Daily Papers

Learning complex dynamics driven by partial differential equations directly from data holds great promise for fast and accurate simulations of complex physical systems. In most cases, this problem can be formulated as an operator learning task, where one aims to learn the operator representing the physics of interest, which entails discretization of the continuous system. However, preserving key continuous properties at the discrete level, such as boundary conditions, and addressing physical systems with complex geometries is challenging for most existing approaches. We introduce a family of operator learning architectures, structure-preserving operator networks (SPONs), that allows to preserve key mathematical and physical properties of the continuous system by leveraging finite element (FE) discretizations of the input-output spaces. SPONs are encode-process-decode architectures that are end-to-end differentiable, where the encoder and decoder follows from the discretizations of the input-output spaces. SPONs can operate on complex geometries, enforce certain boundary conditions exactly, and offer theoretical guarantees. Our framework provides a flexible way of devising structure-preserving architectures tailored to specific applications, and offers an explicit trade-off between performance and efficiency, all thanks to the FE discretization of the input-output spaces. Additionally, we introduce a multigrid-inspired SPON architecture that yields improved performance at higher efficiency. Finally, we release a software to automate the design and training of SPON architectures.

Unsupervised learning with functional data is an emerging paradigm of machine learning research with applications to computer vision, climate modeling and physical systems. A natural way of modeling functional data is by learning operators between infinite dimensional spaces, leading to discretization invariant representations that scale independently of the sample grid resolution. Here we present Variational Autoencoding Neural Operators (VANO), a general strategy for making a large class of operator learning architectures act as variational autoencoders. For this purpose, we provide a novel rigorous mathematical formulation of the variational objective in function spaces for training. VANO first maps an input function to a distribution over a latent space using a parametric encoder and then decodes a sample from the latent distribution to reconstruct the input, as in classic variational autoencoders. We test VANO with different model set-ups and architecture choices for a variety of benchmarks. We start from a simple Gaussian random field where we can analytically track what the model learns and progressively transition to more challenging benchmarks including modeling phase separation in Cahn-Hilliard systems and real world satellite data for measuring Earth surface deformation.

Neural operators learn mappings between infinite-dimensional function spaces and provide a data-driven surrogate modeling paradigm for parametric partial differential equations (PDEs). Existing architectures typically obtain expressivity by parameterizing integral kernels in prescribed transform domains or by applying attention-like interactions over discretized spatial points. While these approaches have achieved substantial progress, they often face a persistent trade-off among physical interpretability, nonlocal spatial communication, mesh scalability, and computational cost. We propose a Light-inspired neural operator(LiNO), an operator-learning architecture whose latent evolution is decomposed into three mechanisms motivated by elementary light transport: reflection, refraction, and scattering. Reflection and refraction act as adaptive pointwise transformations in latent feature space, enabling local feature reorientation and anisotropic modulation, whereas scattering performs input-dependent nonlocal propagation over the physical domain. We first formulate scattering as a normalized pairwise kernel with relative positional bias, and then develop an efficient scattering variant that replaces explicit pairwise interactions with positive-feature global propagation and a local diffusion branch, reducing the dominant spatial complexity from quadratic to linear. This yields a structured neural operator that separates local feature modulation from global spatial communication while retaining a modular and interpretable latent evolution.

Scientific applications increasingly demand real-time surrogate models that can capture the behavior of strongly coupled multiphysics systems driven by multiple input functions, such as in thermo-mechanical and electro-thermal processes. While neural operator frameworks, such as Deep Operator Networks (DeepONets), have shown considerable success in single-physics settings, their extension to multiphysics problems remains poorly understood. In particular, the challenge of learning nonlinear interactions between tightly coupled physical fields has received little systematic attention. This study addresses a foundational question: should the architectural design of a neural operator reflect the strength of physical coupling it aims to model? To answer this, we present the first comprehensive, architecture-aware evaluation of DeepONet variants across three regimes: single-physics, weakly coupled, and strongly coupled multiphysics systems. We consider a reaction-diffusion equation with dual spatial inputs, a nonlinear thermo-electrical problem with bidirectional coupling through temperature-dependent conductivity, and a viscoplastic thermo-mechanical model of steel solidification governed by transient phase-driven interactions. Two operator-learning frameworks, the classical DeepONet and its sequential GRU-based extension, S-DeepONet, are benchmarked using both single-branch and multi-branch (MIONet-style) architectures. Our results demonstrate that architectural alignment with physical coupling is crucial: single-branch networks significantly outperform multi-branch counterparts in strongly coupled settings, whereas multi-branch encodings offer advantages for decoupled or single-physics problems. Once trained, these surrogates achieve full-field predictions up to 1.8e4 times faster than high-fidelity finite-element solvers, without compromising solution accuracy.

A wide range of scientific problems, such as those described by continuous-time dynamical systems and partial differential equations (PDEs), are naturally formulated on function spaces. While function spaces are typically infinite-dimensional, deep learning has predominantly advanced through applications in computer vision and natural language processing that focus on mappings between finite-dimensional spaces. Such fundamental disparities in the nature of the data have limited neural networks from achieving a comparable level of success in scientific applications as seen in other fields. Neural operators are a principled way to generalize neural networks to mappings between function spaces, offering a pathway to replicate deep learning's transformative impact on scientific problems. For instance, neural operators can learn solution operators for entire classes of PDEs, e.g., physical systems with different boundary conditions, coefficient functions, and geometries. A key factor in deep learning's success has been the careful engineering of neural architectures through extensive empirical testing. Translating these neural architectures into neural operators allows operator learning to enjoy these same empirical optimizations. However, prior neural operator architectures have often been introduced as standalone models, not directly derived as extensions of existing neural network architectures. In this paper, we identify and distill the key principles for constructing practical implementations of mappings between infinite-dimensional function spaces. Using these principles, we propose a recipe for converting several popular neural architectures into neural operators with minimal modifications. This paper aims to guide practitioners through this process and details the steps to make neural operators work in practice. Our code can be found at https://github.com/neuraloperator/NNs-to-NOs

