathrm{MoE}$  Key Lab of Artificial Intelligence, AI Institute, Shanghai Jiao Tong University, China  $^{2}\\mathrm{ETH}$  Zürich, Switzerland  $^{3}\\mathrm{Beihang}$  University, China. Correspondence to: Xianglong Liu <xlliu@buaa.edu.cn>.",
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+        "text": "Proceedings of the  $41^{st}$  International Conference on Machine Learning, Vienna, Austria. PMLR 235, 2024. Copyright 2024 by the author(s).",
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+            "FRBT (ours) 115 / 5.4",
+            "Figure 1. Visual samples of image SR  $(\\times 4)$  by lightweight image SR methods. SRResNet (Ledig et al., 2017) is a full-precision (i.e., 32-bit) model and is used as a backbone for binarization (i.e., 1-bit) by BNN (Courbariaux et al., 2016), ReActNet (Liu et al., 2020), and our FRBC. We also binarize SwinIR_S (Liang et al., 2021) and denote it as FRBT. We provide the parameter (i.e., Params (K)) and operation numbers (i.e., Ops (G)). We set the input size as  $3\\times 320\\times 180$  for Ops calculation. Our FRBC and FRBT achieve better visual reconstruction than other binary ones. Our results are more faithful to that of the full-precision SRResNet."
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+        "text": "dates for the same LR input. To address this problem, deep convolutional neural networks (CNNs) and Transformers have been investigated for high-quality reconstructions (Dong et al., 2014; Kim et al., 2016; Lim et al., 2017; Zhang & Patel, 2018; Zhang et al., 2018b; Liang et al., 2021). However, most of them require extensive computational resources, which are usually not friendly for resource-limited devices. In those cases, neural network compression techniques are eagerly needed to significantly reduce model complexity.",
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+    },
+    {
+        "type": "text",
+        "text": "As one of the most promising network compression techniques, binary neural networks (BNNs), where both the network weights and activations are binarized (i.e., 1-bit binarization), are usually chosen for model deployment (Martinez et al., 2020; Rastegari et al., 2016). Theoretically, BNN enjoys  $32 \\times$  parameter compression ratio and up to  $58 \\times$  computation operation reduction (Rastegari et al., 2016). Such practical characteristics make BNN highly efficient for embedded devices (Ding et al., 2019) and friendly for memristor-based hardwares (Liu et al., 2020).",
+        "bbox": [
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+        "page_idx": 0
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+    {
+        "type": "text",
+        "text": "Despite the above-mentioned advantages of BNN, the severe performance drop hinders it from being widely deployed (Liu et al., 2020). Such a problem is particularly critical in binarized image SR, where dense pixel-wise predictions are required and the feature size is usually very large. The performance drop mainly comes from two parts: weights and activations binarization. (1) The weights are binarized from full-precision (i.e., 32-bit) to 1-bit, being hard",
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+    {
+        "type": "page_number",
+        "text": "1",
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+    {
+        "type": "text",
+        "text": "to extract high-frequency information. Even though the activations are full-precision, the SR output would still suffer from heavy degradation (Ma et al., 2019). (2) Binarizing activations (i.e., features) would directly lose high-frequency information, which is the key component that image SR networks try to reconstruct. Moreover, after the computation operations between binarized weights and activations, the output would further lose pixel-wise detailed information with high uncertainty, resulting in worse performance.",
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+    {
+        "type": "text",
+        "text": "To address those issues, we propose a flexible residual binarization (FRB) technique for binarized image SR. (1) To tackle the first issue, we try to reduce the weight error with our second-order residual binarization (SRB). Specifically, we not only binarize the weights as a common practice, we further binarize weight residuals between 1-bit and full-precision weights. Such an SRB practice helps preserve network weight representation capability more effectively than direct binarization only. (2) Furthermore, to compensate the pixel-wise information loss, we propose Distillation-guided Binarization Training (DBT). Specifically, we try to transfer full-precision knowledge to narrow the representation content gap between the binarized and full-precision networks. A normalized attention form is built to uniformly align the contents of different bit-widths.",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We further generalize our FRB to different types of networks and investigate its behaviors. Consequently, we apply our FRB to binarize CNN and Transformer based SR networks respectively, resulting in two binary baselines: FRBC and FRBT. Surprisingly, as shown in Fig. 1, our proposed methods achieve promising results with comparable or much smaller computational resources.",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Our main contributions are summarized as follows:",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "- We propose a simple yet effective method Flexible Residual Binarization (FRB) to accurately binarize full-precision image SR networks during the training.",
+            "- We propose an effective second-order residual binarization (SRB), which binarizes the image SR network with its weight residuals. SRB enhances the representation capacity of the binarized image SR network significantly for pixel-wise reconstruction.",
+            "- We propose Distillation-guided Binarization Training (DBT), which transfers full-precision knowledge to the binarized model. Specifically, we build a normalized attention form to uniformly align the contents of different bit-widths (e.g., 32-bit and 1-bit).",
+            "- We employ our FRB to binarize CNN and Transformer based SR networks respectively, resulting in two binarized baselines: FRBC and FRBT. Our methods achieve superior performance over SOTA binarized SR methods quantitatively and visually."
+        ],
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2. Related Work",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "2.1.Lightweight Image SR",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Lightweight image SR models have recently drawn more and more attention because of their resource-friendly properties. Usually, researchers pursue lightweight networks by architecture design, neural architecture search (NAS), knowledge distillation (KD), and network pruning. Ahn et al. constructed a cascading method upon a residual network (CARN) (Ahn et al., 2018). Hui et al. proposed an information multi-distillation network (IMDN) (Hui et al., 2019). Meantime, model compression methods have been introduced for lightweight SR, too. Chu et al. introduced neural architecture search for image SR in FALSR (Chu et al., 2019). Knowledge distillation was employed to train lighter SR student networks (He et al., 2020; Lee et al., 2020). Using pretrained SR models, Zhang et al. incorporated channel pruning into image SR through aligned structured sparsity learning (ASSL) (Zhang et al., 2021b) or structure-regularized pruning (SRP) (Zhang et al., 2022). Such lightweight network designs and compression techniques have achieved promising performance. They either neglect the fine-grained parameter redundancy or consume a considerable number of additional computations.",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "2.2. Model Quantization",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "There are two main types of quantization methods: Post-Training Quantization (PTQ) and Quantization-Aware Training (QAT). PTQ has become increasingly popular due to its ability to quantize models without the need for retraining, resulting in numerous contributions in the field (Choukroun et al., 2019; Jhunjhunwala et al., 2021; Hubara et al., 2021; Li et al., 2021; Ding et al., 2022). However, this approach only relies on limited expert knowledge and minimal GPU resources to calibrate the model, which significantly restricts its potential for achieving extreme low-bit quantization. Fortunately, QAT provides us with the opportunity to utilize the entire training pipeline to achieve aggressive low-bit quantization, including 1-bit binarization, and demonstrates promising performance (Martinez et al., 2020; Qin et al., 2020; Liu et al., 2020; Zhou et al., 2016; Courbariaux et al., 2016; Rastegari et al., 2016). Qin et al. proposed low-bit quantization for image SR (Qin et al., 2023). This approach allows for more comprehensive model optimization, enabling the model to be trained to perform optimally in the quantized domain. QAT is usually seen as a powerful method for achieving extremely low-bit quantization.",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Recent studies, including (Wang et al., 2020; Simons & Lee, 2019; Wang et al., 2022; Zhang et al., 2021a; Qin et al., 2022), have demonstrated the effectiveness of 1-bit quantization, i.e., binarization, as a highly efficient form of network quantization. This binarization technique compresses networks to achieve extreme computational and storage efficiency by using 1-bit binarized parameters. Compared",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "header",
+        "text": "Flexible Residual Binarization for Image Super-Resolution",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "page_number",
+        "text": "2",
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+        "page_idx": 1
+    },
+    {
+        "type": "image",
+        "img_path": "images/ba67d82697d0f69111a72e688cd8493733c9dc4016231746188200ed38b8d136.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "image",
+        "img_path": "images/d62d10b78caca917bac8fa1c3f33002a61e38f3c798b51290e1974441437354b.jpg",
+        "image_caption": [
+            "Figure 2. Overview of our Flexible Residual Binarization (FRB) for image SR networks. The upper (blue) is the Second-order Residual Binarization, where the SR network weights are binarized in a residual manner. The lower (orange) is the Distillation-guided Binarization Training that uniformly aligns the contents of different bit widths by constructing a normalized attention form."
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "to floating-point models, these quantized models significantly reduce computation resources and save time, and are hardware-friendly for edge devices.",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.3. Binary Neural Networks for Image SR",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Existing SR networks on resource-constrained devices are limited in usage by their high memory requirements and computational overhead. One major challenge is the heavy floating-point storage and operations. Thus room for compression still exists from a bit-width perspective, which gives a strong motivation for the study of 1-bit binarized SR models (Xin et al., 2020; Jiang et al., 2021; Xia et al., 2023). Xin et al. designed a bit-accumulation mechanism to binarize full-precision SR networks (Xin et al., 2020). Xia et al. proposed a basic binary convolution unit for binarized image restoration (Xia et al., 2023). However, they mainly work on binarization for CNNs and lack the investigation about Transformer based binarized SR models.",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3. Flexible Residual Binarization for Binarized Image Super-Resolution",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In this section, we first give an overview of binarization for single image super-resolution (SR) and raise the existing challenges of binarized (i.e., 1-bit) image SR networks. We then introduce our proposed flexible residual binarization (FRB) for image SR. Our FRB consists of two well-designed components: Second-order Residual Binarization (SRB) and Distillation-guided Binarization Training (DBT), which are designed for recovering the representation capacity and",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "aligning the representation context, respectively. Afterward, we show how to utilize FRB for image SR and optimize the binary image SR network (Fig. 2). We finally give more details about implementation.",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.1. Preliminaries: Binarization in SR",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "Here, we give a brief background to the important key components in a general binarized image SR pipeline. Given a full-precision (i.e., 32-bit) low-resolution (LR) image as input  $I_{\\mathrm{LR}}$ , the binary (i.e., 1-bit) super-resolution network aims to obtain its full-precision high-resolution (HR) counterpart  $I_{SR}$ . We formulate such a binarized image SR process with the neural network as follows",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nI _ {S R} = \\mathcal {F} _ {\\mathrm {B S R}} \\left(I _ {\\mathrm {L R}}; \\boldsymbol {\\Theta}\\right), \\tag {1}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where  $\\mathcal{F}_{\\mathrm{BSR}}(\\cdot)$  denotes the binary super-resolution (BSR) network with trainable parameters  $\\Theta$ . Specifically, we binarize the image SR network  $\\mathcal{F}_{\\mathrm{BSR}}(\\cdot)$  by the sign function, which is a standard choice for the task. The forward operation is the standard sign function,",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\operatorname {s i g n} (\\boldsymbol {x}) = \\left\\{ \\begin{array}{l l} 1 & \\text {i f} \\boldsymbol {x} \\geq 0 \\\\ - 1 & \\text {o t h e r w i s e} \\end{array} . \\right. \\tag {2}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Since this standard sign function is not continuous or differentiable, its backward operation can hardly achieved directly. Instead, the backward is replaced by the approximation,",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\frac {\\partial \\text {s i g n}}{\\partial \\boldsymbol {x}} = \\left\\{ \\begin{array}{l l} 1 & \\text {i f} | \\boldsymbol {x} | \\leq 1 \\\\ 0 & \\text {o t h e r w i s e} \\end{array} . \\right. \\tag {3}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "A floating-point precision weight matrix  $\\pmb{w}$  can thus be",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "header",
+        "text": "Flexible Residual Binarization for Image Super-Resolution",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "page_number",
+        "text": "3",
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+        "page_idx": 2
+    },
+    {
+        "type": "image",
+        "img_path": "images/bb39328705e21f2de09730cfea926770130f87f7202482d91e4d5b3be739062a.jpg",
+        "image_caption": [
+            "Figure 3. An computation example of our Second-order Residual Binarization (SRB). Our residual binarization allows the binarized weight representation to retain accurate information, further restoring the functionality of its full-precision counterpart compared to vanilla binarization. And the activation directly uses the sign function to binarize to avoid the extra burden during inference."
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "binarized with the sign function as,",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathbf {B} _ {\\boldsymbol {w}} = \\alpha \\operatorname {s i g n} (\\boldsymbol {w}). \\tag {4}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "A scaling factor  $\\alpha$  is introduced to retain the magnitude of real-value weights. It is computed as",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\alpha = \\frac {1}{n} \\boldsymbol {w} ^ {\\top} \\operatorname {s i g n} (\\boldsymbol {w}) = \\frac {1}{n} \\| \\boldsymbol {w} \\| _ {1}. \\tag {5}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "After binarizing the SR networks, the storage size and computation can be significantly reduced due to the extremely reduced bit-width and highly efficient bitwise XNOR and bitcount operations (Rastegari et al., 2016). We then propose two techniques to improve binarized networks.",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.2. Second-order Residual Binarization for Weight Error Reduction",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "While binarization promises reduced storage and faster inference, it substantially reduces the capacity of the original weights. It causes serious challenges for binarized image SR networks. This can be captured in the error caused by binarizing the continuous weights in Eq. (4) as,",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\epsilon = \\boldsymbol {w} - \\mathbf {B} _ {\\boldsymbol {w}}. \\tag {6}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The error  $\\epsilon$  represents the residual information that is lost in the binarization operation. Intuitively, we want to reduce this error. While this could be done by increasing the number of bits in the discrete representation, it does not allow for the use of efficient binary network operations.",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "In this work, we propose a different approach to reducing binarization errors. We perform a second-order binarization, in order to retrieve information lost in the error Eq. (6). This is performed by binarizing the error Eq. (6) and using it as a residual correction term to approximate continuous weights. Our binarization strategy is thus expressed as,",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathbf {B} _ {\\boldsymbol {w 1}} = \\alpha_ {1} \\operatorname {s i g n} (\\boldsymbol {w}), \\quad \\alpha_ {1} = \\frac {1}{n} \\| \\boldsymbol {w} \\| _ {1}, \\tag {7}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathbf {B} _ {\\boldsymbol {w} \\mathbf {2}} = \\alpha_ {2} \\operatorname {s i g n} (\\boldsymbol {w} - \\mathbf {B} _ {\\boldsymbol {w} \\mathbf {1}}), \\alpha_ {2} = \\frac {1}{n} \\| \\boldsymbol {w} - \\mathbf {B} _ {\\boldsymbol {w} \\mathbf {1}} \\| _ {1}. \\tag {8}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We refer to  $\\mathbf{B}_{w1}$  and  $\\mathbf{B}_{w2}$  as the first and second order binarization, respectively. Note that the scaling factors are computed using the same formula Eq. (5).",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "In Eq. (7), the gradient estimation in the backward propagation for the sign function approximately follows Eq. (3). And for activation, the binarization operation follows the sign binarizer in Liu et al. (2020). Taking the binarized convolution unit as an example, the forward computation process of our second-order residual binarization (SRB) is expressed as,",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\no = \\operatorname {s i g n} (\\boldsymbol {a}) \\otimes \\mathbf {B} _ {\\boldsymbol {w 1}} + \\operatorname {s i g n} (\\boldsymbol {a}) \\otimes \\mathbf {B} _ {\\boldsymbol {w 2}}, \\tag {9}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            557,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where the  $\\otimes$  is the bitwise convolution consisting of XNOR and bitcount instructions (Arm, 2020; AMD, 2022). We also give an example of our technique in Fig. 3.",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Second-order residual binarization (SRB) preserves the representation capability of weights better than direct binarization, while still being able to use bitwise instructions for efficient computation. Moreover, residuals enhance the representation capacity of binarized weights by making them closer to the original values and more diverse in the output space. Such a property can significantly boost the performance of binarized image SR networks.",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.3. Distillation-guided Binarization Training",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "In addition to the decrease in network representation capacity, the high discretization of binarization also leads to severe content distortion of representations.",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "On the other hand, since most image SR models are composed block-by-block (Liang et al., 2021; Lim et al., 2017), for image SR networks, the  $n$ -block  $\\mathcal{F}_{\\mathrm{BSR}}(\\cdot)$  in Eq. (1) can then be reformulated as,",
+        "bbox": [
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+            534
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nI _ {S R} = \\mathcal {F} _ {\\mathrm {B S R}} \\left(I _ {\\mathrm {L R}}; \\boldsymbol {\\Theta}\\right) = \\prod_ {i = 1} ^ {n} B l k _ {\\mathrm {B S R} _ {i}} \\left(I _ {\\mathrm {L R}}; \\boldsymbol {\\Theta}\\right). \\tag {10}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            517,
+            536,
+            885,
+            574
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Here,  $Blk_{\\mathrm{BSR}_i}$  denotes the  $i$ -th inner block of the SR network composed of several binarized computation units, including binarized convolution and linear units. Correspondingly, full-precision models and blocks is denoted as  $\\mathcal{F}_{\\mathrm{SR}}(\\cdot)$  and  $Blk_{\\mathrm{SR}_i}$ . Lastly,  $\\prod$  denotes the composition of blocks.",
+        "bbox": [
+            495,
+            575,
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+            652
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Based on the above formulation and illustrations, the blocklevel ( $k$ -th block) representation distortion caused by binarization can be expressed as,",
+        "bbox": [
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+            657,
+            887,
+            702
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathcal {D} _ {k} = \\prod_ {i = 1} ^ {k} B l k _ {\\mathrm {S R} _ {i}} \\left(I _ {\\mathrm {L R}}; \\boldsymbol {\\Theta}\\right) - \\prod_ {i = 1} ^ {k} B l k _ {\\mathrm {B S R} _ {i}} \\left(I _ {\\mathrm {L R}}; \\boldsymbol {\\Theta}\\right). \\tag {11}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            504,
+            705,
+            885,
+            746
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "To make the binarized SR model perform close to the full-precision level, intuitively, we should reduce the distortion  $\\mathcal{D}_i$  of each block in the model.",
+        "bbox": [
+            495,
+            747,
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Therefore, we propose Distillation-guided Binarization Training (DBT) to align the representation content gap between binarized and full-precision SR networks (as Fig. 4). Inspired by (Martinez et al., 2020), we construct a normalized attention form for block-level representations to uniformly stabilize the contents in networks of different bit-widths. For example, the  $i$ -th block's formed representation",
+        "bbox": [
+            495,
+            799,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "header",
+        "text": "Flexible Residual Binarization for Image Super-Resolution",
+        "bbox": [
+            300,
+            56,
+            673,
+            70
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "page_number",
+        "text": "4",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "image",
+        "img_path": "images/07da1151ea09f08080e236140f5b2003e942a20672786826a89b2038442ac9c0.jpg",
+        "image_caption": [
+            "Figure 4. The computation flow of the loss function considering DBT. During training, the training data is simultaneously fed into the binarized SR network and its well-trained full-precision counterpart, and  $\\mathcal{L}_{\\mathrm{DBT}}$  is calculated according to the block-level intermediate representation (such as Eq. (13)). In the end,  $\\mathcal{L}_{\\mathrm{DBT}}$  participates in the calculation of the total loss and jointly optimizes the binarized SR model with other loss items ( $\\mathcal{L}_{\\mathrm{PIX}}$  in Eq. (14))."