Deep neural networks, despite their success in numerous applications, often function without established theoretical foundations. In this paper, we bridge this gap by drawing parallels between deep learning and classical numerical analysis. By framing neural networks as operators with fixed points representing desired solutions, we develop a theoretical framework grounded in iterative methods for operator equations. Under defined conditions, we present convergence proofs based on fixed point theory. We demonstrate that popular architectures, such as diffusion models and AlphaFold, inherently employ iterative operator learning. Empirical assessments highlight that performing iterations through network operators improves performance. We also introduce an iterative graph neural network, PIGN, that further demonstrates benefits of iterations. Our work aims to enhance the understanding of deep learning by merging insights from numerical analysis, potentially guiding the design of future networks with clearer theoretical underpinnings and improved performance.

Surrogate modeling of wave propagation and scattering (i.e. the wave speed and source to wave field map) in heterogeneous media has significant potential in applications such as seismic imaging and inversion. High-contrast settings, such as subsurface models with salt bodies, exhibit strong scattering and phase sensitivity that challenge existing neural operators. We propose a hybrid architecture that decomposes the scattering operator into two separate contributions: a smooth background propagation and a high-contrast scattering correction. The smooth component is learned with a Fourier Neural Operator (FNO), which produces globally coupled feature tokens encoding background wave propagation; these tokens are then passed to a vision transformer, where attention is used to model the high-contrast scattering correction dominated by strong, spatial interactions. Evaluated on high-frequency Helmholtz problems with strong contrasts, the hybrid model achieves substantially improved phase and amplitude accuracy compared to standalone FNOs or transformers, with favorable accuracy-parameter scaling.

In this paper, we study solution operators of physical field equations on geometric meshes from a function-space perspective. We reveal that Hodge orthogonality fundamentally resolves spectral interference by isolating unlearnable topological degrees of freedom from learnable geometric dynamics, enabling an additive approximation confined to structure-preserving subspaces. Building on Hodge theory and operator splitting, we derive a principled operator-level decomposition. The result is a Hybrid Eulerian-Lagrangian architecture with an algebraic-level inductive bias we call Hodge Spectral Duality (HSD). In our framework, we use discrete differential forms to capture topology-dominated components and an orthogonal auxiliary ambient space to represent complex local dynamics. Our method achieves superior accuracy and efficiency on geometric graphs with enhanced fidelity to physical invariants. Our code is available at https://github.com/ContinuumCoder/Hodge-Spectral-Duality

Learning partial differential equations' (PDEs) solution operators is an essential problem in machine learning. However, there are several challenges for learning operators in practical applications like the irregular mesh, multiple input functions, and complexity of the PDEs' solution. To address these challenges, we propose a general neural operator transformer (GNOT), a scalable and effective transformer-based framework for learning operators. By designing a novel heterogeneous normalized attention layer, our model is highly flexible to handle multiple input functions and irregular meshes. Besides, we introduce a geometric gating mechanism which could be viewed as a soft domain decomposition to solve the multi-scale problems. The large model capacity of the transformer architecture grants our model the possibility to scale to large datasets and practical problems. We conduct extensive experiments on multiple challenging datasets from different domains and achieve a remarkable improvement compared with alternative methods. Our code and data are publicly available at https://github.com/thu-ml/GNOT.

Accurate remote sensing-based crop yield prediction remains a fundamental challenging task due to complex spatial patterns, heterogeneous spectral characteristics, and dynamic agricultural conditions. Existing methods often suffer from limited spatial modeling capacity, weak generalization across crop types and years. To address these challenges, we propose DFYP, a novel Dynamic Fusion framework for crop Yield Prediction, which combines spectral channel attention, edge-adaptive spatial modeling and a learnable fusion mechanism to improve robustness across diverse agricultural scenarios. Specifically, DFYP introduces three key components: (1) a Resolution-aware Channel Attention (RCA) module that enhances spectral representation by adaptively reweighting input channels based on resolution-specific characteristics; (2) an Adaptive Operator Learning Network (AOL-Net) that dynamically selects operators for convolutional kernels to improve edge-sensitive spatial feature extraction under varying crop and temporal conditions; and (3) a dual-branch architecture with a learnable fusion mechanism, which jointly models local spatial details and global contextual information to support cross-resolution and cross-crop generalization. Extensive experiments on multi-year datasets MODIS and multi-crop dataset Sentinel-2 demonstrate that DFYP consistently outperforms current state-of-the-art baselines in RMSE, MAE, and R2 across different spatial resolutions, crop types, and time periods, showcasing its effectiveness and robustness for real-world agricultural monitoring.

A multiplicative dual-encoder network computes a real-valued output for a pair of inputs as the inner product of their separate encodings. This architecture has been developed independently in operator learning, bipartite matching, contrastive vision-language models, retrieval, and other areas, yet no unified theory guides the basic design decisions: how many interaction modes to represent, how to normalize the encoders, and when the architecture should be avoided. We provide such a foundation by introducing the class of functions of low interaction rank, a class whose intrinsic complexity is measured by its interaction spectrum. Within this framework, approximation error decomposes into a spectral truncation term and an encoder-realization term; sample complexity is governed by the sum of the two encoder complexities rather than their product; and a usability criterion based on spectral decay determines when the architecture can succeed. The same framework exposes a central identifiability problem: the encoders are defined only up to a linear gauge symmetry that leaves the learned coordinates arbitrary. We show that normalization is gauge fixing and that whitening pins the interaction modes up to permutation and sign, thereby explaining the uninterpretability of contrastive dimensions and providing a constructive remedy. Experiments on synthetic kernels, operator learning, and CLIP models validate the theoretical predictions: spectral decay rates match the predicted scaling, whitening recovers the true modes, and independently trained CLIP models are related by a single rotation which, after removal by whitening, exposes interpretable concept axes. The code of this paper is provided at https://github.com/RS2002/Mul-Net .