+        ],
+        "image_footnote": [],
+        "bbox": [
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+            80,
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+            289
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "in a binarized IR network can be formulated as",
+        "bbox": [
+            84,
+            406,
+            397,
+            419
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nR _ {\\mathrm {B S R} _ {k}} = \\frac {\\left(\\prod_ {i = 1} ^ {k} B l k _ {\\mathrm {B S R} _ {i}} \\left(I _ {\\mathrm {L R}} ; \\boldsymbol {\\Theta}\\right)\\right) ^ {2}}{\\left\\| \\left(\\prod_ {i = 1} ^ {k} B l k _ {\\mathrm {B S R} _ {i}} \\left(I _ {\\mathrm {L R}} ; \\boldsymbol {\\Theta}\\right)\\right) ^ {2} \\right\\| _ {\\ell 2}}, \\tag {12}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            117,
+            419,
+            475,
+            481
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where  $\\| \\cdot \\|_{\\ell 2}$  denotes the L2 normalization.",
+        "bbox": [
+            84,
+            482,
+            374,
+            498
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Then we distill full-precision representations to binarized ones. We target to consistently push binarized presentations to approach full-precision level representations:",
+        "bbox": [
+            84,
+            505,
+            473,
+            550
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\min  \\mathcal {L} _ {\\mathrm {D B T}} = \\sum_ {i = 1} ^ {n} \\hat {\\mathcal {D}} _ {i} = \\sum_ {i = 1} ^ {n} \\| R _ {\\mathrm {S R} _ {i}} - R _ {\\mathrm {B S R} _ {i}} \\| _ {\\ell 2}. \\tag {13}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            104,
+            551,
+            473,
+            589
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Note that the binarized SR model and the full-precision replica are a pair of natural teachers and students because they have exactly the same architecture and significant differences in computation/storage. We highlight that this fact makes our DBT a flexible and architecture-generic technique, and the blockwise distillation implementation can even be fine-grained to a single computing layer level to suit various architectures. Such a property allows us to practice our compression techniques on various CNN- and Transformer-based image SR networks.",
+        "bbox": [
+            84,
+            590,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.4. FRB for Image SR",
+        "text_level": 1,
+        "bbox": [
+            84,
+            751,
+            250,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Binarized Architectures. For FRB, our proposed SRB technique is allowed to be flexibly applied to various computational units in the SR architecture, such as convolutional and linear units. Therefore, for the image SR architecture using FRB, we apply SRB binarization to all computing units in the body part, which is the most computationally intensive. We maintain the full precision of the head and tail parts. In addition, the ReLU function is replaced by PReLU following Martinez et al. (2020).",
+        "bbox": [
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+            768,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Algorithm 1 Flexible Residual Binarization for Image SR",
+        "text_level": 1,
+        "bbox": [
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+            83,
+            883,
+            99
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "1: Input: Training dataset  $D$ , full-precision model  $\\mathcal{F}_{\\mathrm{SR}}$ , training iterations  $N$ ;",
+            "2: Output: The binarized model  $\\mathcal{F}_{\\mathrm{BSR}}$",
+            "3: Define the binarized  $\\mathcal{F}_{\\mathrm{BSR}}(\\cdot)$  model by binarizing computation units of  $\\mathcal{F}_{\\mathrm{SR}}$  by the SRB as Eq. (7);",
+            "4: for iteration  $i$  in  $[0,N)$  do",
+            "5: Feed data  $D$  in full-precision model  $\\mathcal{F}_{\\mathrm{SR}}$",
+            "6: Feed data  $D$  in binarized model  $\\mathcal{F}_{\\mathrm{BSR}}$ ;",
+            "7: Calculate the loss following Eq. (13) and Eq. (14);",
+            "8: Optimize the binarized model  $\\mathcal{F}_{\\mathrm{BSR}}$",
+            "9: end for",
+            "10: Return  $\\mathcal{F}_{\\mathrm{BSR}}$"
+        ],
+        "bbox": [
+            501,
+            103,
+            887,
+            285
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "SR Model Training. For the given training dataset  $D = \\{I_{\\mathrm{LR}}^i, I_{\\mathrm{HR}}^i\\}_{i=1}^K$  with  $K$  low-resolution inputs and their corresponding HR counterparts, the image SR model with our proposed FRD is optimized by minimizing both the conventional pixel-wise  $\\mathcal{L}_{\\mathrm{PIX}}$  loss and  $\\mathcal{L}_{\\mathrm{DBT}}$  distillation loss:",
+        "bbox": [
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+            324,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathcal {L} _ {\\mathrm {P I X}} = \\frac {1}{n} \\sum_ {i = 1} ^ {k} \\left\\| I _ {\\mathrm {H R}} ^ {i} - I _ {\\mathrm {S R}} ^ {i} \\right\\| _ {\\ell 1}, \\tag {14}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            584,
+            404,
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+            444
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathcal {L} _ {\\text {t o t a l}} = \\mathcal {L} _ {\\mathbf {P I X}} + \\beta \\mathcal {L} _ {\\mathbf {D B T}},\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            583,
+            448,
+            764,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where the  $\\beta$  is a hyperparameter and is set as 1e-4 by default in our FRB, and  $\\mathcal{L}_{\\mathrm{DBT}}$  is in Eq. (13). Figure 4 also presents the computation flow of our training loss. The whole algorithm of our proposed flexible residual binarization for the image SR model can be presented in Algorithm 1.",
+        "bbox": [
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+            465,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4. Experiments",
+        "text_level": 1,
+        "bbox": [
+            496,
+            553,
+            629,
+            570
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4.1. Settings",
+        "text_level": 1,
+        "bbox": [
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+            575,
+            588,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Data. Following the common practice (Lim et al., 2017; Zhang et al., 2018a), we adopt DIV2K (Timofte et al., 2017) as the training data. Five benchmark datasets are used for testing: Set5 (Bevilacqua et al., 2012), Set14 (Zeyde et al., 2010), B100 (Martin et al., 2001), Urban100 (Huang et al., 2015), and Manga109 (Matsui et al., 2017).",
+        "bbox": [
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+            595,
+            888,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Evaluation. To evaluate the reconstruction performance, we calculate PSNR and SSIM (Wang et al., 2004) values on the Y channel of the YCbCr space. For model complexity evaluation, we follow (Rastegari et al., 2016) and report the model size and operations of BNN. Specifically, we calculate the BNN parameters via  $\\text{Params}_-1 = \\text{Params}_-f / 32$ , where  $\\text{Params}_-f$  is the full-precision counterpart parameters. We calculate BNN operations via  $\\text{Ops}_-1 = \\text{Ops}_-f / 64$ , where  $\\text{Ops}_-f$  denotes operations of the full-precision counterpart. Based on  $\\text{Params}_-1$  and  $\\text{Ops}_-1$ , we further provide theoretical compression ratios for parameters and operations.",
+        "bbox": [
+            495,
+            690,
+            888,
+            857
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Proposed Binary Baselines. We apply our FRB to binarize CNN and Transformer based image SR baselines. Specifically, following BAM (Xin et al., 2020) and BTM (Jiang",
+        "bbox": [
+            495,
+            859,
+            888,
+            906
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "header",
+        "text": "Flexible Residual Binarization for Image Super-Resolution",
+        "bbox": [
+            300,
+            56,
+            673,
+            70
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "page_number",
+        "text": "5",
+        "bbox": [
+            480,
+            922,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "table",
+        "img_path": "images/45d965a2f9169cab30b208781742a8c11da54729de4e0380c5767e5d35fc6592.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Method</td><td rowspan=\"2\">Scale</td><td rowspan=\"2\">Bits (W/A)</td><td colspan=\"2\">Set5</td><td colspan=\"2\">Set14</td><td colspan=\"2\">B100</td><td colspan=\"2\">Urban100</td><td colspan=\"2\">Manga109</td></tr><tr><td>PSNR</td><td>SSIM</td><td>PSNR</td><td>SSIM</td><td>PSNR</td><td>SSIM</td><td>PSNR</td><td>SSIM</td><td>PSNR</td><td>SSIM</td></tr><tr><td>Bicubic</td><td>×2</td><td>-/-</td><td>33.66</td><td>0.9299</td><td>30.24</td><td>0.8688</td><td>29.56</td><td>0.8431</td><td>26.88</td><td>0.8403</td><td>30.80</td><td>0.9339</td></tr><tr><td>SRResNet</td><td>×2</td><td>32/32</td><td>38.00</td><td>0.9605</td><td>33.59</td><td>0.9171</td><td>32.19</td><td>0.8997</td><td>32.11</td><td>0.9282</td><td>38.56</td><td>0.9770</td></tr><tr><td>BNN</td><td>×2</td><td>1/1</td><td>32.25</td><td>0.9118</td><td>29.25</td><td>0.8406</td><td>28.68</td><td>0.8104</td><td>25.96</td><td>0.8088</td><td>29.16</td><td>0.9127</td></tr><tr><td>DoReFa</td><td>×2</td><td>1/1</td><td>36.76</td><td>0.9550</td><td>32.44</td><td>0.9072</td><td>31.31</td><td>0.8883</td><td>29.26</td><td>0.8945</td><td>35.81</td><td>0.9682</td></tr><tr><td>Bi-Real</td><td>×2</td><td>1/1</td><td>32.32</td><td>0.9123</td><td>29.47</td><td>0.8424</td><td>28.74</td><td>0.8111</td><td>26.35</td><td>0.8161</td><td>29.64</td><td>0.9167</td></tr><tr><td>IRNet</td><td>×2</td><td>1/1</td><td>37.27</td><td>0.9579</td><td>32.92</td><td>0.9115</td><td>31.76</td><td>0.8941</td><td>30.63</td><td>0.9122</td><td>36.77</td><td>0.9724</td></tr><tr><td>BAM</td><td>×2</td><td>1/1</td><td>37.21</td><td>0.9560</td><td>32.74</td><td>0.9100</td><td>31.60</td><td>0.8910</td><td>30.20</td><td>0.9060</td><td>N/A</td><td>N/A</td></tr><tr><td>BTM</td><td>×2</td><td>1/1</td><td>37.22</td><td>0.9575</td><td>32.93</td><td>0.9118</td><td>31.77</td><td>0.8945</td><td>30.79</td><td>0.9146</td><td>36.76</td><td>0.9724</td></tr><tr><td>ReActNet</td><td>×2</td><td>1/1</td><td>37.26</td><td>0.9579</td><td>32.97</td><td>0.9124</td><td>31.81</td><td>0.8954</td><td>30.85</td><td>0.9156</td><td>36.92</td><td>0.9728</td></tr><tr><td>BBCU-L</td><td>×2</td><td>1/1</td><td>37.58</td><td>0.9590</td><td>33.18</td><td>0.9143</td><td>31.91</td><td>0.8962</td><td>31.12</td><td>0.9179</td><td>37.50</td><td>0.9746</td></tr><tr><td>FRBC (ours)</td><td>×2</td><td>1/1</td><td>37.71</td><td>0.9595</td><td>33.22</td><td>0.9141</td><td>31.95</td><td>0.8968</td><td>31.15</td><td>0.9184</td><td>37.90</td><td>0.9755</td></tr><tr><td>FRBC+ (ours)</td><td>×2</td><td>1/1</td><td>37.85</td><td>0.9600</td><td>33.32</td><td>0.9154</td><td>32.02</td><td>0.8977</td><td>31.29</td><td>0.9198</td><td>38.23</td><td>0.9762</td></tr><tr><td>Bicubic</td><td>×4</td><td>-/-</td><td>28.42</td><td>0.8104</td><td>26.00</td><td>0.7027</td><td>25.96</td><td>0.6675</td><td>23.14</td><td>0.6577</td><td>24.89</td><td>0.7866</td></tr><tr><td>SRResNet</td><td>×4</td><td>32/32</td><td>32.16</td><td>0.8951</td><td>28.60</td><td>0.7822</td><td>27.58</td><td>0.7364</td><td>26.11</td><td>0.7870</td><td>30.46</td><td>0.9089</td></tr><tr><td>BNN</td><td>×4</td><td>1/1</td><td>27.56</td><td>0.7896</td><td>25.51</td><td>0.6820</td><td>25.54</td><td>0.6466</td><td>22.68</td><td>0.6352</td><td>24.19</td><td>0.7670</td></tr><tr><td>DoReFa</td><td>×4</td><td>1/1</td><td>30.33</td><td>0.8601</td><td>27.40</td><td>0.7526</td><td>26.83</td><td>0.7104</td><td>24.29</td><td>0.7175</td><td>27.00</td><td>0.8470</td></tr><tr><td>Bi-Real</td><td>×4</td><td>1/1</td><td>27.75</td><td>0.7935</td><td>25.79</td><td>0.6879</td><td>25.59</td><td>0.6478</td><td>22.91</td><td>0.6450</td><td>24.57</td><td>0.7752</td></tr><tr><td>IRNet</td><td>×4</td><td>1/1</td><td>31.38</td><td>0.8835</td><td>28.08</td><td>0.7679</td><td>27.24</td><td>0.7227</td><td>25.21</td><td>0.7536</td><td>28.97</td><td>0.8863</td></tr><tr><td>BAM</td><td>×4</td><td>1/1</td><td>31.24</td><td>0.8780</td><td>27.97</td><td>0.7650</td><td>27.15</td><td>0.7190</td><td>24.95</td><td>0.7450</td><td>N/A</td><td>N/A</td></tr><tr><td>BTM</td><td>×4</td><td>1/1</td><td>31.43</td><td>0.8850</td><td>28.16</td><td>0.7706</td><td>27.29</td><td>0.7256</td><td>25.34</td><td>0.7605</td><td>29.19</td><td>0.8912</td></tr><tr><td>ReActNet</td><td>×4</td><td>1/1</td><td>31.54</td><td>0.8859</td><td>28.19</td><td>0.7705</td><td>27.31</td><td>0.7252</td><td>25.35</td><td>0.7603</td><td>29.25</td><td>0.8912</td></tr><tr><td>BBCU-L</td><td>×4</td><td>1/1</td><td>31.79</td><td>0.8905</td><td>28.38</td><td>0.7762</td><td>27.41</td><td>0.7303</td><td>25.62</td><td>0.7696</td><td>29.69</td><td>0.8992</td></tr><tr><td>FRBC (ours)</td><td>×4</td><td>1/1</td><td>31.83</td><td>0.8906</td><td>28.39</td><td>0.7763</td><td>27.41</td><td>0.7303</td><td>25.61</td><td>0.7693</td><td>29.71</td><td>0.8989</td></tr><tr><td>FRBC+ (ours)</td><td>×4</td><td>1/1</td><td>31.99</td><td>0.8927</td><td>28.48</td><td>0.7781</td><td>27.47</td><td>0.7319</td><td>25.73</td><td>0.7722</td><td>29.96</td><td>0.9018</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Table 1. Quantitative results in CNN based binarized image SR methods. SRResNet is used as the full-precision backbone. Bits (W/A) denote the bits of weights and activations. The best and second best results are colored with red and cyan.",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "et al., 2021), we use SRResNet (Ledig et al., 2017) as CNN SR backbone, binarize its body part, and name this version as FRBC. We further generalize our FRB to a lightweight Transformer SR backbone, SwinIR_S (Liang et al., 2021). We binarize SwinIR_S and name this version as FRBT. In addition, we use self-ensemble (Lim et al., 2017) to further enhance them and denote as FRBC+ and FRBT+.",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Training Strategy. In the training phase, same as previous work (Lim et al., 2017; Zhang et al., 2018a; Xin et al., 2020; Liang et al., 2021), we conduct data augmentation (random rotation by  $90^{\\circ}$ ,  $180^{\\circ}$ ,  $270^{\\circ}$  and horizontal flip). We train the model for 300K iterations. Each training batch extracts 32 image patches, whose size is  $64 \\times 64$ . We utilize Adam optimizer (Kingma & Ba, 2015) ( $\\beta_{1} = 0.9$ ,  $\\beta_{2} = 0.999$ , and  $\\epsilon = 10^{-8}$ ) during training. The initial learning rate  $2 \\times 10^{-4}$ , which is reduced by half at the  $250K$ -th iteration. PyTorch (Paszke et al., 2017) is employed to conduct all experiments with NVIDIA RTX A6000 GPUs.",
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+    {
+        "type": "text",
+        "text": "4.2. Main Comparisons",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "For CNN-based image SR networks, we choose SRResNet (Ledig et al., 2017) as the full-precision backbone. We then adopt different binary methods: BNN (Courbariaux et al., 2016), DoReFa (Zhou et al., 2016), Bi-Real (Liu et al., 2018), IRNet (Qin et al., 2020), BAM (Xin et al., 2020), BTM (Jiang et al., 2021), ReActNet (Liu et al., 2020), BBCU-L (Xia et al., 2023), and our FRBC.",
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+    },
+    {
+        "type": "text",
+        "text": "Quantitative Results. In Tab. 1, we provide Params, Ops,",
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+    },
+    {
+        "type": "text",
+        "text": "PSNR, and SSIM comparisons with others. When using the same CNN-based full-precision backbone SRResNet, our FRBC achieves comparable or better PSNR/SSIM scores with similar number of Params and Ops as others.",
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+    {
+        "type": "text",
+        "text": "Generalize to Transformer. For Transformer-based image SR networks, we choose the lightweight SwinIR_S (Liang et al., 2021) as the backbone. Due to the more challenging case in Transformer binarization and the performance observation in CNN-based methods, we only apply our FRB to binarize SwinIR_S and name this version as FRBT. We further provide quantitative results of our binarized Transformer baseline, FRBT. In Tab. 2, we can see FRBT reduces the Params and Ops obviously. But the performance gap between FRBT and SwinIR_S is larger than that between FRBC and SRResNet. It means that it is more challenging to binarize Transformer-based image SR networks. However, we investigate firstly the binary behavior in the image SR Transformer. We open the way to further improve binarization performance and narrow the performance gap between the binary and full-precision models.",
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+    },
+    {
+        "type": "text",
+        "text": "Compression Ratio. In Tab. 3, we provide the compression ratio and speedup in terms of Params and Ops respectively. We quantize full-precision CNN- and Transformer-based networks, SRResNet and SwinIR_S, which are stored with data type single precision floating point. Their model size (i.e., Params) and operations (i.e., Ops) can be reduced considerably. Following the practice in BBCU-L (Xin et al., 2020), we only binarize the weights and activations in the",
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+    },
+    {
+        "type": "header",
+        "text": "Flexible Residual Binarization for Image Super-Resolution",
+        "bbox": [
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+    },
+    {
+        "type": "page_number",
+        "text": "6",
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+    {
+        "type": "table",
+        "img_path": "images/8af9fc1f290eb14011a79e5bd2fb0c33c82a000278b6adc05197a663891947a0.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Method</td><td rowspan=\"2\">Scale</td><td rowspan=\"2\">Bits (W/A)</td><td colspan=\"2\">Set5</td><td colspan=\"2\">Set14</td><td colspan=\"2\">B100</td><td colspan=\"2\">Urban100</td><td colspan=\"2\">Manga109</td></tr><tr><td>PSNR</td><td>SSIM</td><td>PSNR</td><td>SSIM</td><td>PSNR</td><td>SSIM</td><td>PSNR</td><td>SSIM</td><td>PSNR</td><td>SSIM</td></tr><tr><td>SwinIR_S</td><td>×2</td><td>32/32</td><td>38.14</td><td>0.9611</td><td>33.86</td><td>0.9206</td><td>32.31</td><td>0.9012</td><td>32.76</td><td>0.9340</td><td>39.12</td><td>0.9783</td></tr><tr><td>FRBT (ours)</td><td>×2</td><td>1/1</td><td>37.69</td><td>0.9594</td><td>33.24</td><td>0.9148</td><td>31.96</td><td>0.8968</td><td>31.13</td><td>0.9184</td><td>37.90</td><td>0.9753</td></tr><tr><td>FRBT+ (ours)</td><td>×2</td><td>1/1</td><td>37.82</td><td>0.9598</td><td>33.32</td><td>0.9156</td><td>32.02</td><td>0.8976</td><td>31.26</td><td>0.9197</td><td>38.23</td><td>0.9762</td></tr><tr><td>SwinIR_S</td><td>×4</td><td>32/32</td><td>32.44</td><td>0.8976</td><td>28.77</td><td>0.7858</td><td>27.69</td><td>0.7406</td><td>26.47</td><td>0.7980</td><td>30.92</td><td>0.9151</td></tr><tr><td>FRBT (ours)</td><td>×4</td><td>1/1</td><td>31.79</td><td>0.8896</td><td>28.35</td><td>0.7757</td><td>27.41</td><td>0.7306</td><td>25.55</td><td>0.7681</td><td>29.68</td><td>0.8988</td></tr><tr><td>FRBT+ (ours)</td><td>×4</td><td>1/1</td><td>31.92</td><td>0.8913</td><td>28.43</td><td>0.7774</td><td>27.47</td><td>0.7320</td><td>25.65</td><td>0.7704</td><td>29.92</td><td>0.9016</td></tr></table>",
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+            "Urban100: img_024  $(\\times 4)$"
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+        "image_caption": [
+            "SwinIR_S"
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+    {
+        "type": "image",
+        "img_path": "images/bf25968d1781c440ad7222803517c638927a718d37d3093eb49f4ba61df8a7b5.jpg",
+        "image_caption": [
+            "Urban100: img_038  $(\\times 4)$",
+            "Urban100: img_095 (×4)",
+            "Figure 5. Visual comparison  $(\\times 4)$  with lightweight and binarized image SR networks on Urban100 dataset. SRResNet and SwinIR_S are full-precision and used as references. Our FRBC performs better than other binarized methods with the same backbone SRResNet."