Generative models in function spaces, situated at the intersection of generative modeling and operator learning, are attracting increasing attention due to their immense potential in diverse scientific and engineering applications. While functional generative models are theoretically domain- and discretization-agnostic, current implementations heavily rely on the Fourier Neural Operator (FNO), limiting their applicability to regular grids and rectangular domains. To overcome these critical limitations, we introduce the Mesh-Informed Neural Operator (MINO). By leveraging graph neural operators and cross-attention mechanisms, MINO offers a principled, domain- and discretization-agnostic backbone for generative modeling in function spaces. This advancement significantly expands the scope of such models to more diverse applications in generative, inverse, and regression tasks. Furthermore, MINO provides a unified perspective on integrating neural operators with general advanced deep learning architectures. Finally, we introduce a suite of standardized evaluation metrics that enable objective comparison of functional generative models, addressing another critical gap in the field.

Scientific Machine Learning (SciML) integrates data-driven inference with physical modeling to solve complex problems in science and engineering. However, the design of SciML architectures, loss formulations, and training strategies remains an expert-driven research process, requiring extensive experimentation and problem-specific insights. Here we introduce AgenticSciML, a collaborative multi-agent system in which over 10 specialized AI agents collaborate to propose, critique, and refine SciML solutions through structured reasoning and iterative evolution. The framework integrates structured debate, retrieval-augmented method memory, and ensemble-guided evolutionary search, enabling the agents to generate and assess new hypotheses about architectures and optimization procedures. Across physics-informed learning and operator learning tasks, the framework discovers solution methods that outperform single-agent and human-designed baselines by up to four orders of magnitude in error reduction. The agents produce novel strategies -- including adaptive mixture-of-expert architectures, decomposition-based PINNs, and physics-informed operator learning models -- that do not appear explicitly in the curated knowledge base. These results show that collaborative reasoning among AI agents can yield emergent methodological innovation, suggesting a path toward scalable, transparent, and autonomous discovery in scientific computing.

Neural operators learn to map initial conditions to the terminal solution of partial differential equations (PDEs), providing a surrogate for the full operator mapping. This enables rapid prediction across different input configurations. While recent neural operator architectures have demonstrated strong performance on diverse PDE tasks, their behavior under structured distribution shifts remains insufficiently understood. To investigate this, we study operator learning in a wave propagation setting governed by a one-dimensional variable-coefficient wave equation, using two representative architectures, the Fourier Neural Operator (FNO) and the Deep Operator Network (DeepONet). To examine their generalization under distribution shifts, we consider structured out-of-distribution (OOD) settings that independently vary input frequency and coefficient smoothness. The results show that under smoothness shifts, both models maintain stable performance, with FNO achieving lower error. In contrast, under frequency shifts, FNO exhibits a sharp increase in error under unseen high-frequency inputs, whereas DeepONet shows milder degradation despite higher overall error. Our analysis reveals that these differences arise from how each architecture represents and responds to variations in frequency structure. Together, these findings highlight a fundamental gap between strong in-distribution performance and generalization under distribution shifts in operator learning, underscoring the role of architectural representation bias in developing more reliable neural operators for physics-based PDE simulations beyond the training distribution.

A variety of infinitely wide neural architectures (e.g., dense NNs, CNNs, and transformers) induce Gaussian process (GP) priors over their outputs. These relationships provide both an accurate characterization of the prior predictive distribution and enable the use of GP machinery to improve the uncertainty quantification of deep neural networks. In this work, we extend this connection to neural operators (NOs), a class of models designed to learn mappings between function spaces. Specifically, we show conditions for when arbitrary-depth NOs with Gaussian-distributed convolution kernels converge to function-valued GPs. Based on this result, we show how to compute the covariance functions of these NO-GPs for two NO parametrizations, including the popular Fourier neural operator (FNO). With this, we compute the posteriors of these GPs in regression scenarios, including PDE solution operators. This work is an important step towards uncovering the inductive biases of current FNO architectures and opens a path to incorporate novel inductive biases for use in kernel-based operator learning methods.

Initial boundary value problems arise commonly in applications with engineering and natural systems governed by nonlinear partial differential equations (PDEs). Operator learning is an emerging field for solving these equations by using a neural network to learn a map between infinite dimensional input and output function spaces. These neural operators are trained using a combination of data (observations or simulations) and PDE-residuals (physics-loss). A major drawback of existing neural approaches is the requirement to retrain with new initial/boundary conditions, and the necessity for a large amount of simulation data for training. We develop a physics-informed transformer neural operator (named PINTO) that efficiently generalizes to unseen initial and boundary conditions, trained in a simulation-free setting using only physics loss. The main innovation lies in our new iterative kernel integral operator units, implemented using cross-attention, to transform the PDE solution's domain points into an initial/boundary condition-aware representation vector, enabling efficient learning of the solution function for new scenarios. The PINTO architecture is applied to simulate the solutions of important equations used in engineering applications: advection, Burgers, and steady and unsteady Navier-Stokes equations (three flow scenarios). For these five test cases, we show that the relative errors during testing under challenging conditions of unseen initial/boundary conditions are only one-fifth to one-third of other leading physics informed operator learning methods. Moreover, our PINTO model is able to accurately solve the advection and Burgers equations at time steps that are not included in the training collocation points. The code is available at https://github.com/quest-lab-iisc/PINTO

Solving partial differential equations (PDEs) by learning the solution operators has emerged as an attractive alternative to traditional numerical methods. However, implementing such architectures presents two main challenges: flexibility in handling irregular and arbitrary input and output formats and scalability to large discretizations. Most existing architectures are limited by their desired structure or infeasible to scale large inputs and outputs. To address these issues, we introduce an attention-based model called an inducing-point operator transformer (IPOT). Inspired by inducing points methods, IPOT is designed to handle any input function and output query while capturing global interactions in a computationally efficient way. By detaching the inputs/outputs discretizations from the processor with a smaller latent bottleneck, IPOT offers flexibility in processing arbitrary discretizations and scales linearly with the size of inputs/outputs. Our experimental results demonstrate that IPOT achieves strong performances with manageable computational complexity on an extensive range of PDE benchmarks and real-world weather forecasting scenarios, compared to state-of-the-art methods.

Understanding and predicting articulated actions is important in robot learning. However, common architectures such as MLPs and Transformers lack inductive biases that reflect the underlying kinematic structure of articulated systems. To this end, we propose the Neural Rodrigues Operator, a learnable generalization of the classical forward kinematics operation, designed to inject kinematics-aware inductive bias into neural computation. Building on this operator, we design the Rodrigues Network (RodriNet), a novel neural architecture specialized for processing actions. We evaluate the expressivity of our network on two synthetic tasks on kinematic and motion prediction, showing significant improvements compared to standard backbones. We further demonstrate its effectiveness in two realistic applications: (i) imitation learning on robotic benchmarks with the Diffusion Policy, and (ii) single-image 3D hand reconstruction. Our results suggest that integrating structured kinematic priors into the network architecture improves action learning in various domains.

In mechanical structures like airplanes, cars and houses, noise is generated and transmitted through vibrations. To take measures to reduce this noise, vibrations need to be simulated with expensive numerical computations. Deep learning surrogate models present a promising alternative to classical numerical simulations as they can be evaluated magnitudes faster, while trading-off accuracy. To quantify such trade-offs systematically and foster the development of methods, we present a benchmark on the task of predicting the vibration of harmonically excited plates. The benchmark features a total of 12,000 plate geometries with varying forms of beadings, material, boundary conditions, load position and sizes with associated numerical solutions. To address the benchmark task, we propose a new network architecture, named Frequency-Query Operator, which predicts vibration patterns of plate geometries given a specific excitation frequency. Applying principles from operator learning and implicit models for shape encoding, our approach effectively addresses the prediction of highly variable frequency response functions occurring in dynamic systems. To quantify the prediction quality, we introduce a set of evaluation metrics and evaluate the method on our vibrating-plates benchmark. Our method outperforms DeepONets, Fourier Neural Operators and more traditional neural network architectures and can be used for design optimization. Code, dataset and visualizations: https://github.com/ecker-lab/Learning_Vibrating_Plates

The efficacy of deep residual networks is fundamentally predicated on the identity shortcut connection. While this mechanism effectively mitigates the vanishing gradient problem, it imposes a strictly additive inductive bias on feature transformations, thereby limiting the network's capacity to model complex state transitions. In this paper, we introduce Deep Delta Learning (DDL), a novel architecture that generalizes the standard residual connection by modulating the identity shortcut with a learnable, data-dependent geometric transformation. This transformation, termed the Delta Operator, constitutes a rank-1 perturbation of the identity matrix, parameterized by a reflection direction vector k(X) and a gating scalar β(X). We provide a spectral analysis of this operator, demonstrating that the gate β(X) enables dynamic interpolation between identity mapping, orthogonal projection, and geometric reflection. Furthermore, we restructure the residual update as a synchronous rank-1 injection, where the gate acts as a dynamic step size governing both the erasure of old information and the writing of new features. This unification empowers the network to explicitly control the spectrum of its layer-wise transition operator, enabling the modeling of complex, non-monotonic dynamics while preserving the stable training characteristics of gated residual architectures.

Recent advancements in operator-type neural networks have shown promising results in approximating the solutions of spatiotemporal Partial Differential Equations (PDEs). However, these neural networks often entail considerable training expenses, and may not always achieve the desired accuracy required in many scientific and engineering disciplines. In this paper, we propose a new Spatiotemporal Fourier Neural Operator (SFNO) that learns maps between Bochner spaces, and a new learning framework to address these issues. This new paradigm leverages wisdom from traditional numerical PDE theory and techniques to refine the pipeline of commonly adopted end-to-end neural operator training and evaluations. Specifically, in the learning problems for the turbulent flow modeling by the Navier-Stokes Equations (NSE), the proposed architecture initiates the training with a few epochs for SFNO, concluding with the freezing of most model parameters. Then, the last linear spectral convolution layer is fine-tuned without the frequency truncation. The optimization uses a negative Sobolev norm for the first time as the loss in operator learning, defined through a reliable functional-type a posteriori error estimator whose evaluation is almost exact thanks to the Parseval identity. This design allows the neural operators to effectively tackle low-frequency errors while the relief of the de-aliasing filter addresses high-frequency errors. Numerical experiments on commonly used benchmarks for the 2D NSE demonstrate significant improvements in both computational efficiency and accuracy, compared to end-to-end evaluation and traditional numerical PDE solvers.

Deep Neural Networks (DNNs) are becoming an important tool in modern computing applications. Accelerating their training is a major challenge and techniques range from distributed algorithms to low-level circuit design. In this survey, we describe the problem from a theoretical perspective, followed by approaches for its parallelization. We present trends in DNN architectures and the resulting implications on parallelization strategies. We then review and model the different types of concurrency in DNNs: from the single operator, through parallelism in network inference and training, to distributed deep learning. We discuss asynchronous stochastic optimization, distributed system architectures, communication schemes, and neural architecture search. Based on those approaches, we extrapolate potential directions for parallelism in deep learning.