+        ],
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+            "DoReFa",
+            "HQ"
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+            "Bicubic"
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+        "img_path": "images/3f603c07b95ab17104cc211609a396e8b7315462b9750a2051b976211543e4be.jpg",
+        "image_caption": [
+            "BNN"
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+    {
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+        "img_path": "images/c7f5509322a213b0bb3226a63840f4fe8830868ad76da8a2ff849f06876e46c2.jpg",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/609ae6af3ecd8531754a18fea60ceb3e363ae295834f4aafb7c26427c8293fdc.jpg",
+        "image_caption": [
+            "SwinIR_S"
+        ],
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+    },
+    {
+        "type": "table",
+        "img_path": "images/b449e61f60d3563a53b0517a52ecf30e80a0d44b38059749bdb82e5f5cdda186.jpg",
+        "table_caption": [
+            "Table 2. Quantitative results in Transformer based binarized image SR methods. We use SwinIR_S as the backbone. We find quantization of Transformer models causes a significant quality loss. This is an interesting problem for future work."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Method</td><td rowspan=\"2\">Bits (W/A)</td><td rowspan=\"2\">Params (K) (↓ Compr. Ratio)</td><td rowspan=\"2\">Ops (G) (↓ Compr. Ratio)</td><td colspan=\"2\">Urban100</td></tr><tr><td>PSNR</td><td>SSIM</td></tr><tr><td>SRResNet</td><td>32 / 32</td><td>1367 (0%)</td><td>85.4 (0%)</td><td>32.11</td><td>0.9282</td></tr><tr><td>FRBC (ours)</td><td>1 / 1</td><td>225 (↓ 83.5%)</td><td>18.6 (↓ 78.2%)</td><td>31.00</td><td>0.9164</td></tr><tr><td>SwinIR.S</td><td>32 / 32</td><td>910 (0%)</td><td>62.4 (0%)</td><td>32.76</td><td>0.9340</td></tr><tr><td>FRBT (ours)</td><td>1 / 1</td><td>95 (↓ 89.6%)</td><td>4.3 (↓ 93.1%)</td><td>31.02</td><td>0.9173</td></tr></table>",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Table 3. Compression ratio of SRResNet and SwinIR_S ( $\\times 2$ ). Bits (W/A) denote the bit number of weights and activations. We set the input size as  $3 \\times 320 \\times 180$  for Ops calculation. Our Transformer baseline FRBT performs better than the CNN one FRBC with a larger compression ratio. Performance drop is denoted as  $\\downarrow drop'$ .",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "body part module. But, we calculate the compression ratio and speedup over the whole model. Our FRBC and FRBT still achieve around  $80\\%$  compression ratio. The reconstruction performance could drop, but binary quantization can significantly save the model size and operations.",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Visual Results. In Fig. 5, we provide visual results of representative and recently leading methods with scale  $\\times 4$  in terms of some challenging cases. For each case, we compare",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "with several BNN methods, like BNN, DoReFa, Bi-Real, and ReActNet. Our FRBC obtains obviously better results than others on the same CNN-based SR backbone. On the other hand, we further consider full-precision models (i.e., SRResNet and SwinIR_S) and their corresponding binary counterparts (i.e., FRBC and FRBT). Their visual difference is small. These visual comparisons further demonstrate the effectiveness of our FRBC and FRBT, which is consistent with the observations in Tabs. 1 and 2.",
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+    {
+        "type": "text",
+        "text": "4.3. Ablation Study",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "To demonstrate the effectiveness of our contributions, we conduct ablation studies about second-order residual binarization (SRB) and Distillation-guided Binarization Training (DBT). To save training time and resources, we reduce the input size to  $48 \\times 48$  and train 200K iterations. We use SR-ResNet (Ledig et al., 2017) as the image SR backbone. We use the well-known and basic binary method DoReFa (Zhou",
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+    },
+    {
+        "type": "header",
+        "text": "Flexible Residual Binarization for Image Super-Resolution",
+        "bbox": [
+            300,
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+    {
+        "type": "page_number",
+        "text": "7",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/bba45f5ac3f417c6a62260a058b108a0e7427b042dcd1a78e2bd7ecc673372af.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Method</td><td colspan=\"2\">B100</td><td colspan=\"2\">Urban100</td><td colspan=\"2\">Manga109</td></tr><tr><td>PSNR</td><td>SSIM</td><td>PSNR</td><td>SSIM</td><td>PSNR</td><td>SSIM</td></tr><tr><td>DoReFa</td><td>31.25</td><td>0.8873</td><td>29.15</td><td>0.8929</td><td>35.66</td><td>0.9676</td></tr><tr><td>SRB</td><td>31.77</td><td>0.8939</td><td>30.56</td><td>0.9113</td><td>37.51</td><td>0.9739</td></tr><tr><td>DBT</td><td>31.26</td><td>0.8873</td><td>29.18</td><td>0.8929</td><td>35.77</td><td>0.9678</td></tr><tr><td>FRB</td><td>31.83</td><td>0.8948</td><td>30.74</td><td>0.9138</td><td>37.64</td><td>0.9744</td></tr></table>",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Table 4. Ablation study  $(\\times 2)$  about our proposed Second-order Residual Binarization (SRB), Distillation-guided Binarization Training (DBT), and flexible residual binarization (FRB). The SR backbone is SRResNet (Ledig et al., 2017).",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "et al., 2016) as a baseline. We then equip SRB or/and DBT to SRResNet and binarize it. We report PSNR/SSIM values on B100, Urban100, and Manga109 in Tab. 4.",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Second-order Residual Binarization (SRB). As a vanilla version of binary method, DoReFa (Zhou et al., 2016) has shown the basic SR performance. We conduct second-order residual binarization (SRB) for the weights in the computation unit. In Tab. 4, we can see our proposed SRB significantly boosts the performance of the binary network and reduces the performance drop. Our SRB achieves around  $0.4\\sim 1.8$  dB and  $0.0066\\sim 0.0184$  in terms of PSNR and SSIM. In image SR, residual learning or residual feature usually extracts high-frequency information, which contributes much to high-quality reconstruction. On the other hand, feature size usually is very large or has an arbitrary size, which consumes lots of computational resources. Instead, we turn to enhancing the representation capacity with SRB. This performance gain from SRB over DoReFa indicates that binarizing weights residually is an efficient way to reduce the performance gap in binary SR models.",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Distillation-guided Binarization Training (DBT). During the image SR network training, there are still full-precision weights for binarization. It is straightforward to utilize full-precision information as guidance. As shown in Tab. 4, using DBT would only increase the performance by marginal gains, except for Manga109 (i.e., 0.11 dB PSNR gain). Such an observation gives us two thoughts. (1). Our proposed DBT is effective in boosting the binary SR performance independently. This is mainly because DBT leads to better representation content alignment in the image SR process. (2). Knowledge distillation can hardly achieve notable gains without considering the specific property of image superresolution (SR). Then, we are inspired to jointly integrate SRB and DBT together by aiming to reconstruct more high-frequency information effectively.",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Flexible Residual Binarization (FRB). When we jointly train the SR network with reconstruction and distillation losses, we reach flexible residual binarization (FRB). Considering the whole data lines in Tab. 4, we find that our proposed FRB achieves even higher performance over the vanilla binary baseline DoReFa (Zhou et al., 2016), resulting in larger gains than those obtained by using SRB and DBT separately. These observations demonstrate that our pro",
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+    {
+        "type": "text",
+        "text": "posed FRB can well extract more valuable information (e.g., structure and texture information) with residual binarized weights and also transfer full-precision knowledge more or less to the binary image SR network.",
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+    {
+        "type": "text",
+        "text": "5. Discussions and Future Works",
+        "text_level": 1,
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+        "page_idx": 7
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+    {
+        "type": "text",
+        "text": "Theoretical vs. Practical Speedup. As demonstrated in XNOR-Net (Rastegari et al., 2016), the Params and Ops can be compressed tens of times compared to the original full precision model. However, in practice, the real training and/or inference time may not be accelerated accordingly. Our FRB provides acceleration potential, which would also need more efficient models and hardware implementations.",
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+    },
+    {
+        "type": "text",
+        "text": "Efficient Model and Hardware Design. The real-valued compact neural networks are still required and can further compensate for the performance in binarized SR networks. Such efficient models also need specific hardware design (e.g. FPGA, ASIC, CPU, and GPU implementations).",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "More Flexible Compression. In our investigations, we also find that binarization only is not enough for large image models. Because the compression ratio upper bound of binarization is limited. In this case, we believe a more flexible compression technique is more desired. For example, jointly compressing a large network by network pruning (e.g., structured and unstructured pruning) and quantization (e.g., binarization) simultaneously.",
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+    {
+        "type": "text",
+        "text": "6. Conclusion",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "In this work, we propose a flexible residual binarization (FRB) technique to dramatically reduce the parameters and operations of full-precision (i.e., 32-bit) image superresolution (SR) networks. To extract more high-frequency information for better image reconstruction, we propose a second-order residual binarization (SRB). Our proposed SRB binarizes (i.e., 1-bit) the residual weights, which has been demonstrated to be pretty effective over binarizing weights directly. At the same time, to make the binarized image SR model perform closer to its full-precision counterpart, we transfer full-precision knowledge to guide the training of binary SR networks. Specifically, we propose Distillation-guided Binarization Training (DBT), which uniformly aligns the contents of different bit-widths. We finally apply our FRB to binarize both CNN- and transformer-based SR methods, resulting in two binarized baselines: FRBC and FRBT. We conduct extensive ablation studies and main experiments to show the effectiveness of our proposed components. Surprisingly, we find that our FRBT obtains comparable or even better performance than FRBC with much fewer Params and Ops. To this end, our proposed FRB opens the way to compress BNNs with efficient hardware, like FPGA, CPU, and GPU.",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Acknowledgement. This work was supported by Shang-",
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+    },
+    {
+        "type": "header",
+        "text": "Flexible Residual Binarization for Image Super-Resolution",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "page_number",
+        "text": "8",
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+    },
+    {
+        "type": "text",
+        "text": "hai Municipal Science and Technology Major Project (2021SHZDZX0102), the Fundamental Research Funds for the Central Universities, National Science and Technology Major Project (2022ZD0116405), National Natural Science Foundation of China (No. 62306025, No. 92367204), and Huawei Technologies Oy (Finland) Project.",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Impact Statement",
+        "text_level": 1,
+        "bbox": [
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "This paper presents work whose goal is to advance the field of Machine Learning. There are many potential societal consequences of our work, none which we feel must be specifically highlighted here.",
+        "bbox": [
+            84,
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+    },
+    {
+        "type": "text",
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+# Flexible Residual Binarization for Image Super-Resolution
+
+Yulun Zhang *1 Haotong Qin *2 Zixiang Zhao 2 Xianglong Liu 3 Martin Danelljan 2 Fisher Yu 2
+
+# Abstract
+
+Binarized image super-resolution (SR) has attracted much research attention due to its potential to drastically reduce parameters and operations. However, most binary SR works binarize network weights directly, which hinders high-frequency information extraction. Furthermore, as a pixel-wise reconstruction task, binarization often results in heavy representation content distortion. To address these issues, we propose a flexible residual binarization (FRB) method for image SR. We first propose a second-order residual binarization (SRB), to counter the information loss caused by binarization. In addition to the primary weight binarization, we also binarize the reconstruction error, which is added as a residual term in the prediction. Furthermore, to narrow the representation content gap between the binarized and full-precision networks, we propose Distillation-guided Binarization Training (DBT). We uniformly align the contents of different bit widths by constructing a normalized attention form. Finally, we generalize our method by applying our FRB to binarize convolution and Transformer-based SR networks, resulting in two binary baselines: FRBC and FRBT. We conduct extensive experiments and comparisons with recent leading binarization methods. Our proposed baselines, FRBC and FRBT, achieve superior performance both quantitatively and visually.
+
+# 1. Introduction
+
+Given a full-precision low-resolution (LR) input, single image super-resolution (SR) aims to obtain its high-resolution (HR) counterpart by reconstructing more details. Essentially, image SR is ill-posed, as there exist multiple HR candidi
+
+*Equal contribution  $^{1}\mathrm{MoE}$  Key Lab of Artificial Intelligence, AI Institute, Shanghai Jiao Tong University, China  $^{2}\mathrm{ETH}$  Zürich, Switzerland  $^{3}\mathrm{Beihang}$  University, China. Correspondence to: Xianglong Liu <xlliu@buaa.edu.cn>.
+
+Proceedings of the  $41^{st}$  International Conference on Machine Learning, Vienna, Austria. PMLR 235, 2024. Copyright 2024 by the author(s).
+
+![](images/ce29b14d7be6cd48c20b7af8d409bb4bc47667a9999a223f01f4bbb7a7929ffa.jpg)  
+Urban100: img_001  
+Params (K) / Ops (G)
+
+![](images/c7f1f9a06166440abff87ed4d1c9068a6786c1cb479c4329a91d49acf9eb806e.jpg)  
+ReActNet 372 / 79.2
+
+![](images/00a6df1e5f861180ff1c28f7b95f9c6d98e2d51087d321afd61877600c3a44c2.jpg)  
+FRBC (ours) 372 / 79.2
+
+![](images/41a36a3289e79010607a8b55d1b68d07a429e0fbcc271165b9f725ab51b972f5.jpg)  
+FRBT (ours) 115 / 5.4  
+Figure 1. Visual samples of image SR  $(\times 4)$  by lightweight image SR methods. SRResNet (Ledig et al., 2017) is a full-precision (i.e., 32-bit) model and is used as a backbone for binarization (i.e., 1-bit) by BNN (Courbariaux et al., 2016), ReActNet (Liu et al., 2020), and our FRBC. We also binarize SwinIR_S (Liang et al., 2021) and denote it as FRBT. We provide the parameter (i.e., Params (K)) and operation numbers (i.e., Ops (G)). We set the input size as  $3\times 320\times 180$  for Ops calculation. Our FRBC and FRBT achieve better visual reconstruction than other binary ones. Our results are more faithful to that of the full-precision SRResNet.
+
+dates for the same LR input. To address this problem, deep convolutional neural networks (CNNs) and Transformers have been investigated for high-quality reconstructions (Dong et al., 2014; Kim et al., 2016; Lim et al., 2017; Zhang & Patel, 2018; Zhang et al., 2018b; Liang et al., 2021). However, most of them require extensive computational resources, which are usually not friendly for resource-limited devices. In those cases, neural network compression techniques are eagerly needed to significantly reduce model complexity.
+
+As one of the most promising network compression techniques, binary neural networks (BNNs), where both the network weights and activations are binarized (i.e., 1-bit binarization), are usually chosen for model deployment (Martinez et al., 2020; Rastegari et al., 2016). Theoretically, BNN enjoys  $32 \times$  parameter compression ratio and up to  $58 \times$  computation operation reduction (Rastegari et al., 2016). Such practical characteristics make BNN highly efficient for embedded devices (Ding et al., 2019) and friendly for memristor-based hardwares (Liu et al., 2020).
+
+Despite the above-mentioned advantages of BNN, the severe performance drop hinders it from being widely deployed (Liu et al., 2020). Such a problem is particularly critical in binarized image SR, where dense pixel-wise predictions are required and the feature size is usually very large. The performance drop mainly comes from two parts: weights and activations binarization. (1) The weights are binarized from full-precision (i.e., 32-bit) to 1-bit, being hard
+
+to extract high-frequency information. Even though the activations are full-precision, the SR output would still suffer from heavy degradation (Ma et al., 2019). (2) Binarizing activations (i.e., features) would directly lose high-frequency information, which is the key component that image SR networks try to reconstruct. Moreover, after the computation operations between binarized weights and activations, the output would further lose pixel-wise detailed information with high uncertainty, resulting in worse performance.
+
+To address those issues, we propose a flexible residual binarization (FRB) technique for binarized image SR. (1) To tackle the first issue, we try to reduce the weight error with our second-order residual binarization (SRB). Specifically, we not only binarize the weights as a common practice, we further binarize weight residuals between 1-bit and full-precision weights. Such an SRB practice helps preserve network weight representation capability more effectively than direct binarization only. (2) Furthermore, to compensate the pixel-wise information loss, we propose Distillation-guided Binarization Training (DBT). Specifically, we try to transfer full-precision knowledge to narrow the representation content gap between the binarized and full-precision networks. A normalized attention form is built to uniformly align the contents of different bit-widths.
+
+We further generalize our FRB to different types of networks and investigate its behaviors. Consequently, we apply our FRB to binarize CNN and Transformer based SR networks respectively, resulting in two binary baselines: FRBC and FRBT. Surprisingly, as shown in Fig. 1, our proposed methods achieve promising results with comparable or much smaller computational resources.
+
+Our main contributions are summarized as follows:
+
+- We propose a simple yet effective method Flexible Residual Binarization (FRB) to accurately binarize full-precision image SR networks during the training.  
+- We propose an effective second-order residual binarization (SRB), which binarizes the image SR network with its weight residuals. SRB enhances the representation capacity of the binarized image SR network significantly for pixel-wise reconstruction.  
+- We propose Distillation-guided Binarization Training (DBT), which transfers full-precision knowledge to the binarized model. Specifically, we build a normalized attention form to uniformly align the contents of different bit-widths (e.g., 32-bit and 1-bit).  
+- We employ our FRB to binarize CNN and Transformer based SR networks respectively, resulting in two binarized baselines: FRBC and FRBT. Our methods achieve superior performance over SOTA binarized SR methods quantitatively and visually.
+
+# 2. Related Work
+
+# 2.1.Lightweight Image SR
+
+Lightweight image SR models have recently drawn more and more attention because of their resource-friendly properties. Usually, researchers pursue lightweight networks by architecture design, neural architecture search (NAS), knowledge distillation (KD), and network pruning. Ahn et al. constructed a cascading method upon a residual network (CARN) (Ahn et al., 2018). Hui et al. proposed an information multi-distillation network (IMDN) (Hui et al., 2019). Meantime, model compression methods have been introduced for lightweight SR, too. Chu et al. introduced neural architecture search for image SR in FALSR (Chu et al., 2019). Knowledge distillation was employed to train lighter SR student networks (He et al., 2020; Lee et al., 2020). Using pretrained SR models, Zhang et al. incorporated channel pruning into image SR through aligned structured sparsity learning (ASSL) (Zhang et al., 2021b) or structure-regularized pruning (SRP) (Zhang et al., 2022). Such lightweight network designs and compression techniques have achieved promising performance. They either neglect the fine-grained parameter redundancy or consume a considerable number of additional computations.
+
+# 2.2. Model Quantization
+
+There are two main types of quantization methods: Post-Training Quantization (PTQ) and Quantization-Aware Training (QAT). PTQ has become increasingly popular due to its ability to quantize models without the need for retraining, resulting in numerous contributions in the field (Choukroun et al., 2019; Jhunjhunwala et al., 2021; Hubara et al., 2021; Li et al., 2021; Ding et al., 2022). However, this approach only relies on limited expert knowledge and minimal GPU resources to calibrate the model, which significantly restricts its potential for achieving extreme low-bit quantization. Fortunately, QAT provides us with the opportunity to utilize the entire training pipeline to achieve aggressive low-bit quantization, including 1-bit binarization, and demonstrates promising performance (Martinez et al., 2020; Qin et al., 2020; Liu et al., 2020; Zhou et al., 2016; Courbariaux et al., 2016; Rastegari et al., 2016). Qin et al. proposed low-bit quantization for image SR (Qin et al., 2023). This approach allows for more comprehensive model optimization, enabling the model to be trained to perform optimally in the quantized domain. QAT is usually seen as a powerful method for achieving extremely low-bit quantization.
+
+Recent studies, including (Wang et al., 2020; Simons & Lee, 2019; Wang et al., 2022; Zhang et al., 2021a; Qin et al., 2022), have demonstrated the effectiveness of 1-bit quantization, i.e., binarization, as a highly efficient form of network quantization. This binarization technique compresses networks to achieve extreme computational and storage efficiency by using 1-bit binarized parameters. Compared
+
+![](images/ba67d82697d0f69111a72e688cd8493733c9dc4016231746188200ed38b8d136.jpg)
+
+![](images/d62d10b78caca917bac8fa1c3f33002a61e38f3c798b51290e1974441437354b.jpg)  
+Figure 2. Overview of our Flexible Residual Binarization (FRB) for image SR networks. The upper (blue) is the Second-order Residual Binarization, where the SR network weights are binarized in a residual manner. The lower (orange) is the Distillation-guided Binarization Training that uniformly aligns the contents of different bit widths by constructing a normalized attention form.
+
+to floating-point models, these quantized models significantly reduce computation resources and save time, and are hardware-friendly for edge devices.
+
+# 2.3. Binary Neural Networks for Image SR
+
+Existing SR networks on resource-constrained devices are limited in usage by their high memory requirements and computational overhead. One major challenge is the heavy floating-point storage and operations. Thus room for compression still exists from a bit-width perspective, which gives a strong motivation for the study of 1-bit binarized SR models (Xin et al., 2020; Jiang et al., 2021; Xia et al., 2023). Xin et al. designed a bit-accumulation mechanism to binarize full-precision SR networks (Xin et al., 2020). Xia et al. proposed a basic binary convolution unit for binarized image restoration (Xia et al., 2023). However, they mainly work on binarization for CNNs and lack the investigation about Transformer based binarized SR models.
+
+# 3. Flexible Residual Binarization for Binarized Image Super-Resolution
+
+In this section, we first give an overview of binarization for single image super-resolution (SR) and raise the existing challenges of binarized (i.e., 1-bit) image SR networks. We then introduce our proposed flexible residual binarization (FRB) for image SR. Our FRB consists of two well-designed components: Second-order Residual Binarization (SRB) and Distillation-guided Binarization Training (DBT), which are designed for recovering the representation capacity and
+
+aligning the representation context, respectively. Afterward, we show how to utilize FRB for image SR and optimize the binary image SR network (Fig. 2). We finally give more details about implementation.
+
+# 3.1. Preliminaries: Binarization in SR
+
+Here, we give a brief background to the important key components in a general binarized image SR pipeline. Given a full-precision (i.e., 32-bit) low-resolution (LR) image as input  $I_{\mathrm{LR}}$ , the binary (i.e., 1-bit) super-resolution network aims to obtain its full-precision high-resolution (HR) counterpart  $I_{SR}$ . We formulate such a binarized image SR process with the neural network as follows
+
+$$
+I _ {S R} = \mathcal {F} _ {\mathrm {B S R}} \left(I _ {\mathrm {L R}}; \boldsymbol {\Theta}\right), \tag {1}
+$$
+
+where  $\mathcal{F}_{\mathrm{BSR}}(\cdot)$  denotes the binary super-resolution (BSR) network with trainable parameters  $\Theta$ . Specifically, we binarize the image SR network  $\mathcal{F}_{\mathrm{BSR}}(\cdot)$  by the sign function, which is a standard choice for the task. The forward operation is the standard sign function,
+
+$$
+\operatorname {s i g n} (\boldsymbol {x}) = \left\{ \begin{array}{l l} 1 & \text {i f} \boldsymbol {x} \geq 0 \\ - 1 & \text {o t h e r w i s e} \end{array} . \right. \tag {2}
+$$
+
+Since this standard sign function is not continuous or differentiable, its backward operation can hardly achieved directly. Instead, the backward is replaced by the approximation,
+
+$$
+\frac {\partial \text {s i g n}}{\partial \boldsymbol {x}} = \left\{ \begin{array}{l l} 1 & \text {i f} | \boldsymbol {x} | \leq 1 \\ 0 & \text {o t h e r w i s e} \end{array} . \right. \tag {3}
+$$
+
+A floating-point precision weight matrix  $\pmb{w}$  can thus be
+
+![](images/bb39328705e21f2de09730cfea926770130f87f7202482d91e4d5b3be739062a.jpg)  
+Figure 3. An computation example of our Second-order Residual Binarization (SRB). Our residual binarization allows the binarized weight representation to retain accurate information, further restoring the functionality of its full-precision counterpart compared to vanilla binarization. And the activation directly uses the sign function to binarize to avoid the extra burden during inference.
+
+binarized with the sign function as,
+
+$$
+\mathbf {B} _ {\boldsymbol {w}} = \alpha \operatorname {s i g n} (\boldsymbol {w}). \tag {4}
+$$
+
+A scaling factor  $\alpha$  is introduced to retain the magnitude of real-value weights. It is computed as
+
+$$
+\alpha = \frac {1}{n} \boldsymbol {w} ^ {\top} \operatorname {s i g n} (\boldsymbol {w}) = \frac {1}{n} \| \boldsymbol {w} \| _ {1}. \tag {5}
+$$
+
+After binarizing the SR networks, the storage size and computation can be significantly reduced due to the extremely reduced bit-width and highly efficient bitwise XNOR and bitcount operations (Rastegari et al., 2016). We then propose two techniques to improve binarized networks.
+
+# 3.2. Second-order Residual Binarization for Weight Error Reduction
+
+While binarization promises reduced storage and faster inference, it substantially reduces the capacity of the original weights. It causes serious challenges for binarized image SR networks. This can be captured in the error caused by binarizing the continuous weights in Eq. (4) as,
+
+$$
+\epsilon = \boldsymbol {w} - \mathbf {B} _ {\boldsymbol {w}}. \tag {6}
+$$
+
+The error  $\epsilon$  represents the residual information that is lost in the binarization operation. Intuitively, we want to reduce this error. While this could be done by increasing the number of bits in the discrete representation, it does not allow for the use of efficient binary network operations.
+
+In this work, we propose a different approach to reducing binarization errors. We perform a second-order binarization, in order to retrieve information lost in the error Eq. (6). This is performed by binarizing the error Eq. (6) and using it as a residual correction term to approximate continuous weights. Our binarization strategy is thus expressed as,
+
+$$
+\mathbf {B} _ {\boldsymbol {w 1}} = \alpha_ {1} \operatorname {s i g n} (\boldsymbol {w}), \quad \alpha_ {1} = \frac {1}{n} \| \boldsymbol {w} \| _ {1}, \tag {7}
+$$
+
+$$
+\mathbf {B} _ {\boldsymbol {w} \mathbf {2}} = \alpha_ {2} \operatorname {s i g n} (\boldsymbol {w} - \mathbf {B} _ {\boldsymbol {w} \mathbf {1}}), \alpha_ {2} = \frac {1}{n} \| \boldsymbol {w} - \mathbf {B} _ {\boldsymbol {w} \mathbf {1}} \| _ {1}. \tag {8}
+$$
+
+We refer to  $\mathbf{B}_{w1}$  and  $\mathbf{B}_{w2}$  as the first and second order binarization, respectively. Note that the scaling factors are computed using the same formula Eq. (5).
+
+In Eq. (7), the gradient estimation in the backward propagation for the sign function approximately follows Eq. (3). And for activation, the binarization operation follows the sign binarizer in Liu et al. (2020). Taking the binarized convolution unit as an example, the forward computation process of our second-order residual binarization (SRB) is expressed as,
+
+$$
+o = \operatorname {s i g n} (\boldsymbol {a}) \otimes \mathbf {B} _ {\boldsymbol {w 1}} + \operatorname {s i g n} (\boldsymbol {a}) \otimes \mathbf {B} _ {\boldsymbol {w 2}}, \tag {9}
+$$
+
+where the  $\otimes$  is the bitwise convolution consisting of XNOR and bitcount instructions (Arm, 2020; AMD, 2022). We also give an example of our technique in Fig. 3.
+
+Second-order residual binarization (SRB) preserves the representation capability of weights better than direct binarization, while still being able to use bitwise instructions for efficient computation. Moreover, residuals enhance the representation capacity of binarized weights by making them closer to the original values and more diverse in the output space. Such a property can significantly boost the performance of binarized image SR networks.
+
+# 3.3. Distillation-guided Binarization Training
+
+In addition to the decrease in network representation capacity, the high discretization of binarization also leads to severe content distortion of representations.
+
+On the other hand, since most image SR models are composed block-by-block (Liang et al., 2021; Lim et al., 2017), for image SR networks, the  $n$ -block  $\mathcal{F}_{\mathrm{BSR}}(\cdot)$  in Eq. (1) can then be reformulated as,
+
+$$
+I _ {S R} = \mathcal {F} _ {\mathrm {B S R}} \left(I _ {\mathrm {L R}}; \boldsymbol {\Theta}\right) = \prod_ {i = 1} ^ {n} B l k _ {\mathrm {B S R} _ {i}} \left(I _ {\mathrm {L R}}; \boldsymbol {\Theta}\right). \tag {10}
+$$
+
+Here,  $Blk_{\mathrm{BSR}_i}$  denotes the  $i$ -th inner block of the SR network composed of several binarized computation units, including binarized convolution and linear units. Correspondingly, full-precision models and blocks is denoted as  $\mathcal{F}_{\mathrm{SR}}(\cdot)$  and  $Blk_{\mathrm{SR}_i}$ . Lastly,  $\prod$  denotes the composition of blocks.
+
+Based on the above formulation and illustrations, the blocklevel ( $k$ -th block) representation distortion caused by binarization can be expressed as,
+
+$$
+\mathcal {D} _ {k} = \prod_ {i = 1} ^ {k} B l k _ {\mathrm {S R} _ {i}} \left(I _ {\mathrm {L R}}; \boldsymbol {\Theta}\right) - \prod_ {i = 1} ^ {k} B l k _ {\mathrm {B S R} _ {i}} \left(I _ {\mathrm {L R}}; \boldsymbol {\Theta}\right). \tag {11}
+$$
+
+To make the binarized SR model perform close to the full-precision level, intuitively, we should reduce the distortion  $\mathcal{D}_i$  of each block in the model.
+
+Therefore, we propose Distillation-guided Binarization Training (DBT) to align the representation content gap between binarized and full-precision SR networks (as Fig. 4). Inspired by (Martinez et al., 2020), we construct a normalized attention form for block-level representations to uniformly stabilize the contents in networks of different bit-widths. For example, the  $i$ -th block's formed representation
+
+![](images/07da1151ea09f08080e236140f5b2003e942a20672786826a89b2038442ac9c0.jpg)  
+Figure 4. The computation flow of the loss function considering DBT. During training, the training data is simultaneously fed into the binarized SR network and its well-trained full-precision counterpart, and  $\mathcal{L}_{\mathrm{DBT}}$  is calculated according to the block-level intermediate representation (such as Eq. (13)). In the end,  $\mathcal{L}_{\mathrm{DBT}}$  participates in the calculation of the total loss and jointly optimizes the binarized SR model with other loss items ( $\mathcal{L}_{\mathrm{PIX}}$  in Eq. (14)).
+
+in a binarized IR network can be formulated as
+
+$$
+R _ {\mathrm {B S R} _ {k}} = \frac {\left(\prod_ {i = 1} ^ {k} B l k _ {\mathrm {B S R} _ {i}} \left(I _ {\mathrm {L R}} ; \boldsymbol {\Theta}\right)\right) ^ {2}}{\left\| \left(\prod_ {i = 1} ^ {k} B l k _ {\mathrm {B S R} _ {i}} \left(I _ {\mathrm {L R}} ; \boldsymbol {\Theta}\right)\right) ^ {2} \right\| _ {\ell 2}}, \tag {12}
+$$
+
+where  $\| \cdot \|_{\ell 2}$  denotes the L2 normalization.
+
+Then we distill full-precision representations to binarized ones. We target to consistently push binarized presentations to approach full-precision level representations:
+
+$$
+\min  \mathcal {L} _ {\mathrm {D B T}} = \sum_ {i = 1} ^ {n} \hat {\mathcal {D}} _ {i} = \sum_ {i = 1} ^ {n} \| R _ {\mathrm {S R} _ {i}} - R _ {\mathrm {B S R} _ {i}} \| _ {\ell 2}. \tag {13}
+$$
+
+Note that the binarized SR model and the full-precision replica are a pair of natural teachers and students because they have exactly the same architecture and significant differences in computation/storage. We highlight that this fact makes our DBT a flexible and architecture-generic technique, and the blockwise distillation implementation can even be fine-grained to a single computing layer level to suit various architectures. Such a property allows us to practice our compression techniques on various CNN- and Transformer-based image SR networks.
+
+# 3.4. FRB for Image SR
+
+Binarized Architectures. For FRB, our proposed SRB technique is allowed to be flexibly applied to various computational units in the SR architecture, such as convolutional and linear units. Therefore, for the image SR architecture using FRB, we apply SRB binarization to all computing units in the body part, which is the most computationally intensive. We maintain the full precision of the head and tail parts. In addition, the ReLU function is replaced by PReLU following Martinez et al. (2020).
+
+# Algorithm 1 Flexible Residual Binarization for Image SR
+
+1: Input: Training dataset  $D$ , full-precision model  $\mathcal{F}_{\mathrm{SR}}$ , training iterations  $N$ ;  
+2: Output: The binarized model  $\mathcal{F}_{\mathrm{BSR}}$  
+3: Define the binarized  $\mathcal{F}_{\mathrm{BSR}}(\cdot)$  model by binarizing computation units of  $\mathcal{F}_{\mathrm{SR}}$  by the SRB as Eq. (7);  
+4: for iteration  $i$  in  $[0,N)$  do  
+5: Feed data  $D$  in full-precision model  $\mathcal{F}_{\mathrm{SR}}$  
+6: Feed data  $D$  in binarized model  $\mathcal{F}_{\mathrm{BSR}}$ ;  
+7: Calculate the loss following Eq. (13) and Eq. (14);  
+8: Optimize the binarized model  $\mathcal{F}_{\mathrm{BSR}}$  
+9: end for  
+10: Return  $\mathcal{F}_{\mathrm{BSR}}$
+
+SR Model Training. For the given training dataset  $D = \{I_{\mathrm{LR}}^i, I_{\mathrm{HR}}^i\}_{i=1}^K$  with  $K$  low-resolution inputs and their corresponding HR counterparts, the image SR model with our proposed FRD is optimized by minimizing both the conventional pixel-wise  $\mathcal{L}_{\mathrm{PIX}}$  loss and  $\mathcal{L}_{\mathrm{DBT}}$  distillation loss:
+
+$$
+\mathcal {L} _ {\mathrm {P I X}} = \frac {1}{n} \sum_ {i = 1} ^ {k} \left\| I _ {\mathrm {H R}} ^ {i} - I _ {\mathrm {S R}} ^ {i} \right\| _ {\ell 1}, \tag {14}
+$$
+
+$$
+\mathcal {L} _ {\text {t o t a l}} = \mathcal {L} _ {\mathbf {P I X}} + \beta \mathcal {L} _ {\mathbf {D B T}},
+$$
+
+where the  $\beta$  is a hyperparameter and is set as 1e-4 by default in our FRB, and  $\mathcal{L}_{\mathrm{DBT}}$  is in Eq. (13). Figure 4 also presents the computation flow of our training loss. The whole algorithm of our proposed flexible residual binarization for the image SR model can be presented in Algorithm 1.
+
+# 4. Experiments
+
+# 4.1. Settings
+
+Data. Following the common practice (Lim et al., 2017; Zhang et al., 2018a), we adopt DIV2K (Timofte et al., 2017) as the training data. Five benchmark datasets are used for testing: Set5 (Bevilacqua et al., 2012), Set14 (Zeyde et al., 2010), B100 (Martin et al., 2001), Urban100 (Huang et al., 2015), and Manga109 (Matsui et al., 2017).
+
+Evaluation. To evaluate the reconstruction performance, we calculate PSNR and SSIM (Wang et al., 2004) values on the Y channel of the YCbCr space. For model complexity evaluation, we follow (Rastegari et al., 2016) and report the model size and operations of BNN. Specifically, we calculate the BNN parameters via  $\text{Params}_-1 = \text{Params}_-f / 32$ , where  $\text{Params}_-f$  is the full-precision counterpart parameters. We calculate BNN operations via  $\text{Ops}_-1 = \text{Ops}_-f / 64$ , where  $\text{Ops}_-f$  denotes operations of the full-precision counterpart. Based on  $\text{Params}_-1$  and  $\text{Ops}_-1$ , we further provide theoretical compression ratios for parameters and operations.