Recent work has made clear that the residual pathway is not mere optimization plumbing; it is part of the model's representational machinery. We agree, but argue that the cleanest way to organize this design space is through a two-axis view of the Transformer. A decoder evolves information along two ordered dimensions: sequence position and layer depth. Self-attention already provides adaptive mixing along the sequence axis, whereas the residual stream usually performs fixed addition along the depth axis. If we fix a token position and treat layer index as the ordered variable, then a causal depth-wise residual attention read is exactly the same local operator as causal short sliding-window attention (ShortSWA), except written over depth rather than over sequence. This is the core residual stream duality behind Transformer^2. This perspective also clarifies the recent literature. ELC-BERT and DenseFormer already show that learned aggregation over depth can outperform uniform residual accumulation, while Vertical Attention, DeepCrossAttention (DCA), MUDDFormer, and Attention Residuals move further toward explicit attention-based routing over earlier layers. The key point, however, is that operator-level duality does not imply systems-level symmetry. For large-scale autoregressive models, sequence-axis ShortSWA is usually the more hardware-friendly placement because it reuses token-side sliding-window kernels, KV-cache layouts, and chunked execution. If the goal is instead to change the shortcut itself, Deep Delta Learning (DDL) is the cleaner intervention because it modifies the residual operator directly rather than adding a separate cross-layer retrieval path. Our recommendation is therefore simple: use DDL when the shortcut is the object of interest, and use sequence-axis ShortSWA when the goal is local adaptive mixing.

Numerical simulations play a critical role in design and development of engineering products and processes. Traditional computational methods, such as CFD, can provide accurate predictions but are computationally expensive, particularly for complex geometries. Several machine learning (ML) models have been proposed in the literature to significantly reduce computation time while maintaining acceptable accuracy. However, ML models often face limitations in terms of accuracy and scalability and depend on significant mesh downsampling, which can negatively affect prediction accuracy and generalization. In this work, we propose a novel ML model architecture, DoMINO (Decomposable Multi-scale Iterative Neural Operator) developed in NVIDIA Modulus to address the various challenges of machine learning based surrogate modeling of engineering simulations. DoMINO is a point cloudbased ML model that uses local geometric information to predict flow fields on discrete points. The DoMINO model is validated for the automotive aerodynamics use case using the DrivAerML dataset. Through our experiments we demonstrate the scalability, performance, accuracy and generalization of our model to both in-distribution and out-of-distribution testing samples. Moreover, the results are analyzed using a range of engineering specific metrics important for validating numerical simulations.

Transformer-based neural operators have achieved substantial progress in solving Partial Differential Equations (PDEs) by projecting spatial observations into compact latent tokens and learning physical interactions in latent spaces. However, we reveal that existing learnable projection mechanisms cannot ensure stable and balanced assignments from observation points to latent tokens, causing some latent tokens to be over-assigned while others remain underutilized. This limitation further restricts the design of hierarchical architectures, as assignment imbalance is continuously inherited and amplified across latent spaces, eventually causing severe token collapse in deeper spaces. To address these issues, we propose MoNo (Multiscale Optimal Transport Neural Operator), a progressive multiscale neural operator that efficiently solves PDEs on general geometries through stable latent-space construction. At its core is CoTAP (Cross-scale Optimal Transport Assignment and Projection), a novel latent-space construction method that formulates cross-space assignment between adjacent spaces as an entropy-regularized optimal transport problem, thereby constructing balanced bidirectional projections and stable latent spaces. CoTAP also ensures stable information transfer across multiple latent spaces, further enabling multiscale architectures on general geometries, which in turn support more efficient learning of long-range physical interactions. Extensive experiments demonstrate that MoNo outperforms existing state-of-the-art neural operators in both prediction performance and computational efficiency. Code is available at https://github.com/ZijiangY1116/MoNo.

Deep models have achieved impressive progress in solving partial differential equations (PDEs). A burgeoning paradigm is learning neural operators to approximate the input-output mappings of PDEs. While previous deep models have explored the multiscale architectures and various operator designs, they are limited to learning the operators as a whole in the coordinate space. In real physical science problems, PDEs are complex coupled equations with numerical solvers relying on discretization into high-dimensional coordinate space, which cannot be precisely approximated by a single operator nor efficiently learned due to the curse of dimensionality. We present Latent Spectral Models (LSM) toward an efficient and precise solver for high-dimensional PDEs. Going beyond the coordinate space, LSM enables an attention-based hierarchical projection network to reduce the high-dimensional data into a compact latent space in linear time. Inspired by classical spectral methods in numerical analysis, we design a neural spectral block to solve PDEs in the latent space that approximates complex input-output mappings via learning multiple basis operators, enjoying nice theoretical guarantees for convergence and approximation. Experimentally, LSM achieves consistent state-of-the-art and yields a relative gain of 11.5% averaged on seven benchmarks covering both solid and fluid physics. Code is available at https://github.com/thuml/Latent-Spectral-Models.

Sparse-view Computed Tomography (CT) reconstructs images from a limited number of X-ray projections to reduce radiation and scanning time, which is an ill-posed inverse problem. Existing methods achieve high-fidelity reconstructions but overfit to a fixed acquisition setup, failing to generalize well across sampling rates. For example, convolutional neural networks (CNNs) use the same kernels across resolutions, leading to artifacts when data resolution changes. This is a critical limitation in clinical practice, where acquisition sampling settings vary across organs and diagnostic protocols. We propose Computed Tomography neural Operator (CTO), the first neural operator (NO) framework for CT reconstruction. CTO extends learning from fixed discretized grids to continuous function space, enabling a single model to generalize across measurement sampling rates without retraining. We also propose new NO architectural designs for CT: (i) a dual-domain NO architecture in both sinogram and image spaces, capturing complementary spatial-frequency information, and (ii) rotation-equivariant DIScrete-COntinuous convolutions (DISCO) that exploit the rotational structure inherent in tomographic acquisition. Empirically, CTO outperforms CNNs (> 3.4dB PSNR gain) and other baselines in multi-resolution settings across multiple CT datasets. Compared to state-of-the-art diffusion methods, CTO has 500x faster inference with an average 3dB gain. CTO further demonstrates strong out-of-distribution robustness, maintaining gains under cross-dataset transfer and noisy sinogram conditions. Ablation studies also validate each design choice. CTO establishes neural operators as a principled and practical paradigm for flexible, discretization-agnostic CT reconstruction. Our code is available at https://github.com/neuraloperator/sparse_ct.