+
+Proposed Binary Baselines. We apply our FRB to binarize CNN and Transformer based image SR baselines. Specifically, following BAM (Xin et al., 2020) and BTM (Jiang
+
+<table><tr><td rowspan="2">Method</td><td rowspan="2">Scale</td><td rowspan="2">Bits (W/A)</td><td colspan="2">Set5</td><td colspan="2">Set14</td><td colspan="2">B100</td><td colspan="2">Urban100</td><td colspan="2">Manga109</td></tr><tr><td>PSNR</td><td>SSIM</td><td>PSNR</td><td>SSIM</td><td>PSNR</td><td>SSIM</td><td>PSNR</td><td>SSIM</td><td>PSNR</td><td>SSIM</td></tr><tr><td>Bicubic</td><td>×2</td><td>-/-</td><td>33.66</td><td>0.9299</td><td>30.24</td><td>0.8688</td><td>29.56</td><td>0.8431</td><td>26.88</td><td>0.8403</td><td>30.80</td><td>0.9339</td></tr><tr><td>SRResNet</td><td>×2</td><td>32/32</td><td>38.00</td><td>0.9605</td><td>33.59</td><td>0.9171</td><td>32.19</td><td>0.8997</td><td>32.11</td><td>0.9282</td><td>38.56</td><td>0.9770</td></tr><tr><td>BNN</td><td>×2</td><td>1/1</td><td>32.25</td><td>0.9118</td><td>29.25</td><td>0.8406</td><td>28.68</td><td>0.8104</td><td>25.96</td><td>0.8088</td><td>29.16</td><td>0.9127</td></tr><tr><td>DoReFa</td><td>×2</td><td>1/1</td><td>36.76</td><td>0.9550</td><td>32.44</td><td>0.9072</td><td>31.31</td><td>0.8883</td><td>29.26</td><td>0.8945</td><td>35.81</td><td>0.9682</td></tr><tr><td>Bi-Real</td><td>×2</td><td>1/1</td><td>32.32</td><td>0.9123</td><td>29.47</td><td>0.8424</td><td>28.74</td><td>0.8111</td><td>26.35</td><td>0.8161</td><td>29.64</td><td>0.9167</td></tr><tr><td>IRNet</td><td>×2</td><td>1/1</td><td>37.27</td><td>0.9579</td><td>32.92</td><td>0.9115</td><td>31.76</td><td>0.8941</td><td>30.63</td><td>0.9122</td><td>36.77</td><td>0.9724</td></tr><tr><td>BAM</td><td>×2</td><td>1/1</td><td>37.21</td><td>0.9560</td><td>32.74</td><td>0.9100</td><td>31.60</td><td>0.8910</td><td>30.20</td><td>0.9060</td><td>N/A</td><td>N/A</td></tr><tr><td>BTM</td><td>×2</td><td>1/1</td><td>37.22</td><td>0.9575</td><td>32.93</td><td>0.9118</td><td>31.77</td><td>0.8945</td><td>30.79</td><td>0.9146</td><td>36.76</td><td>0.9724</td></tr><tr><td>ReActNet</td><td>×2</td><td>1/1</td><td>37.26</td><td>0.9579</td><td>32.97</td><td>0.9124</td><td>31.81</td><td>0.8954</td><td>30.85</td><td>0.9156</td><td>36.92</td><td>0.9728</td></tr><tr><td>BBCU-L</td><td>×2</td><td>1/1</td><td>37.58</td><td>0.9590</td><td>33.18</td><td>0.9143</td><td>31.91</td><td>0.8962</td><td>31.12</td><td>0.9179</td><td>37.50</td><td>0.9746</td></tr><tr><td>FRBC (ours)</td><td>×2</td><td>1/1</td><td>37.71</td><td>0.9595</td><td>33.22</td><td>0.9141</td><td>31.95</td><td>0.8968</td><td>31.15</td><td>0.9184</td><td>37.90</td><td>0.9755</td></tr><tr><td>FRBC+ (ours)</td><td>×2</td><td>1/1</td><td>37.85</td><td>0.9600</td><td>33.32</td><td>0.9154</td><td>32.02</td><td>0.8977</td><td>31.29</td><td>0.9198</td><td>38.23</td><td>0.9762</td></tr><tr><td>Bicubic</td><td>×4</td><td>-/-</td><td>28.42</td><td>0.8104</td><td>26.00</td><td>0.7027</td><td>25.96</td><td>0.6675</td><td>23.14</td><td>0.6577</td><td>24.89</td><td>0.7866</td></tr><tr><td>SRResNet</td><td>×4</td><td>32/32</td><td>32.16</td><td>0.8951</td><td>28.60</td><td>0.7822</td><td>27.58</td><td>0.7364</td><td>26.11</td><td>0.7870</td><td>30.46</td><td>0.9089</td></tr><tr><td>BNN</td><td>×4</td><td>1/1</td><td>27.56</td><td>0.7896</td><td>25.51</td><td>0.6820</td><td>25.54</td><td>0.6466</td><td>22.68</td><td>0.6352</td><td>24.19</td><td>0.7670</td></tr><tr><td>DoReFa</td><td>×4</td><td>1/1</td><td>30.33</td><td>0.8601</td><td>27.40</td><td>0.7526</td><td>26.83</td><td>0.7104</td><td>24.29</td><td>0.7175</td><td>27.00</td><td>0.8470</td></tr><tr><td>Bi-Real</td><td>×4</td><td>1/1</td><td>27.75</td><td>0.7935</td><td>25.79</td><td>0.6879</td><td>25.59</td><td>0.6478</td><td>22.91</td><td>0.6450</td><td>24.57</td><td>0.7752</td></tr><tr><td>IRNet</td><td>×4</td><td>1/1</td><td>31.38</td><td>0.8835</td><td>28.08</td><td>0.7679</td><td>27.24</td><td>0.7227</td><td>25.21</td><td>0.7536</td><td>28.97</td><td>0.8863</td></tr><tr><td>BAM</td><td>×4</td><td>1/1</td><td>31.24</td><td>0.8780</td><td>27.97</td><td>0.7650</td><td>27.15</td><td>0.7190</td><td>24.95</td><td>0.7450</td><td>N/A</td><td>N/A</td></tr><tr><td>BTM</td><td>×4</td><td>1/1</td><td>31.43</td><td>0.8850</td><td>28.16</td><td>0.7706</td><td>27.29</td><td>0.7256</td><td>25.34</td><td>0.7605</td><td>29.19</td><td>0.8912</td></tr><tr><td>ReActNet</td><td>×4</td><td>1/1</td><td>31.54</td><td>0.8859</td><td>28.19</td><td>0.7705</td><td>27.31</td><td>0.7252</td><td>25.35</td><td>0.7603</td><td>29.25</td><td>0.8912</td></tr><tr><td>BBCU-L</td><td>×4</td><td>1/1</td><td>31.79</td><td>0.8905</td><td>28.38</td><td>0.7762</td><td>27.41</td><td>0.7303</td><td>25.62</td><td>0.7696</td><td>29.69</td><td>0.8992</td></tr><tr><td>FRBC (ours)</td><td>×4</td><td>1/1</td><td>31.83</td><td>0.8906</td><td>28.39</td><td>0.7763</td><td>27.41</td><td>0.7303</td><td>25.61</td><td>0.7693</td><td>29.71</td><td>0.8989</td></tr><tr><td>FRBC+ (ours)</td><td>×4</td><td>1/1</td><td>31.99</td><td>0.8927</td><td>28.48</td><td>0.7781</td><td>27.47</td><td>0.7319</td><td>25.73</td><td>0.7722</td><td>29.96</td><td>0.9018</td></tr></table>
+
+Table 1. Quantitative results in CNN based binarized image SR methods. SRResNet is used as the full-precision backbone. Bits (W/A) denote the bits of weights and activations. The best and second best results are colored with red and cyan.
+
+et al., 2021), we use SRResNet (Ledig et al., 2017) as CNN SR backbone, binarize its body part, and name this version as FRBC. We further generalize our FRB to a lightweight Transformer SR backbone, SwinIR_S (Liang et al., 2021). We binarize SwinIR_S and name this version as FRBT. In addition, we use self-ensemble (Lim et al., 2017) to further enhance them and denote as FRBC+ and FRBT+.
+
+Training Strategy. In the training phase, same as previous work (Lim et al., 2017; Zhang et al., 2018a; Xin et al., 2020; Liang et al., 2021), we conduct data augmentation (random rotation by  $90^{\circ}$ ,  $180^{\circ}$ ,  $270^{\circ}$  and horizontal flip). We train the model for 300K iterations. Each training batch extracts 32 image patches, whose size is  $64 \times 64$ . We utilize Adam optimizer (Kingma & Ba, 2015) ( $\beta_{1} = 0.9$ ,  $\beta_{2} = 0.999$ , and  $\epsilon = 10^{-8}$ ) during training. The initial learning rate  $2 \times 10^{-4}$ , which is reduced by half at the  $250K$ -th iteration. PyTorch (Paszke et al., 2017) is employed to conduct all experiments with NVIDIA RTX A6000 GPUs.
+
+# 4.2. Main Comparisons
+
+For CNN-based image SR networks, we choose SRResNet (Ledig et al., 2017) as the full-precision backbone. We then adopt different binary methods: BNN (Courbariaux et al., 2016), DoReFa (Zhou et al., 2016), Bi-Real (Liu et al., 2018), IRNet (Qin et al., 2020), BAM (Xin et al., 2020), BTM (Jiang et al., 2021), ReActNet (Liu et al., 2020), BBCU-L (Xia et al., 2023), and our FRBC.
+
+Quantitative Results. In Tab. 1, we provide Params, Ops,
+
+PSNR, and SSIM comparisons with others. When using the same CNN-based full-precision backbone SRResNet, our FRBC achieves comparable or better PSNR/SSIM scores with similar number of Params and Ops as others.
+
+Generalize to Transformer. For Transformer-based image SR networks, we choose the lightweight SwinIR_S (Liang et al., 2021) as the backbone. Due to the more challenging case in Transformer binarization and the performance observation in CNN-based methods, we only apply our FRB to binarize SwinIR_S and name this version as FRBT. We further provide quantitative results of our binarized Transformer baseline, FRBT. In Tab. 2, we can see FRBT reduces the Params and Ops obviously. But the performance gap between FRBT and SwinIR_S is larger than that between FRBC and SRResNet. It means that it is more challenging to binarize Transformer-based image SR networks. However, we investigate firstly the binary behavior in the image SR Transformer. We open the way to further improve binarization performance and narrow the performance gap between the binary and full-precision models.
+
+Compression Ratio. In Tab. 3, we provide the compression ratio and speedup in terms of Params and Ops respectively. We quantize full-precision CNN- and Transformer-based networks, SRResNet and SwinIR_S, which are stored with data type single precision floating point. Their model size (i.e., Params) and operations (i.e., Ops) can be reduced considerably. Following the practice in BBCU-L (Xin et al., 2020), we only binarize the weights and activations in the
+
+<table><tr><td rowspan="2">Method</td><td rowspan="2">Scale</td><td rowspan="2">Bits (W/A)</td><td colspan="2">Set5</td><td colspan="2">Set14</td><td colspan="2">B100</td><td colspan="2">Urban100</td><td colspan="2">Manga109</td></tr><tr><td>PSNR</td><td>SSIM</td><td>PSNR</td><td>SSIM</td><td>PSNR</td><td>SSIM</td><td>PSNR</td><td>SSIM</td><td>PSNR</td><td>SSIM</td></tr><tr><td>SwinIR_S</td><td>×2</td><td>32/32</td><td>38.14</td><td>0.9611</td><td>33.86</td><td>0.9206</td><td>32.31</td><td>0.9012</td><td>32.76</td><td>0.9340</td><td>39.12</td><td>0.9783</td></tr><tr><td>FRBT (ours)</td><td>×2</td><td>1/1</td><td>37.69</td><td>0.9594</td><td>33.24</td><td>0.9148</td><td>31.96</td><td>0.8968</td><td>31.13</td><td>0.9184</td><td>37.90</td><td>0.9753</td></tr><tr><td>FRBT+ (ours)</td><td>×2</td><td>1/1</td><td>37.82</td><td>0.9598</td><td>33.32</td><td>0.9156</td><td>32.02</td><td>0.8976</td><td>31.26</td><td>0.9197</td><td>38.23</td><td>0.9762</td></tr><tr><td>SwinIR_S</td><td>×4</td><td>32/32</td><td>32.44</td><td>0.8976</td><td>28.77</td><td>0.7858</td><td>27.69</td><td>0.7406</td><td>26.47</td><td>0.7980</td><td>30.92</td><td>0.9151</td></tr><tr><td>FRBT (ours)</td><td>×4</td><td>1/1</td><td>31.79</td><td>0.8896</td><td>28.35</td><td>0.7757</td><td>27.41</td><td>0.7306</td><td>25.55</td><td>0.7681</td><td>29.68</td><td>0.8988</td></tr><tr><td>FRBT+ (ours)</td><td>×4</td><td>1/1</td><td>31.92</td><td>0.8913</td><td>28.43</td><td>0.7774</td><td>27.47</td><td>0.7320</td><td>25.65</td><td>0.7704</td><td>29.92</td><td>0.9016</td></tr></table>
+
+![](images/5f0d99a39e0e4c1b07a4a68bcead3b26330e4c508501a46ac42bc0cea09926ac.jpg)
+
+![](images/b64228adad28867bd6f6cb5cd7593031c9219d636cb5a9894d5948442ba9e01b.jpg)  
+HQ
+
+![](images/f24bd0af7053e4ad56c7238e3e470857a4940a4bc524e40e08d409158990888d.jpg)  
+Bicubic
+
+![](images/3750291b907f3b4952575787923f108eaf1fa66d63b6a06b24c2be24476632be.jpg)  
+BNN
+
+![](images/b17a4f37aa3d82e77240b5cbc272127e1055cd8491647a44d91b90e26d557fe7.jpg)  
+SRResNet
+
+![](images/259a58ec73e6da1ba452bee41e44521df3e91a648fae3c6889f017dce5b73f50.jpg)  
+SwinIR_S
+
+![](images/a99a2572fde03114eb68a07724c2da147bb5a5e2e1ddcb8e5e5be41ae0992798.jpg)  
+Urban100: img_024  $(\times 4)$
+
+![](images/374ebe58ba83477dcf5d4959a14333816e7348916c45bf8f64d48bcdcd3bbb4a.jpg)
+
+![](images/403a622b273a417916ea4282cfe3ce0dfc232563231ef88cd23814f5d10c8905.jpg)  
+Bicubic
+
+![](images/8d002e4c35c865d22ffc2339e915a63be1222216cefbd658c505d465c409fc27.jpg)  
+BNN
+
+![](images/5ca97ba59c0442017f52f0567139d062e025615d028565479f8d06bf1946f962.jpg)  
+SRResNet
+
+![](images/e2dd477b06c659d815052c0dd3d25fd12f91aa2f23e3ea3108e6b3d7399fef16.jpg)  
+SwinIR_S
+
+![](images/bf25968d1781c440ad7222803517c638927a718d37d3093eb49f4ba61df8a7b5.jpg)  
+Urban100: img_038  $(\times 4)$  
+Urban100: img_095 (×4)  
+Figure 5. Visual comparison  $(\times 4)$  with lightweight and binarized image SR networks on Urban100 dataset. SRResNet and SwinIR_S are full-precision and used as references. Our FRBC performs better than other binarized methods with the same backbone SRResNet.
+
+![](images/0de7cbb199b9bf15d6bb07d4d06ca23d3dab9d10bbf0e810d8b01a6cc6afa9fb.jpg)  
+DoReFa  
+HQ
+
+![](images/97edcb15e04fe4ecb4778144b4ea052ccb41252fae8a1eb119a0cd0de7751c30.jpg)  
+Bicubic
+
+![](images/3f603c07b95ab17104cc211609a396e8b7315462b9750a2051b976211543e4be.jpg)  
+BNN
+
+![](images/c7f5509322a213b0bb3226a63840f4fe8830868ad76da8a2ff849f06876e46c2.jpg)  
+SRResNet
+
+![](images/609ae6af3ecd8531754a18fea60ceb3e363ae295834f4aafb7c26427c8293fdc.jpg)  
+SwinIR_S
+
+Table 2. Quantitative results in Transformer based binarized image SR methods. We use SwinIR_S as the backbone. We find quantization of Transformer models causes a significant quality loss. This is an interesting problem for future work.  
+
+<table><tr><td rowspan="2">Method</td><td rowspan="2">Bits (W/A)</td><td rowspan="2">Params (K) (↓ Compr. Ratio)</td><td rowspan="2">Ops (G) (↓ Compr. Ratio)</td><td colspan="2">Urban100</td></tr><tr><td>PSNR</td><td>SSIM</td></tr><tr><td>SRResNet</td><td>32 / 32</td><td>1367 (0%)</td><td>85.4 (0%)</td><td>32.11</td><td>0.9282</td></tr><tr><td>FRBC (ours)</td><td>1 / 1</td><td>225 (↓ 83.5%)</td><td>18.6 (↓ 78.2%)</td><td>31.00</td><td>0.9164</td></tr><tr><td>SwinIR.S</td><td>32 / 32</td><td>910 (0%)</td><td>62.4 (0%)</td><td>32.76</td><td>0.9340</td></tr><tr><td>FRBT (ours)</td><td>1 / 1</td><td>95 (↓ 89.6%)</td><td>4.3 (↓ 93.1%)</td><td>31.02</td><td>0.9173</td></tr></table>
+
+Table 3. Compression ratio of SRResNet and SwinIR_S ( $\times 2$ ). Bits (W/A) denote the bit number of weights and activations. We set the input size as  $3 \times 320 \times 180$  for Ops calculation. Our Transformer baseline FRBT performs better than the CNN one FRBC with a larger compression ratio. Performance drop is denoted as  $\downarrow drop'$ .
+
+body part module. But, we calculate the compression ratio and speedup over the whole model. Our FRBC and FRBT still achieve around  $80\%$  compression ratio. The reconstruction performance could drop, but binary quantization can significantly save the model size and operations.
+
+Visual Results. In Fig. 5, we provide visual results of representative and recently leading methods with scale  $\times 4$  in terms of some challenging cases. For each case, we compare
+
+with several BNN methods, like BNN, DoReFa, Bi-Real, and ReActNet. Our FRBC obtains obviously better results than others on the same CNN-based SR backbone. On the other hand, we further consider full-precision models (i.e., SRResNet and SwinIR_S) and their corresponding binary counterparts (i.e., FRBC and FRBT). Their visual difference is small. These visual comparisons further demonstrate the effectiveness of our FRBC and FRBT, which is consistent with the observations in Tabs. 1 and 2.
+
+# 4.3. Ablation Study
+
+To demonstrate the effectiveness of our contributions, we conduct ablation studies about second-order residual binarization (SRB) and Distillation-guided Binarization Training (DBT). To save training time and resources, we reduce the input size to  $48 \times 48$  and train 200K iterations. We use SR-ResNet (Ledig et al., 2017) as the image SR backbone. We use the well-known and basic binary method DoReFa (Zhou
+
+<table><tr><td rowspan="2">Method</td><td colspan="2">B100</td><td colspan="2">Urban100</td><td colspan="2">Manga109</td></tr><tr><td>PSNR</td><td>SSIM</td><td>PSNR</td><td>SSIM</td><td>PSNR</td><td>SSIM</td></tr><tr><td>DoReFa</td><td>31.25</td><td>0.8873</td><td>29.15</td><td>0.8929</td><td>35.66</td><td>0.9676</td></tr><tr><td>SRB</td><td>31.77</td><td>0.8939</td><td>30.56</td><td>0.9113</td><td>37.51</td><td>0.9739</td></tr><tr><td>DBT</td><td>31.26</td><td>0.8873</td><td>29.18</td><td>0.8929</td><td>35.77</td><td>0.9678</td></tr><tr><td>FRB</td><td>31.83</td><td>0.8948</td><td>30.74</td><td>0.9138</td><td>37.64</td><td>0.9744</td></tr></table>
+
+Table 4. Ablation study  $(\times 2)$  about our proposed Second-order Residual Binarization (SRB), Distillation-guided Binarization Training (DBT), and flexible residual binarization (FRB). The SR backbone is SRResNet (Ledig et al., 2017).
+
+et al., 2016) as a baseline. We then equip SRB or/and DBT to SRResNet and binarize it. We report PSNR/SSIM values on B100, Urban100, and Manga109 in Tab. 4.
+
+Second-order Residual Binarization (SRB). As a vanilla version of binary method, DoReFa (Zhou et al., 2016) has shown the basic SR performance. We conduct second-order residual binarization (SRB) for the weights in the computation unit. In Tab. 4, we can see our proposed SRB significantly boosts the performance of the binary network and reduces the performance drop. Our SRB achieves around  $0.4\sim 1.8$  dB and  $0.0066\sim 0.0184$  in terms of PSNR and SSIM. In image SR, residual learning or residual feature usually extracts high-frequency information, which contributes much to high-quality reconstruction. On the other hand, feature size usually is very large or has an arbitrary size, which consumes lots of computational resources. Instead, we turn to enhancing the representation capacity with SRB. This performance gain from SRB over DoReFa indicates that binarizing weights residually is an efficient way to reduce the performance gap in binary SR models.
+
+Distillation-guided Binarization Training (DBT). During the image SR network training, there are still full-precision weights for binarization. It is straightforward to utilize full-precision information as guidance. As shown in Tab. 4, using DBT would only increase the performance by marginal gains, except for Manga109 (i.e., 0.11 dB PSNR gain). Such an observation gives us two thoughts. (1). Our proposed DBT is effective in boosting the binary SR performance independently. This is mainly because DBT leads to better representation content alignment in the image SR process. (2). Knowledge distillation can hardly achieve notable gains without considering the specific property of image superresolution (SR). Then, we are inspired to jointly integrate SRB and DBT together by aiming to reconstruct more high-frequency information effectively.
+
+Flexible Residual Binarization (FRB). When we jointly train the SR network with reconstruction and distillation losses, we reach flexible residual binarization (FRB). Considering the whole data lines in Tab. 4, we find that our proposed FRB achieves even higher performance over the vanilla binary baseline DoReFa (Zhou et al., 2016), resulting in larger gains than those obtained by using SRB and DBT separately. These observations demonstrate that our pro
+
+posed FRB can well extract more valuable information (e.g., structure and texture information) with residual binarized weights and also transfer full-precision knowledge more or less to the binary image SR network.
+
+# 5. Discussions and Future Works
+
+Theoretical vs. Practical Speedup. As demonstrated in XNOR-Net (Rastegari et al., 2016), the Params and Ops can be compressed tens of times compared to the original full precision model. However, in practice, the real training and/or inference time may not be accelerated accordingly. Our FRB provides acceleration potential, which would also need more efficient models and hardware implementations.
+
+Efficient Model and Hardware Design. The real-valued compact neural networks are still required and can further compensate for the performance in binarized SR networks. Such efficient models also need specific hardware design (e.g. FPGA, ASIC, CPU, and GPU implementations).
+
+More Flexible Compression. In our investigations, we also find that binarization only is not enough for large image models. Because the compression ratio upper bound of binarization is limited. In this case, we believe a more flexible compression technique is more desired. For example, jointly compressing a large network by network pruning (e.g., structured and unstructured pruning) and quantization (e.g., binarization) simultaneously.
+
+# 6. Conclusion
+
+In this work, we propose a flexible residual binarization (FRB) technique to dramatically reduce the parameters and operations of full-precision (i.e., 32-bit) image superresolution (SR) networks. To extract more high-frequency information for better image reconstruction, we propose a second-order residual binarization (SRB). Our proposed SRB binarizes (i.e., 1-bit) the residual weights, which has been demonstrated to be pretty effective over binarizing weights directly. At the same time, to make the binarized image SR model perform closer to its full-precision counterpart, we transfer full-precision knowledge to guide the training of binary SR networks. Specifically, we propose Distillation-guided Binarization Training (DBT), which uniformly aligns the contents of different bit-widths. We finally apply our FRB to binarize both CNN- and transformer-based SR methods, resulting in two binarized baselines: FRBC and FRBT. We conduct extensive ablation studies and main experiments to show the effectiveness of our proposed components. Surprisingly, we find that our FRBT obtains comparable or even better performance than FRBC with much fewer Params and Ops. To this end, our proposed FRB opens the way to compress BNNs with efficient hardware, like FPGA, CPU, and GPU.
+
+Acknowledgement. This work was supported by Shang-
+
+hai Municipal Science and Technology Major Project (2021SHZDZX0102), the Fundamental Research Funds for the Central Universities, National Science and Technology Major Project (2022ZD0116405), National Natural Science Foundation of China (No. 62306025, No. 92367204), and Huawei Technologies Oy (Finland) Project.
+
+# Impact Statement
+
+This paper presents work whose goal is to advance the field of Machine Learning. There are many potential societal consequences of our work, none which we feel must be specifically highlighted here.
+
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+# Ruisi Cai $^{1,2}$  Saurav Muralidharan $^{1}$  Greg Heinrich $^{1}$  Hongxu Yin $^{1}$  Zhangyang Wang $^{2}$  Jan Kautz $^{1}$  Pavlo Molchanov $^{1}$
+
+# Abstract
+
+Training modern LLMs is extremely resource intensive, and customizing them for various deployment scenarios characterized by limited compute and memory resources through repeated training is impractical. In this paper, we introduce FLEXTRON, a network architecture and post-training model optimization framework supporting flexible model deployment. The FLEXTRON architecture utilizes a nested elastic structure to rapidly adapt to specific user-defined latency and accuracy targets during inference with no additional fine-tuning required. It is also input-adaptive, and can automatically route tokens through its sub-networks for improved performance and efficiency. We present a sample-efficient training method and associated routing algorithms for systematically transforming an existing trained LLM into a FLEXTRON model. We evaluate FLEXTRON on the GPT-3 and LLama-2 family of LLMs, and demonstrate superior performance over multiple end-to-end trained variants and other state-of-the-art elastic networks, all with a single pretraining run that consumes a mere  $7.63\%$  tokens compared to original pretraining.
+
+# 1. Introduction
+
+Large language models (LLMs) have revolutionized real-world natural language processing applications and have showed impressive proficiency in understanding difficult contexts (Brown et al., 2020; OpenAI et al., 2023; Wei et al., 2022; Touvron et al., 2023). Nonetheless, the substantial size of these models, typically running into several billion parameters, imposes significant constraints on their utilization in scenarios characterized by limited memory and computational resources. To address this limitation, model
+
+![](images/c88ee98c9b3cb971896afc2b7780d3e81969deb39e4e4f49c70ae2b3243e5300.jpg)  
+Multiple Models in One during deployment
+
+![](images/e0b5519bcad7d2c8ea0b5c9c07816f4b3b3a7a85e03bcb5b335f2eb6d902ac9c.jpg)  
+Converting trained LLM into Flextron:  
+Figure 1. High-level overview of the FLEXTRON framework. The top half of the figure demonstrates how we convert a trained LLM into an elastic network with input-adaptive routing. As shown in the bottom half of the Figure, FLEXTRON enables fast, zero-shot generation of hardware and input-adaptive sub-networks targeting various accuracy, latency and parameter constraints.
+
+providers typically train multiple model variants for users to choose from (depending on system memory and computational constraints) before trying to find model(s) satisfying the trade-off between efficiency and accuracy. For instance, the Llama-2 model family (Touvron et al., 2023) includes three different variants with 7 billion, 13 billion, and 70 billion parameters, while the Pythia family (Biderman et al., 2023) offers a selection of eight models with sizes ranging from 80 million to 12 billion parameters.
+
+Training multiple multi-billion parameter models is demanding in time, data, and resources. Adopting a single, customizable model with multiple sub-networks for varied budgets, as seen in Once-for-all (Cai et al., 2019), SortedNet (Valipour et al., 2023), Matformer (Kudugunta et al.,
+
+2023) and (Stamoulis et al., 2019), simplifies this. These models typically use a supernet with elastic, nested components, but require non-standard, costly architectures with even longer training than a single model.
+
+Mixture-of-Expert (MoE) networks, while more efficient than dense models (Fedus et al., 2022; Riquelme et al., 2021; Jiang et al., 2024), are generally restricted to feedforward layers and fixed budgets. The Pathways architecture (Dean, 2021; Zhou et al., 2022) highlights the potential of heterogeneous expert networks. We advocate for input-adaptive sub-network selection of different sizes to maximize performance and efficiency.
+
+In this paper, we present FLEXTRON, a network architecture and a post-training model optimization framework that takes the best from MoEs, elastic models and dynamic inference. The architecture extends the idea of MoE to attention and feed forward layers. Experts are heterogeneous and have different sizes via a nested elastic structure to support efficient model storage, memory bandwidth savings and ease-of-use. Particular experts are selected via a router conditioned on input data and target deployment constraints. FLEXTRON is a single model that provides Multiple Models in One during deployment with no additional finetuning. Finally, we present a framework where a standard trained LLM such as GPT-3 and Llama-2 can be efficiently converted to FLEXTRON while using a small fraction of the training time. Figure 1 provides a high-level overview.
+
+We found that training a router that allows adaptive computation is challenging due to gradient vanishing. Similar issues arise in normal MoE training, known as expert collapse (Chi et al., 2022), where routers constantly pick the same path or learns similar experts. To address this issue, we propose to train a Surrogate Model (SM) that predicts an LLM's language loss value given only router choices; once trained, we freeze it and tune routers to minimize the language loss solely on SM feedback.
+
+This paper makes the following contributions:
+
+- A novel architecture, called FLEXTRON, that flexibly adapts to different latency and accuracy targets during inference with no additional fine-tuning.  
+- A post-training optimization framework for systematically transforming existing trained LLMs into dynamic (input-adaptive) elastic networks.  
+- New static and dynamic routing algorithms that automatically select the optimal sub-network given a latency target and/or input token. We introduce a novel surrogate model for effective training of our routers.  
+- An efficient sampling-based training method for elastic networks that requires significantly less compute than existing methods.
+
+# 2. Background and Notation
+
+Given a model with  $N$  layers, each layer can be formalized as  $\mathbf{Y}_i = f_i(\mathbf{X}_i, \mathbf{W}_i)$ , where  $i \in [1, N]$  refers to the layer index,  $\mathbf{X}_i$  denotes the layer input, with dimensions of  $B \times C$  representing batch  $\times$  embedding dimension, and  $\mathbf{W}_i$  denotes the parameters of the layer. We define an elastic network as one that can flexibly adapt its layers to target specific user-defined objectives such as latency, memory, accuracy, etc. In this paper, we define each layer of an elastic network as follows:  $\mathbf{Y}_i = f_i(\mathbf{X}_i, \mathbf{W}_i^j)$  where each  $\mathbf{W}_i^j$ ,  $j \in [1, K]$  represents a different parameter matrix for the same operation  $f_i$  for layer  $i$ . By substituting the original layer with a candidate layer, we are able to generate an exponential number of elastic sub-networks ( $K^N$  choices for the formulation above, assuming  $K$  candidates per layer), each with different runtime and accuracy characteristics.
+
+Elastic Multi-Layer Perceptron (MLP). FLEXTRON models utilize a nested structure for elastic MLP layers, inspired by the Matformer work (Kudugunta et al., 2023). Nesting enables hidden neurons to be shared between layerwise candidates using simple indexing operations, saving memory and improving efficiency. Formally, elastic MLP candidates with 2 linear layers have the following format:
+
+$$
+\operatorname {M L P} ^ {j} (x) = \sigma \left(\mathbf {X} \cdot \left(\mathbf {I} _ {d _ {j}} \boldsymbol {W} ^ {(1)}\right) ^ {T}\right) \cdot \left(\mathbf {I} _ {d _ {j}} \boldsymbol {W} ^ {(2)}\right), \tag {1}
+$$
+
+where,  $\mathbf{I}_{d_j}$  is a diagonal matrix of size  $D\times D$  where the first  $d_{j}$  diagonal elements being 1 and the rest being 0, with  $D$  being the maximum hidden dimension. In this way, the  $j^{th}$  MLP candidate will only utilize the first  $d_{j}$  hidden neurons from the corresponding shared matrices  $\pmb{W}$ .  $\pmb{W}^{(1)}$  and  $\pmb{W}^{(2)}$  are the associated two weight matrices in MLP layers, with  $\pmb{W}^{(1)},\pmb{W}^{(2)}\in \mathbb{R}^{D\times C}$ ;  $\sigma (\cdot)$  refers to the non-linear activation function. For implementation, the diagonal matrix  $\mathbf{I}$  can be replaced with a slicing operator that selects only the first  $d_{j}$  rows:  $\mathbf{I}_{d_j}\pmb{W}^{(1)} = \pmb{W}^{(1)}[0:d_j,:]$ . We constrain  $d_1 < d_2 < \ldots < d_K$ , where  $d_{K} = D$ , to formulate the nested structure of  $K$  experts. Note that the MLP can take a more complex form when employing SwiGLU activation.
+
+Elastic Multi-Head Attention (MHA). MHA layers constitute a significant proportion of LLM runtime and memory usage (for KV cache), and making them elastic will improve overall efficiency. To the best of our knowledge, FLEXTRON is the first work that supports both elastic MLP and elastic MHA layers, enabling a richer candidate operation search space. An elastic MHA candidate uses a subset of attention heads. Formally, given hidden states  $\mathbf{X}$ , we define elastic MHA as follows:
+
+$$
+\begin{array}{r l} \mathrm {M H A} ^ {j} (x) = \operatorname {C o n c a t} \left(\text {h e a d} _ {1}, \dots \text {h e a d} _ {d _ {j}}\right) \cdot \left(\mathbf {I} _ {d _ {j} H} \boldsymbol {W} ^ {O}\right), \\ \text {h e a d} _ {i} = \operatorname {A t t n} \left(\boldsymbol {\mathbf {X}} \boldsymbol {\mathbf {W}} ^ {Q, i}, \boldsymbol {\mathbf {X}} \boldsymbol {\mathbf {W}} ^ {K, i}, \boldsymbol {\mathbf {X}} \boldsymbol {\mathbf {W}} ^ {V, i}\right), \end{array} \tag {2}
+$$
+
+where,  $\mathbf{I}_{d_jH}$  is a diagonal matrix with the first  $d_{j}H$  elements being 1, and the rest are 0s;  $d_{j}$  - number of heads selected,  $H$  - size of a single head,  $L$  - total number of heads;  $\pmb {W}^{Q,i},\pmb{W}^{K,i},\pmb{W}^{V,i}\in \mathbb{R}^{H\times C}$  and  $\pmb {W}^O\in \mathbb{R}^{LH\times C}$ . Different heads can be computed/selected via weight slicing.
+
+# 3. FLEXTRON Framework
+
+We now describe the elastic network continued-tuning process, and provide more details on automatic sub-network selection from the trained elastic network.
+
+# 3.1. Elastic Network Continued-Tuning
+
+We start the elastic continued-tuning process by taking an existing trained LLM and performing importance ranking for each neuron/head. Here, using a small set of data samples, we compute the importance of each neuron/head based on the accumulated magnitude of activations. For MHA layers, the importance of each head is calculated as
+
+$$
+F _ {\text {h e a d}} ^ {(i)} = \sum_ {\mathbf {X}} \| \operatorname {A t t n} \left(\mathbf {X} \boldsymbol {W} ^ {Q, i}, \mathbf {X} \boldsymbol {W} ^ {K, i}, \mathbf {X} \boldsymbol {W} ^ {V, i}\right) \| _ {1}. \tag {3}
+$$
+
+For MLP layers:
+
+$$
+F _ {\text {n e u r o n}} ^ {(i)} = \sum_ {\mathbf {X}} \| \mathbf {X} \left(\boldsymbol {W} ^ {(1), r}\right) ^ {T} \| _ {1}, \tag {4}
+$$
+
+here  $W^{(1),r}$  refers to the  $r^{\mathrm{th}}$  row of the weight matrix  $W^{(1)}$ . In practice, only a small dataset comprising 512 samples is sufficient (see Section 4 for more details). Once importance is computed, we permute the respective weight matrices in the MLP and MHA layers such that neurons/heads are stored in decreasing order of importance for every individual layer. Sub-networks can now be constructed by simply indexing the first several neurons/heads in each layer, thus preserving essential knowledge encoded in important channels. In this way, we construct nested elastic layers with parameter sharing, with channels/heads sorted by importance, such that the first channels are the most important.
+
+Next, we train all elastic network candidates simultaneously using a combined loss term as in (Kudugunta et al., 2023). Since the number of such candidates can be prohibitively large (for example, there are  $4^{64}$  possible combinations for the 32-layer LLaMa2-7B model (Touvron et al., 2023)), we randomly sample a smaller subset of  $k$  networks from the candidate pool to keep the total pretraining time tractable. Specifically, we randomly generate a one-hot vector  $s_i$  for each layer  $i$  and use it to construct a candidate network  $\mathcal{M}_j$ ; here,  $s_i \in \mathbb{R}^{K_i}$  and  $K_i$  represents the number of candidate MLP/MHA operations in layer  $i$ .  $\mathcal{M}_j$  is the random model indexed by  $j$ , where  $j \in [0, K - 1]$ . The training loss is:
+
+$$
+\mathcal {L} _ {\text {j o i n t}} = \sum_ {j = 0} ^ {k - 1} \mathcal {L} \left(\mathcal {M} _ {j} (\mathbf {x}), \mathbf {y}\right), \tag {5}
+$$
+
+![](images/8e3149e457e367182d46f14c6adb5ae6751d99d2888b360177b5fd001375d8b0.jpg)  
+Figure 2. Illustration of the elastic continued-tuning phase.
+
+Figure 2 provides an overview of elastic continued-tuning with random sampling. We provide additional details on pretraining, including choice of hyper-parameter values and datasets, in Section 4.
+
+# 3.2. Automatic Network Selection
+
+Given a large number of possible sub-networks to choose from, each with different latency, parameter, and accuracy trade-offs, a natural question arises: can we automatically determine Pareto-optimal sub-networks given specific constraints? In this section, we introduce FLEXTRON's router architecture and describe how it helps us automatically select optimal sub-networks for a given constraint.