Network binarization emerges as one of the most promising compression approaches offering extraordinary computation and memory savings by minimizing the bit-width. However, recent research has shown that applying existing binarization algorithms to diverse tasks, architectures, and hardware in realistic scenarios is still not straightforward. Common challenges of binarization, such as accuracy degradation and efficiency limitation, suggest that its attributes are not fully understood. To close this gap, we present BiBench, a rigorously designed benchmark with in-depth analysis for network binarization. We first carefully scrutinize the requirements of binarization in the actual production and define evaluation tracks and metrics for a comprehensive and fair investigation. Then, we evaluate and analyze a series of milestone binarization algorithms that function at the operator level and with extensive influence. Our benchmark reveals that 1) the binarized operator has a crucial impact on the performance and deployability of binarized networks; 2) the accuracy of binarization varies significantly across different learning tasks and neural architectures; 3) binarization has demonstrated promising efficiency potential on edge devices despite the limited hardware support. The results and analysis also lead to a promising paradigm for accurate and efficient binarization. We believe that BiBench will contribute to the broader adoption of binarization and serve as a foundation for future research. The code for our BiBench is released https://github.com/htqin/BiBench .

Image copy detection is an important task for content moderation. We introduce SSCD, a model that builds on a recent self-supervised contrastive training objective. We adapt this method to the copy detection task by changing the architecture and training objective, including a pooling operator from the instance matching literature, and adapting contrastive learning to augmentations that combine images. Our approach relies on an entropy regularization term, promoting consistent separation between descriptor vectors, and we demonstrate that this significantly improves copy detection accuracy. Our method produces a compact descriptor vector, suitable for real-world web scale applications. Statistical information from a background image distribution can be incorporated into the descriptor. On the recent DISC2021 benchmark, SSCD is shown to outperform both baseline copy detection models and self-supervised architectures designed for image classification by huge margins, in all settings. For example, SSCD out-performs SimCLR descriptors by 48% absolute. Code is available at https://github.com/facebookresearch/sscd-copy-detection.

In this work, we propose a concise neural operator architecture for operator learning. Drawing an analogy with a conventional fully connected neural network, we define the neural operator as follows: the output of the i-th neuron in a nonlinear operator layer is defined by mathcal O_i(u) = sigmaleft( sum_j mathcal W_{ij} u + mathcal B_{ij}right). Here, mathcal W_{ij} denotes the bounded linear operator connecting j-th input neuron to i-th output neuron, and the bias mathcal B_{ij} takes the form of a function rather than a scalar. Given its new universal approximation property, the efficient parameterization of the bounded linear operators between two neurons (Banach spaces) plays a critical role. As a result, we introduce MgNO, utilizing multigrid structures to parameterize these linear operators between neurons. This approach offers both mathematical rigor and practical expressivity. Additionally, MgNO obviates the need for conventional lifting and projecting operators typically required in previous neural operators. Moreover, it seamlessly accommodates diverse boundary conditions. Our empirical observations reveal that MgNO exhibits superior ease of training compared to other CNN-based models, while also displaying a reduced susceptibility to overfitting when contrasted with spectral-type neural operators. We demonstrate the efficiency and accuracy of our method with consistently state-of-the-art performance on different types of partial differential equations (PDEs).

A large class of inverse problems for PDEs are only well-defined as mappings from operators to functions. Existing operator learning frameworks map functions to functions and need to be modified to learn inverse maps from data. We propose a novel architecture termed Neural Inverse Operators (NIOs) to solve these PDE inverse problems. Motivated by the underlying mathematical structure, NIO is based on a suitable composition of DeepONets and FNOs to approximate mappings from operators to functions. A variety of experiments are presented to demonstrate that NIOs significantly outperform baselines and solve PDE inverse problems robustly, accurately and are several orders of magnitude faster than existing direct and PDE-constrained optimization methods.

Neural operators have become increasingly popular in solving partial differential equations (PDEs) due to their superior capability to capture intricate mappings between function spaces over complex domains. However, the data-hungry nature of operator learning inevitably poses a bottleneck for their widespread applications. At the core of the challenge lies the absence of transferability of neural operators to new geometries. To tackle this issue, we propose operator learning with domain decomposition, a local-to-global framework to solve PDEs on arbitrary geometries. Under this framework, we devise an iterative scheme Schwarz Neural Inference (SNI). This scheme allows for partitioning of the problem domain into smaller subdomains, on which local problems can be solved with neural operators, and stitching local solutions to construct a global solution. Additionally, we provide a theoretical analysis of the convergence rate and error bound. We conduct extensive experiments on several representative PDEs with diverse boundary conditions and achieve remarkable geometry generalization compared to alternative methods. These analysis and experiments demonstrate the proposed framework's potential in addressing challenges related to geometry generalization and data efficiency.

Single-operator learning involves training a deep neural network to learn a specific operator, whereas recent work in multi-operator learning uses an operator embedding structure to train a single neural network on data from multiple operators. Thus, multi-operator learning is capable of predicting a range of operators within one model. In this work, we propose pretraining and fine-tuning strategies for solving PDEs using multi-operator learning. One key aspect is that by increasing the number of families of operators used in pretraining, a PDE foundation model can be fine-tuned to downstream tasks involving new PDEs with a limited number of samples, thus outperforming single operator neural networks. Specifically, a multi-operator learning model pre-trained with data from diverse PDE families can predict unseen operators after fine-tuning with only a limited number of operators from the new family, enabling them to serve as a data-free PDE solver. We also show that the proposed training and fine-tuning method is able to predict new operators in zero-shot prediction without samples. Additionally, we introduce a PDE-agnostic meta-learning algorithm to improve the adaptability of the model to various PDEs by providing a better parameter initialization process. To address the needs of applications with limited computing resources, we explore low-rank adaptation methods that reduce computational costs while enhancing solver accuracy. Lastly, by examining the scaling law with respect to the number of operator families, we establish and highlight its potential for broad adaptation in PDE-solving tasks.