+
+The problem can be formalized as follows:
+
+$$
+\min  _ {\boldsymbol {S} _ {t}} \sum_ {t} \mathcal {L} _ {C E} \left(\mathcal {M} _ {s _ {t}}\right), \quad \text {s . t . L a t e n c y} \left(\mathcal {M} _ {s _ {t}}\right) \leq T _ {t}, \tag {6}
+$$
+
+$$
+\mathcal {M} _ {s _ {t}} = \mathcal {G} (\mathcal {M}, \boldsymbol {S} _ {T _ {t}}),
+$$
+
+where  $\mathcal{M}$  is original network topology,  $T_{t}$  refers to a latency constraint of index  $t$ , and  $S_{T_t}$  denotes the related selection matrix;  $\mathcal{M}_{s_t}$  defines the selected topology based on latency constraint,  $\mathcal{G}(\cdot)$  is a function for selecting network topology;  $\mathcal{L}_{CE}$  refers to the cross-entropy loss. We use a Lagrange multiplier and impose a constraint to convert the aforementioned optimization problem into directly minimizing the following loss term:
+
+$$
+\mathcal {L} = \sum_ {t} \mathcal {L} _ {C E} \left(\mathcal {M} _ {s _ {t}}\right) + \lambda \cdot \mathcal {T} _ {T} \left(\mathcal {M} _ {s _ {t}}\right), \tag {7}
+$$
+
+where  $\mathcal{T}_T$  represents the target constraint loss. In what follows as an example, we explain latency loss between the constraint  $T_{t}$  and actual model latency  $\mathrm{Latency}(\mathcal{M}_{s_t})$ :
+
+$$
+\mathcal {T} _ {T} \left(\mathcal {M} _ {s _ {t}}\right) = \sum_ {t} \max  (\text {L a t e n c y} \left(\mathcal {M} _ {s _ {t}}\right) - T _ {t}, 0) \tag {8}
+$$
+
+Note that the loss can also be extended to other constraints such as GPU memory as we show later in our experiments.
+
+The model requires an architecture selection mechanism to support multiple budgets with maximized accuracy. Inspired
+
+![](images/3dd267811e779064ccb8207cfa009f000dbb330da7c914ad3728f683c116a7ff.jpg)  
+Figure 3. Illustration of how routers are trained via a Surrogate Model (SM). The Surrogate Model is trained to approximate the LLM language loss value given only routers logits. If the error of the SM is smaller than a predefined threshold, the routers are updated. Updates are based on (i) the latency loss, ensuring the requested latency matches the real overall latency via a Lookup Table (LUT), and (ii) the loss from minimization of the SM output. The SM serves as a proxy for the full model's language loss and allows for simpler backpropagation due to its smaller size. Once the routers are trained, we discard the SM and finetune the LLM and routers jointly.
+
+by MoEs, we use routers. We define two routing scenarios: static, where the output depends only on the input latency; and dynamic, where it is additionally conditioned on the hidden state. We observe that training routers, even after the elastic continued-tuning stage, is challenging due to limited gradient propagation from the final model's output loss. As a remedy, we propose using a surrogate model to predict the LLM's performance based solely on router outputs. Given this prediction, routers can be trained to minimize the expected LM loss of the surrogate model. We provide additional details on the surrogate model in the following sections.
+
+FLEXTRON-Static: Static Model Selection. We first tackle the problem of static model selection, which refers to automatically selecting sub-networks given only a target latency  $T$  (no input-adaptivity). Here, we insert layer-wise learnable routers; each router takes the latency requirement  $T$  as input and outputs the choice  $h_i$  for layer  $i$ , thereby deciding the number of channels/heads to be used for that layer via expert groups. The router picks the expert with the following formulation:
+
+$$
+s _ {i} = \operatorname {a r g m a x} \left(\mathcal {R} _ {i} (T)\right), \tag {9}
+$$
+
+where  $\mathcal{R}$  is a small MLP that embeds a scalar value  $T$  (latency) into logits of the size of the predefined set of expert candidates (in our paper selected to be 4).
+
+To provide a strong and stable signal to the router, we propose to use a Surrogate Model (SM). Its task is to predict the value of the full LLM language loss given only logits at the outputs of the routers. It becomes a proxy for the full model output error. Once it is learned, we can optimize routers to minimize the output of the SM. The basic idea is to use the SM as a loss term that can be minimized. The SM is defined
+
+as a two layer MLP:
+
+$$
+r = \operatorname {C o n c a t} \left(\mathcal {R} _ {0} (T), \mathcal {R} _ {1} (T), \dots \mathcal {R} _ {N - 1} (T)\right), \tag {10}
+$$
+
+$$
+\boldsymbol {\mathcal {S}} (r) = \sigma (r \boldsymbol {W} _ {\mathcal {S} _ {1}} ^ {T}) \boldsymbol {W} _ {\mathcal {S} _ {2}},
+$$
+
+where  $\mathbf{W}_{\mathcal{S}_1}$  and  $\mathbf{W}_{\mathcal{S}_2}$  are weights of size  $\mathbf{W}_{\mathcal{S}_1} \in \mathbb{R}^{P \times K \cdot N}$  and  $\mathbf{W}_{\mathcal{S}_2} \in \mathbb{R}^{P \times 1}$ ;  $P$  is an internal hidden dimension.
+
+FLEXTRON-Adaptive: Dynamic Model Selection. Recent work on sparsely-activated MoE networks has demonstrated that an ensemble of different sub-networks ("experts"), each specializing in particular input domains, performs better and more efficiently than dense baselines (Fedus et al., 2022; Zhou et al., 2022). Drawing inspiration from previous studies, FLEXTRON introduces an input-adaptive routing mechanism to dynamically select optimal sub-networks based on latency and input, reducing memory and communication overheads through weight sharing and array-based indexing.
+
+For input adaptivity, we modify the router design to also incorporate the current hidden states  $h_i$  as follows:
+
+$$
+s _ {i} = \operatorname {a r g m a x} \left(\mathcal {R} _ {i} \left(T, h _ {i}\right)\right),
+$$
+
+$$
+\text {w h e r e} \mathcal {R} _ {i} (T, h _ {i}) = \sigma \left(T \cdot \boldsymbol {W} + h _ {i} \boldsymbol {W} _ {H _ {i}} ^ {T}\right) \boldsymbol {W} _ {\mathcal {R} _ {i}} \tag {11}
+$$
+
+Here, the current hidden features  $h_i$  are projected into the embedding space of dimension  $U$  by an MLP layer parameterized by  $WH_i$ . Similarly,  $T$  is projected via simple scaling of the matrix  $W$ . We limit  $U$  to 128. Token-wise routing decisions are generated by aggregating the latency embedding vector with the hidden feature embedding vector, and passing them through a linear layer.
+
+We also extend the surrogate model format described in Eq. (10) to additionally incorporate the final hidden states  $h_N$ .
+
+Hidden states are projected to the dimension of  $P$  using a linear matrix. This projection is then summed with the latency embedding before applying the activation function.
+
+Training. Figure 3 provides an overview of training routers via SM. Initially, the main LLM is frozen. Routers are always updated with gradients from the latency loss defined in Eq. (8). The Surrogate Model is updated to minimize the predicted LM loss (via the MSE objective). If the MSE is below a predefined threshold, then routers are additionally updated with gradients from the output of the SM to minimize the predicted LM loss. In this way, routers learn to minimize the LM loss in an indirect way, via SM. Once the routers are trained, we disregard the SM and fine-tune both the routers and the LLM parameters.
+
+# 4. Experiments
+
+# 4.1. Experimental Settings
+
+Model and Dataset. We perform our evaluation on the GPT-3 and Llama-2 (Touvron et al., 2023) model families. GPT-3 is a representative multilingual model family (Shoeybi et al., 2019) with 2 and 8 billion parameter variants (among others); these are pretrained with the NeMo framework (Kuchaiev et al., 2019). The total number of trainable parameters for GPT-3-2B is 2 billion, with 1.2 billion non-embedding parameters. The model contains 24 layers, with a hidden dimension of 2048. Each MHA layer possesses 16 heads. GPT-3-8B comprises 8 billion parameters, of which 6.4 billion are non-embedding parameters. The model contains 32 layers, each with a hidden dimension of 4096. Each MHA layer possesses 32 heads. Both models are trained on 1.1 trillion tokens, where data is obtained from publicly available data sources, comprising 53 languages and code. Both models support a maximum context length of 4096. We further validate our approach using the Llama2-7B model (Touvron et al., 2023), a widely used open-source pre-trained model with 6.5 billion non-embedding parameters. This model employs a 32-layer transformer architecture with a hidden dimension of 4096, incorporating 32 attention heads for each Multi-Head Attention (MHA) mechanism. We perform neuron/head sorting, elastic continued-training and automatic network selection with the same domain data, unless otherwise specified.
+
+Baselines. We compare our method to Matformer (Kudugunta et al., 2023), which adopts a nested structure on MLPs, obtaining 4 variants per MLP layer by training once. Instead of training the Matformer models from scratch, we adopt the pretraining strategy described in Section 3.1. We also compare our method with smaller pretrained models obtained with the same data and training recipe. We choose GPT3-2B and GPT3-8B, our base model, and GPT3-843M, a smaller version of
+
+GPT3 with embedding size of 1024, 24 layers and 16 heads. We additionally compare our method with representative open-source model families, including Pythia (Biderman et al., 2023), OpenLLaMA (Geng & Liu, 2023), and models generated by post-hoc compression methods, including Sheared-LLaMA (Xia et al., 2023), Compresso (Guo et al., 2023), LLM-Pruner (Ma et al., 2023), SliceGPT (Ashkboos et al., 2024), and LaCo (Yang et al., 2024).
+
+Training. As described in Section 3.1, during elastic network pretraining, we first perform importance sorting of each head/neuron in MHA/MLP layers using a tiny fraction (512 samples) of the full training set. We then perform training of the sorted and permuted elastic model. We use a batch-size of 256, and tune the model for 80000 steps. At each step, we randomly construct 3 sub-models together with the full model; perform gradient accumulation for all 4 models for a single update. We perform lightweight tuning for automatic network selection: we freeze the backbone parameters and only tune the routers and surrogate models for 1000 steps using a batch size of 256. For static router tuning, we observe a consistent performance ranking over multiple data domains for sub-models, and thus use only single domain data (Wikipedia (Foundation)). During the input-adaptive router training, which is harder, we use the subset of pretraining dataset.
+
+# 4.2. Results
+
+FLEXTRON Performance. We validate the effectiveness of FLEXTRON framework on multiple representative downstream tasks in Table 1. These tasks include: ARCEasy (Clark et al., 2018), LAMBADA (Paperno et al., 2016), PIQA (Bisk et al., 2020), WinoGrande (Sakaguchi et al., 2021), MMLU(Hendrycks et al., 2020), and HellaSwag (Zellers et al., 2019). We follow the practice in Xia et al. (2023) and report the 5-shot and 10-shot performance for MMLU and Hellaswag, respectively. We report zero-shot performance for other tasks. In Table 1, FLEXTRON-8B and FLEXTRON-Llama2-7B denote the Flextron models built upon GPT3-8B and Llama2-7B, respectively. "Dynamic" refers to the model with input-adaptive router while "static" represent the static case where all tokens select the same sub-network given the latency/memory requirements.  $\times$  suffix indicates the remaining latency of the model.
+
+Additionally, we measure the latency of FLEXTRON models in Table 2, with latency measured using TensorRT-LLM (NVIDIA, 2023). It's worth noting that FLEXTRON8B models are multi-lingual models with a vocabulary size of 320,000, and the embedding operation incurs a latency of 1.82s, constituting  $17.4\%$  of full latency. For comparison, the embedding layer of Llama2-7B incurs a latency of 0.69s, constituting  $7.2\%$  of the full latency. All results are tested on the NVIDIA A100 80GB GPU, with latency measured
+
+Table 1. Downstream task evaluation of FLEXTRON family models and comparison with representative open-source models and compression methods. We report the zero-shot performance of ARC-easy (Clark et al., 2018), LAMBADA (Paperno et al., 2016), PIQA (Bisk et al., 2020), and WinoGrande (Sakaguchi et al., 2021). We also report the 5-shot performance of MMLU(Hendrycks et al., 2020), and the 10-shot performance of HellaSwag (Zellers et al., 2019). Here, #params refers to the number of non-embedding parameters. Note that for dynamic FLEXTRON models, we use the averaged number of activated non-embedding parameters.  
+
+<table><tr><td></td><td></td><td># Params</td><td>ARC-E</td><td>LAMBADA</td><td>PIQA</td><td>Winogrande</td><td>MMLU (5)</td><td>Hellaswag (10)</td><td>Avg.</td></tr><tr><td rowspan="7">FLEXTRON-8B</td><td>Full</td><td>6.4 B</td><td>71.7%</td><td>69.7%</td><td>79.4%</td><td>68.8%</td><td>35.4%</td><td>75.9%</td><td>66.8%</td></tr><tr><td>Static-0.7×</td><td>4.1 B</td><td>66.7%</td><td>62.9%</td><td>75.1%</td><td>63.9%</td><td>28.7%</td><td>70.6%</td><td>61.3%</td></tr><tr><td>Dynamic-0.7×</td><td>4.3 B</td><td>67.0%</td><td>64.8%</td><td>75.9%</td><td>64.1%</td><td>30.0%</td><td>70.4%</td><td>62.0%</td></tr><tr><td>Static-0.6×</td><td>3.9 B</td><td>66.2%</td><td>62.8%</td><td>75.6%</td><td>62.7%</td><td>28.8%</td><td>68.8%</td><td>60.8%</td></tr><tr><td>Dynamic-0.6×</td><td>3.9 B</td><td>66.2%</td><td>63.7%</td><td>76.1%</td><td>62.7%</td><td>29.1%</td><td>69.2%</td><td>61.2%</td></tr><tr><td>Static-0.5×</td><td>3.4 B</td><td>64.2%</td><td>62.0%</td><td>74.9%</td><td>61.7%</td><td>25.1%</td><td>66.8%</td><td>59.1%</td></tr><tr><td>Dynamic-0.5×</td><td>3.3 B</td><td>65.0%</td><td>62.5%</td><td>75.8%</td><td>61.8%</td><td>27.1%</td><td>67.8%</td><td>60.0%</td></tr><tr><td rowspan="7">FLEXTRON-Llama2-7B</td><td>Full</td><td>6.5 B</td><td>75.1%</td><td>71.5%</td><td>77.5%</td><td>69.1%</td><td>43.0%</td><td>78.1%</td><td>69.1%</td></tr><tr><td>Static-0.7×</td><td>4.2 B</td><td>65.8%</td><td>64.2%</td><td>75.6%</td><td>62.3%</td><td>32.5%</td><td>67.1%</td><td>61.3%</td></tr><tr><td>Dynamic-0.7×</td><td>4.1 B</td><td>68.6%</td><td>65.1%</td><td>76.1%</td><td>63.7%</td><td>35.9%</td><td>69.4%</td><td>63.1%</td></tr><tr><td>Static-0.6×</td><td>4.0 B</td><td>66.1%</td><td>63.8%</td><td>75.0%</td><td>62.1%</td><td>31.9%</td><td>68.0%</td><td>61.2%</td></tr><tr><td>Dynamic-0.6×</td><td>3.9 B</td><td>67.1%</td><td>63.8%</td><td>74.9%</td><td>62.2%</td><td>34.5%</td><td>69.7%</td><td>62.0%</td></tr><tr><td>Static-0.5×</td><td>3.5 B</td><td>65.9%</td><td>61.7%</td><td>74.8%</td><td>61.9%</td><td>30.3%</td><td>67.6%</td><td>60.4%</td></tr><tr><td>Dynamic-0.5×</td><td>3.4 B</td><td>66.5%</td><td>62.9%</td><td>74.1%</td><td>62.0%</td><td>33.4%</td><td>68.5%</td><td>61.2%</td></tr><tr><td rowspan="7">Open-Source</td><td>Llama2-7B</td><td>6.5 B</td><td>75.2%</td><td>68.2%</td><td>78.8%</td><td>69.2%</td><td>45.3%</td><td>78.6%</td><td>69.2%</td></tr><tr><td>OpenLLaMA-3Bv2</td><td>3.2 B</td><td>63.7%</td><td>59.1%</td><td>78.1%</td><td>63.3%</td><td>25.7%</td><td>71.6%</td><td>60.3%</td></tr><tr><td>OpenLLaMA-7Bv2</td><td>6.5 B</td><td>69.5%</td><td>63.8%</td><td>79.9%</td><td>66.0%</td><td>40.4%</td><td>76.6%</td><td>66.0%</td></tr><tr><td>GPT3-8B</td><td>6.4 B</td><td>70.1%</td><td>70.5%</td><td>79.7%</td><td>69.8%</td><td>40.2%</td><td>77.7%</td><td>68.0%</td></tr><tr><td>Pythia-1.4B</td><td>1.2 B</td><td>53.9%</td><td>46.8%</td><td>70.6%</td><td>57.1%</td><td>25.6%</td><td>52.2%</td><td>51.0%</td></tr><tr><td>Pythia-2.8B</td><td>2.5 B</td><td>57.9%</td><td>50.1%</td><td>73.8%</td><td>58.6%</td><td>26.8%</td><td>60.0%</td><td>54.5%</td></tr><tr><td>Pythia-6.9B</td><td>6.4 B</td><td>60.2%</td><td>47.1%</td><td>75.2%</td><td>59.9%</td><td>25.5%</td><td>64.4%</td><td>55.4%</td></tr><tr><td rowspan="7">Compressed</td><td>Sheared-LLaMA-1.3B</td><td>1.2 B</td><td>61.5%</td><td>61.0%</td><td>73.4%</td><td>57.9%</td><td>25.7%</td><td>60.7%</td><td>56.7%</td></tr><tr><td>Sheared-LLaMA-2.7B</td><td>2.5 B</td><td>67.0%</td><td>68.4%</td><td>75.8%</td><td>64.2%</td><td>26.4%</td><td>70.8%</td><td>62.1%</td></tr><tr><td>NutePrune</td><td>3.2 B</td><td>51.7%</td><td>-</td><td>71.0%</td><td>57.5%</td><td>-</td><td>55.9%</td><td>-</td></tr><tr><td>LLM-Pruner</td><td>4.5 B</td><td>59.2%</td><td>-</td><td>73.4%</td><td>64.2%</td><td>23.9%</td><td>56.5%</td><td>-</td></tr><tr><td>Compresso</td><td>4.5 B</td><td>66.0%</td><td>-</td><td>72.9%</td><td>63.4%</td><td>25.9%</td><td>-</td><td>-</td></tr><tr><td>LaCo</td><td>4.7 B</td><td>-</td><td>-</td><td>69.8%</td><td>-</td><td>26.5%</td><td>55.7%</td><td>-</td></tr><tr><td>SliceGPT</td><td>4.8 B</td><td>-</td><td>-</td><td>66.2%</td><td>-</td><td>28.9%</td><td>50.3%</td><td>-</td></tr></table>
+
+when the prompting length and generation length is set to 8 and 512, respectively. We use the batch size of 1.
+
+Table 2. Latency of FLEXTRON family models. The latency is measured based on TensorRT-LLM (NVIDIA, 2023) and NVIDIA A100 80GB GPU. We measure the latency when the prompting length and generation length is set to 8 and 512, respectively. We use the batch size of 1. The reported numbers present (# non-embedding params)/(latency).  
+
+<table><tr><td></td><td>Full</td><td>0.7×</td><td>0.6×</td><td>0.5×</td></tr><tr><td>FLEXTRON-8B</td><td>6.4B / 10.43s</td><td>4.1B / 8.02s</td><td>3.9B / 6.39s</td><td>3.4B / 5.48s</td></tr><tr><td>FLEXTRON-Llama2-7B</td><td>6.5B / 9.64s</td><td>4.1B / 7.09s</td><td>3.9B / 5.41s</td><td>3.4B / 4.91s</td></tr></table>
+
+Neural Scaling Laws. Recent work (Kaplan et al., 2020; Hoffmann et al., 2022) has empirically demonstrated scaling laws for LLMs with respect to model size. Specifically, model capacity scales as follows:
+
+$$
+L (N) = \left(N / N _ {c}\right) ^ {- \alpha_ {N}} + E _ {N}, \tag {12}
+$$
+
+here,  $N$  denotes the number of non-embedding model parameters, and  $N_{c}$ ,  $\alpha_{N}$ , and  $E_N$  are model-dependent coeffi
+
+cients. This curve typically utilizes multiple independently trained models to capture the correlation between model size and validation loss. For FLEXTRON, we extend the model scaling law along two dimensions: (1) we observe that the model's capacity, which grows with the number of sub-model parameters, follows the existing model scaling law, and (2) we establish a power law relationship between the model's capacity and the input latency.
+
+Figure 5 plots the trade-off between validation loss and latency (left) / number of non-embedding parameters (right) for both FLEXTRON-Static and input-adaptive FLEXTRON-Adaptive routing of the trained elastic model. MHA layers typically introduce fewer parameters but incur high latency; as such, elastic MHA is favorable in the higher latency regimes. This is evident when comparing FLEXTRON's performance to Matformer (Kudugunta et al., 2023), which only leverages elastic MLP. In Figure 5, we fit the data points of sub-networks with Equation 12, and provide the fitted parameters for the scaling equation in Table 3; this suggests a useful guideline for model practitioners to choose
+
+![](images/68b8bd235ce3cf204086b4a603d2aeca13050ec6470e8977add60ad0cdccfe3b.jpg)  
+Figure 4. The Flextron-Llama2-7B model family demonstrates superior MMLU (Hendrycks et al., 2020) performance compared to both open-source models and existing post-hoc compression methods. Specifically, we compare against models from the Pythia (Biderman et al., 2023) family and the OpenLLaMA-v2 (Geng & Liu, 2023) family. Additionally, our method is compared with Sheared-LLaMA (Xia et al., 2023), Compresso (Guo et al., 2023), LLM-Pruner (Ma et al., 2023), SliceGPT (Ashkboos et al., 2024), and LaCo (Yang et al., 2024).  $\times$  suffix indicates the remaining number of non-embedding parameters of the model.
+
+the proper model that simultaneously meets latency, number of parameters, and model capacity constraints.
+
+Table 3. Fitted parameters for Equation 12.  
+
+<table><tr><td></td><td>NC</td><td>αN</td><td>EN</td></tr><tr><td>Matformer (Kudugunta et al., 2023)</td><td>1.680</td><td>52.74</td><td>1.729</td></tr><tr><td>FLEXTRON (Static)</td><td>1.465</td><td>38.57</td><td>1.733</td></tr><tr><td>FLEXTRON (Input-adaptive)</td><td>1.289</td><td>25.52</td><td>1.729</td></tr></table>
+
+Training Efficiency. FLEXTRON demonstrates excellent training efficiency, as detailed in Table 4. During elastic continued-training, we utilize only 89.9 billion tokens for both the GPT-3 and LLama2 models, while for router tuning, we use 1.049 billion tokens.
+
+Table 4. Flextron training costs compared to pretraining cost. We report the number of tokens to illustrate the training cost.  
+
+<table><tr><td rowspan="2"></td><td colspan="2">Flextron Training Cost</td><td rowspan="2">Pretraining Cost</td></tr><tr><td>Elastic Continued-Tuning</td><td>Router Tuning</td></tr><tr><td>GPT-3</td><td>89.9 B (7.54%)</td><td>1.049 B (0.09%)</td><td>1.1T</td></tr><tr><td>Llama2</td><td>89.9 B (4.50%)</td><td>1.049 B (0.05%)</td><td>2T</td></tr></table>
+
+# 5. Analysis
+
+# 5.1. FLEXTRON Learnings & Insights
+
+Routers Assign More Computation to Deeper MLP layers. During inference, MHA and MLP layers have similar latency, despite having different sizes in terms of parameters. For instance, in the GPT3-2B model, processing each MHA and MLP layer requires 3.830 ms and 3.016 ms, respectively. In practical scenarios where low latency is crucial, understanding how to distribute compute and the number of model parameters among MLP and MHA layers becomes
+
+essential. FLEXTRON provides us with a test-bed. In Figure 6, for GPT3-2B model, we replace the full MHA/MLP layer with elastic candidates and calculate the performance degradation in terms of averaged LM loss. We compute the averaged LM loss over 7 data domains, similar to previous sections. Two conclusions can be drawn: (1) replacing the full MLP layers results in higher performance degradation; (2) replacing deep layers, especially deep MLP layers, significantly hurts performance. Additionally, we visualize two Llama2-7B-based models with different latency targets, optimized by learnable static routers in Figure 14. We observe that the learned structure aligns with the previous conclusions. We visualize other optimized architectures in the Appendix D and provide guidelines of architecture designs.
+
+Input-adaptive Routers Assign More Computation to Hard Samples. The necessity of input-adaptive routing naturally comes from data diversity. Typically, "easy" datasets only need small-scale models for good performance, while "hard" datasets require large-scale models. This observation motivated us to include support for input-adaptive routing in FLEXTRON. We evaluate this hypothesis in Figure 8. Here, we evaluate the sub-networks optimized by routers, with different latency, across multiple data domains. We mainly test on three categories: (1) English datasets: Arxiv, Books3 (Gao et al., 2020), Wikipedia (Foundation), (2) multilingual datasets: Korean, German, and (3) code data: HTML, JAVA. On GPT3-2B, we visualize the performance degradation of networks, calculated by  $\mathrm{(PPL_{sub} / PPL_{full})}$ , and plot their correlation with latency. As a concrete example, notice that the curves for code datasets are much flatter than others, indicating that the task only requires a relatively small number of parameters. Conversely, multilingual datasets require more model parameters.
+
+The input-adaptive models exhibit similar behavior. In Figure 9, we selected the model with  $61.8\%$  latency and obtained the router decision statistics for the first layer. For the HTML dataset, almost half of the tokens select the smallest elastic candidate, while tokens from the Books3 dataset (Gao et al., 2020) tend to choose the full layer.
+
+# 5.2. Effectiveness of Learned Routers
+
+To demonstrate the effectiveness of our learned routers, we compare the learned sub-models with randomly picked ones. In Figure 10, we first randomly sample sub-models at different latencies from GPT3-2B, and measure their performance. We use box plots to visualize the distributions of their LM loss. As seen from the Figure, the majority of randomly selected sub-models have unpredictable performance. For instance, sub-models at  $65\%$  latency have averaged LM loss ranging from 2.32 to 3.06. We compare them to sub-models found by routers (blue and yellow lines in the Figure), demonstrating that FLEXTRON effectively identifies
+
+![](images/fb6a3e984568a2447514132d4e319fb6fa1386277269789dcb53ed1374418831.jpg)  
+Figure 5. Pareto curves for language modeling loss vs latency (left) and # non-embedding parameters (right). The curve is fitted by the model scaling equation. FLEXTRON achieves superior performance to Matformer and even end-to-end-trained smaller models (843M). The performance of the model is evaluated by language modeling validation loss and averaged over 7 representative datasets: (1) English datasets: Arxiv, Books3 (Gao et al., 2020), Wikipedia (Foundation), (2) multilingual datasets: Korean, German languages, and (3) code data: HTML, JAVA. We measure model latency with the Megatron framework (Shoeybi et al., 2019) using a batch size of 2 and sequence length of 4096 in the context prefilling stage on NVIDIA A100 GPU.
+
+![](images/5f56bbcce5de6b0f2c4c5006ffdef7c4d001ca97f3f7e3b4c5762a3c616a384c.jpg)
+
+![](images/f2b2e1b801e19ea3c0ee49d7855975fd24b31aec1bb6bc73c47bff4d39dd1014.jpg)  
+Figure 6. Performance degradation introduced by replacing the full MHA/MLP layer with elastic candidates; specifically, the effect of replacing the full layer with  $75\%$  and  $50\%$  of the layer width. We observe that (1) using lightweight MHA layers could preserve more model performance, and (2) it's crucial to use full MLP layers deeper in the network. The experiment is based on GPT3-2B model.
+
+![](images/3dd09d75b59dd754ada77f9360339a943bf395a9fdff703df36d05d2765abcaa.jpg)
+
+optimal sub-models.
+
+# 6. Related Work
+
+Elastic Inference. The idea of obtaining multiple models from a single trained model has been explored extensively in the convolutional neural network (CNN) literature; in particular, Yu et al. (2018); Yu & Huang (2019) introduced slimmable neural networks, which support deployment of the same model with varying numbers of convolutional filters. Li et al. (2021) leverage a gating mechanism to dynamically identify sample difficulty and adjust the percentage of activated filters accordingly. Finally, Cai et al. (2019) generalized pruning methods to derive a single model adaptable to different configurations. Recent work has explored the application of slimmable models to Transformer-derived architectures; specifically, Rao et al. (2021) and Yin et al. (2022) explore the mechanism of slimmable token removal for adaptive token dropping. Kusupati et al. (2022) introduce a nested weight structure for Transformer networks,
+
+![](images/d784f841e13e9b31194efb587671fef90b631e73cd4d98ef212374be0d781b10.jpg)  
+Figure 7. Obtained architectures for  $50\%$  and  $70\%$  latency targets.
+
+![](images/06da02f45015c424cd83b1796e12adca02a66ea005f9991d3045f0a1d4f7333d.jpg)
+
+and Kudugunta et al. (2023) use this formalization in the Matformer architecture. Valipour et al. (2023) additionally utilize a sampling-based training strategy to train multiple models via gradient accumulation. While FLEXTRON shares Matformer's nested weight structure, it uniquely extends it by offering elasticity in both MLP and MHA layers, a larger pool of operations, efficient pretraining for sub-linear training times, and automatic input-adaptive sub-network selection based on latency for enhanced efficiency.
+
+Input Adaptivity. Sparse Mixture-of-Expert networks (MoEs) utilize input adaptivity to achieve efficient model scaling by collectively utilizing multiple specialized subnetworks (Fedus et al., 2022; Riquelme et al., 2021; Zhou
+
+![](images/ad99bef410c55365c4c2cafd208f49e6725bdef4b153c610907dff75493578aa.jpg)  
+Figure 8. Performance degradation on sub-networks of different latency, on different data domains.
+
+![](images/9740a7c649d5d225bbf842a28f51123cf94638304859c0d9acaf1ba5b4b98587.jpg)  
+Figure 9. Router allocation vs data domain. The y-axis depicts elastic operator remaining computations and horizontal axis depicts the frequency of the operator being chosen. On GPT3-2B, we observe that "hard data" (such as data from Books3 (Gao et al., 2020), with a PPL of 11.64 on GPT3-2B) tend to utilize full layers more frequently. Almost half of the tokens on "easy data" (e.g., HTML dataset, with a PPL of 1.571) select the smaller layers.
+
+et al., 2022; Jiang et al., 2024), to handle data from diverse domains (Li et al., 2022; Zhang et al., 2023). Tokens in MoE networks only pass through the most relevant sub-networks, identified by learnable routers. Recent work has started challenging the traditional definition of MoEs by introducing heterogeneous experts (Wang et al., 2020; Dean, 2021; Zhou et al., 2022) and in-situ adaptiveness (Chen et al., 2023; Cai et al., 2023). However, all existing MoE designs that we are aware of store expert weights separately, with no notion of weight sharing. This design introduces significant memory and communication overheads, especially at larger batch sizes with higher expert utilization. In FLEXTRON, all the "experts" in a layer share the same weight matrix, and different sub-networks are selected through simple array indexing, thus relieving most of the pressure from the memory and networking interconnect. Additionally, FLEXTRON includes provisions for routing decisions to be dictated by a latency target, a feature absent from most existing MoE networks.
+
+Static Acceleration. A vast body of work has also demonstrated the efficacy of static acceleration methods
+
+![](images/95dc21d2f8dace3314b9dca4e88b7a285ff27c0181af1edf80d232419621968e.jpg)  
+Figure 10. Effectiveness of our automatic network selection algorithm. The box-plot visualizes the performance distribution of randomly selected models. The blue and yellow lines denote performance of FLEXTRON's routers. Performance is evaluated on Wikipedia (Foundation) and GPT3-2B Flextron models.
+
+on transformers, including weight and activation quantization Lin et al. (2023); Frantar et al. (2022), patterned 2:4 sparsity Mishra et al. (2021), neural architecture search (NAS) Wang et al. (2020); Wu et al. (2021), token eviction Zhang et al. (2024), and hardware-aware structural pruning Yang et al. (2023). Besides, Ma et al. (2023); Xia et al. (2023); Wang et al. (2023a;b) aim to re-use pre-trained checkpoints to avoid repeated computation. These methods are orthogonal to the dynamic inference literature and can provide further opportunities for performance improvement.
+
+# 7. Conclusion
+
+This paper has presented FLEXTRON, a novel network architecture and post-training optimization framework. FLEXTRON models flexibly adapt to different latency and accuracy targets during inference with no additional fine-tuning, and come with built-in support for input-adaptive routing to maximize performance. We have also presented a posttraining framework for systematically converting standard trained LLMs such as GPT-3 and Llama2 into FLEXTRON models using a sample-efficient training procedure. FLEXTRON demonstrates superior zero-shot performance over multiple smaller end-to-end trained variants on the GPT-3 family and Llama-2-7B model; FLEXTRON also outperforms the state-of-the-art Matformer framework (Kudugunta et al., 2023). FLEXTRON achieves this through a single pretraining run that consumes a mere  $7.63\%$  of training tokens of full pretraining cost.
+
+# Impact Statement
+
+This paper presents work whose goal is to advance the field of Machine Learning. There are many potential societal consequences of our work, none which we feel must be specifically highlighted here.
+
+# References
+
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+
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+
+# A. Training Trajectory of elastic continued-tuning.
+
+In Figure 11, we visualize the validation loss of sub-models of different sizes, during the elastic continued-tuning process. We draw the following observations: (1) elastic continued-tuning does not negatively impact the performance of the full model (i.e., employing all attention heads in MHA and full hidden size in MLP), as demonstrated by the purple curve. (2) Throughout training, all sub-networks converge synchronously, while the larger sub-models lead to smaller validation losses overall. To validate, we depict validation loss of randomly selected sub-networks using the blue, orange, green, and red curves, incurring  $46\%$ ,  $52\%$ ,  $58\%$ ,  $64\%$  of the full latency, respectively. Note that the sub-models are randomly picked independently at each validation step. (3) The middle-sized sub-models converge more stably, as indicated by the smoother curves of the green and orange compared to the blue and red ones. This stability could potentially be attributed to the fact that middle-sized models are sampled more frequently during elastic continued-tuning.
+
+![](images/dacb33a5be01ef949f5fe4f69ca56307ca6381a5b8e608afc854eed15c30cfc9.jpg)  
+Figure 11. Visualization of validation loss for sub-models of varying sizes during elastic continued-tuning.
+
+We additionally depict the validation loss trajectory of models
+
+employing a uniform elastic selection strategy in Figure 12. For instance, in the first sub-figure, the “ $50\%$ #Heads  $75\%$  #Channels” refers to the model selecting the first half of the attention heads and  $75\%$  of the Channels for all layers. The figure echos the observation in Section 5.1 that adopting lightweight MHAs, characterized by a reduced number of heads, is more advantageous in limited resources.
+
+![](images/4c719102a6263d3ee868d1c74228c2b5aec4e9e623be31ab4917492688d20b74.jpg)  
+Figure 12. Training trajectory of models performing uniform elastic selection strategy.
+
+![](images/76de7a2acaf5ec002478e84ec7e38cd02f9a830d0ef3adb01de9b513e45bd49d.jpg)
+
+![](images/bb2cb31994cf832719f54bd1af91cea36e4364d44cba1a18efb5c591ecd7f430.jpg)
+
+![](images/0e865cd7f70fb46f92d16294317f252aa85a7585a21f6e34265eb0592bb27592.jpg)
+
+# B. Router Training Dynamics.
+
+During router training, we introduce the surrogate model (SM), to estimate the language modeling loss (LM Loss) based on router logits, providing a stable signal for router training. As detailed in Figure 3, when the SM is not accurate enough, the "L2 Loss" is utilized, where "L2 Loss" refers to the MSE loss between the "ground truth" language modeling loss and the estimated LM loss via SM. When the SM error is smaller than the threshold, the router will be optimized based on the estimated LM loss. The dynamics of router training are depicted in Figure 13. Router training can be roughly divided into three stages: (1) SM tuning: the "L2 Loss" quickly drops, during which the LM Loss slightly increases; (2) Joint tuning: both losses decrease simultaneously; (3) Router tuning: the LM Loss continues to decrease while the "L2 Loss" remains below the threshold.
+
+![](images/1655f61f238e67a0cd8915219c27fbbfe01916787eae279e549d79dc9c05d6c4.jpg)  
+Figure 13. Router training dynamics. We visualize the curve of losses ("L2 Loss" and "LM Loss") during the router training.
+
+# C. Necessity of Weight Permutation
+
+We assess the effectiveness of weight permutation by testing the performance of the original and permuted models using their first-half (50%) MLP neurons/MHA heads as is. As Table 5 demonstrates, the perplexity on Wikipedia (Foundation) significantly improves post-permutation, with the un-permuted model's perplexity exceeding 1000, while the permuted model's perplexity was 193.6. We observe a similar enhancement for MHA modules.
+
+Table 5. Ablation on permuting the base model by channel/neuron importance score (Equation 3 and Eqn 4 in Section 3.1) as the initialization. Numbers correspond to the Wikipedia perplexity of pretrained models cut to the first half neurons/heads. Note that we report the zero-shot performance of the permuted model.  
+
+<table><tr><td>Setup</td><td>MHA</td><td>MLP</td></tr><tr><td>Full (baseline)</td><td>9.144</td><td>9.144</td></tr><tr><td>50% Operator w/o Elastic Sorting</td><td>184.5</td><td>1902.0</td></tr><tr><td>50% Operator w/ Elastic Sorting (ours)</td><td>179.9</td><td>193.6</td></tr></table>
+
+# D. Architecture Visualization
+
+We provide searched architectures in Figure 14, based on two variants of GPT3 family. The observation validates our previous heuristic entailed in Sec 5.1.
+
+![](images/fed2b98e67b91ca44750ba9a3e4ac023c20b78fcd90b27e8512f602dfd9926d5.jpg)  
+GPT3-2B
+
+![](images/d1285ec9bae26be76824d4ef053afd00ab2a3f03c60cdfdefd44a6ffcb78a145.jpg)  
+70% latency
+
+![](images/4cda32a488ddd5354aa420e6cc38b671d822e19352c86e97c26e0dcc658a2040.jpg)  
+GPT3-8B
+
+![](images/2247a632d05eab25cc7f7012e96e4c1f16ad7c992889172b8974e23361dc999d.jpg)  
+70% latency  
+Figure 14. Obtained architectures for  $50\%$  and  $70\%$  latency targets based on GPT-3 family.

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