Expanding on neural operators, we propose a novel framework for stochastic process learning across arbitrary domains. In particular, we develop operator flow matching (OFM) for learning stochastic process priors on function spaces. OFM provides the probability density of the values of any collection of points and enables mathematically tractable functional regression at new points with mean and density estimation. Our method outperforms state-of-the-art models in stochastic process learning, functional regression, and prior learning.

While it is widely known that neural networks are universal approximators of continuous functions, a less known and perhaps more powerful result is that a neural network with a single hidden layer can approximate accurately any nonlinear continuous operator. This universal approximation theorem is suggestive of the potential application of neural networks in learning nonlinear operators from data. However, the theorem guarantees only a small approximation error for a sufficient large network, and does not consider the important optimization and generalization errors. To realize this theorem in practice, we propose deep operator networks (DeepONets) to learn operators accurately and efficiently from a relatively small dataset. A DeepONet consists of two sub-networks, one for encoding the input function at a fixed number of sensors x_i, i=1,dots,m (branch net), and another for encoding the locations for the output functions (trunk net). We perform systematic simulations for identifying two types of operators, i.e., dynamic systems and partial differential equations, and demonstrate that DeepONet significantly reduces the generalization error compared to the fully-connected networks. We also derive theoretically the dependence of the approximation error in terms of the number of sensors (where the input function is defined) as well as the input function type, and we verify the theorem with computational results. More importantly, we observe high-order error convergence in our computational tests, namely polynomial rates (from half order to fourth order) and even exponential convergence with respect to the training dataset size.

Although very successfully used in conventional machine learning, convolution based neural network architectures -- believed to be inconsistent in function space -- have been largely ignored in the context of learning solution operators of PDEs. Here, we present novel adaptations for convolutional neural networks to demonstrate that they are indeed able to process functions as inputs and outputs. The resulting architecture, termed as convolutional neural operators (CNOs), is designed specifically to preserve its underlying continuous nature, even when implemented in a discretized form on a computer. We prove a universality theorem to show that CNOs can approximate operators arising in PDEs to desired accuracy. CNOs are tested on a novel suite of benchmarks, encompassing a diverse set of PDEs with possibly multi-scale solutions and are observed to significantly outperform baselines, paving the way for an alternative framework for robust and accurate operator learning. Our code is publicly available at https://github.com/bogdanraonic3/ConvolutionalNeuralOperator

We introduce Topological Neural Operators (TNOs), a principled framework for operator learning on cell complexes that lifts neural operators (NOs) from functions on points and/or edges to topological domains. TNOs represent data as features defined on cells of varying dimension and model their interactions through Discrete Exterior Calculus, enabling explicit cross-dimensional coupling via gradient-, curl-, and divergence-type operators. The key design principle is to decouple where information flows, as governed by fixed topological operators, from how it is transformed (which is learned), yielding models that respect the geometric support of physical quantities and expose conservation and compatibility structure. We further propose Hierarchical TNOs (HTNOs), which incorporate learned coarse complexes to propagate long-range and topology-dependent information. Our framework subsumes existing NOs as a special case, providing a unified perspective on operator learning across discretizations. Across a range of PDE benchmarks, including irregular-geometry flow problems, TNOs and HTNOs improve accuracy; controlled studies further isolate the benefits of native higher-rank and topological structure. Project page: https://circle-group.github.io/research/TNO

Learning mappings between infinite-dimensional function spaces, or operator learning, is essential for many machine learning applications. Although transformer-based operators are popular, they often rely on token-wise attention. These methods treat continuous fields as discrete tokens and usually ignore the global functional structure. We introduce Functional Attention, which reinterprets attention as a functional correspondence between adaptive bases. Inspired by geometric functional maps, our method replaces softmax affinities with structured linear operators. This yields a compact, generalizable, resolution-invariant representation that explicitly captures global dependencies. Experiments demonstrate that Functional Attention can match state-of-the-art performance in many operator learning tasks, including solving PDEs, 3D segmentation, and regression, while remaining robust to varying discretizations. Project page is available at https://github.com/xjffff/FUNCATTN.

In this paper, we apply the self-attention from the state-of-the-art Transformer in Attention Is All You Need for the first time to a data-driven operator learning problem related to partial differential equations. An effort is put together to explain the heuristics of, and to improve the efficacy of the attention mechanism. By employing the operator approximation theory in Hilbert spaces, it is demonstrated for the first time that the softmax normalization in the scaled dot-product attention is sufficient but not necessary. Without softmax, the approximation capacity of a linearized Transformer variant can be proved to be comparable to a Petrov-Galerkin projection layer-wise, and the estimate is independent with respect to the sequence length. A new layer normalization scheme mimicking the Petrov-Galerkin projection is proposed to allow a scaling to propagate through attention layers, which helps the model achieve remarkable accuracy in operator learning tasks with unnormalized data. Finally, we present three operator learning experiments, including the viscid Burgers' equation, an interface Darcy flow, and an inverse interface coefficient identification problem. The newly proposed simple attention-based operator learner, Galerkin Transformer, shows significant improvements in both training cost and evaluation accuracy over its softmax-normalized counterparts.

We propose PROSE-FD, a zero-shot multimodal PDE foundational model for simultaneous prediction of heterogeneous two-dimensional physical systems related to distinct fluid dynamics settings. These systems include shallow water equations and the Navier-Stokes equations with incompressible and compressible flow, regular and complex geometries, and different buoyancy settings. This work presents a new transformer-based multi-operator learning approach that fuses symbolic information to perform operator-based data prediction, i.e. non-autoregressive. By incorporating multiple modalities in the inputs, the PDE foundation model builds in a pathway for including mathematical descriptions of the physical behavior. We pre-train our foundation model on 6 parametric families of equations collected from 13 datasets, including over 60K trajectories. Our model outperforms popular operator learning, computer vision, and multi-physics models, in benchmark forward prediction tasks. We test our architecture choices with ablation studies.

In recent years, large convolutional neural networks have been widely used as tools for image deblurring, because of their ability in restoring images very precisely. It is well known that image deblurring is mathematically modeled as an ill-posed inverse problem and its solution is difficult to approximate when noise affects the data. Really, one limitation of neural networks for deblurring is their sensitivity to noise and other perturbations, which can lead to instability and produce poor reconstructions. In addition, networks do not necessarily take into account the numerical formulation of the underlying imaging problem, when trained end-to-end. In this paper, we propose some strategies to improve stability without losing to much accuracy to deblur images with deep-learning based methods. First, we suggest a very small neural architecture, which reduces the execution time for training, satisfying a green AI need, and does not extremely amplify noise in the computed image. Second, we introduce a unified framework where a pre-processing step balances the lack of stability of the following, neural network-based, step. Two different pre-processors are presented: the former implements a strong parameter-free denoiser, and the latter is a variational model-based regularized formulation of the latent imaging problem. This framework is also formally characterized by mathematical analysis. Numerical experiments are performed to verify the accuracy and stability of the proposed approaches for image deblurring when unknown or not-quantified noise is present; the results confirm that they improve the network stability with respect to noise. In particular, the model-based framework represents the most reliable trade-off between visual precision and robustness.

Neural operators, as an efficient surrogate model for learning the solutions of PDEs, have received extensive attention in the field of scientific machine learning. Among them, attention-based neural operators have become one of the mainstreams in related research. However, existing approaches overfit the limited training data due to the considerable number of parameters in the attention mechanism. To address this, we develop an orthogonal attention based on the eigendecomposition of the kernel integral operator and the neural approximation of eigenfunctions. The orthogonalization naturally poses a proper regularization effect on the resulting neural operator, which aids in resisting overfitting and boosting generalization. Experiments on six standard neural operator benchmark datasets comprising both regular and irregular geometries show that our method can outperform competing baselines with decent margins.

Neural operators learn mappings between function spaces, which is practical for learning solution operators of PDEs and other scientific modeling applications. Among them, the Fourier neural operator (FNO) is a popular architecture that performs global convolutions in the Fourier space. However, such global operations are often prone to over-smoothing and may fail to capture local details. In contrast, convolutional neural networks (CNN) can capture local features but are limited to training and inference at a single resolution. In this work, we present a principled approach to operator learning that can capture local features under two frameworks by learning differential operators and integral operators with locally supported kernels. Specifically, inspired by stencil methods, we prove that we obtain differential operators under an appropriate scaling of the kernel values of CNNs. To obtain local integral operators, we utilize suitable basis representations for the kernels based on discrete-continuous convolutions. Both these approaches preserve the properties of operator learning and, hence, the ability to predict at any resolution. Adding our layers to FNOs significantly improves their performance, reducing the relative L2-error by 34-72% in our experiments, which include a turbulent 2D Navier-Stokes and the spherical shallow water equations.

Neural Module Networks, originally proposed for the task of visual question answering, are a class of neural network architectures that involve human-specified neural modules, each designed for a specific form of reasoning. In current formulations of such networks only the parameters of the neural modules and/or the order of their execution is learned. In this work, we further expand this approach and also learn the underlying internal structure of modules in terms of the ordering and combination of simple and elementary arithmetic operators. Our results show that one is indeed able to simultaneously learn both internal module structure and module sequencing without extra supervisory signals for module execution sequencing. With this approach, we report performance comparable to models using hand-designed modules.

Transformers, and the attention mechanism in particular, have become ubiquitous in machine learning. Their success in modeling nonlocal, long-range correlations has led to their widespread adoption in natural language processing, computer vision, and time series problems. Neural operators, which map spaces of functions into spaces of functions, are necessarily both nonlinear and nonlocal if they are universal; it is thus natural to ask whether the attention mechanism can be used in the design of neural operators. Motivated by this, we study transformers in the function space setting. We formulate attention as a map between infinite dimensional function spaces and prove that the attention mechanism as implemented in practice is a Monte Carlo or finite difference approximation of this operator. The function space formulation allows for the design of transformer neural operators, a class of architectures designed to learn mappings between function spaces. In this paper, we state and prove the first universal approximation result for transformer neural operators, using only a slight modification of the architecture implemented in practice. The prohibitive cost of applying the attention operator to functions defined on multi-dimensional domains leads to the need for more efficient attention-based architectures. For this reason we also introduce a function space generalization of the patching strategy from computer vision, and introduce a class of associated neural operators. Numerical results, on an array of operator learning problems, demonstrate the promise of our approaches to function space formulations of attention and their use in neural operators.

The classical development of neural networks has primarily focused on learning mappings between finite dimensional Euclidean spaces or finite sets. We propose a generalization of neural networks to learn operators, termed neural operators, that map between infinite dimensional function spaces. We formulate the neural operator as a composition of linear integral operators and nonlinear activation functions. We prove a universal approximation theorem for our proposed neural operator, showing that it can approximate any given nonlinear continuous operator. The proposed neural operators are also discretization-invariant, i.e., they share the same model parameters among different discretization of the underlying function spaces. Furthermore, we introduce four classes of efficient parameterization, viz., graph neural operators, multi-pole graph neural operators, low-rank neural operators, and Fourier neural operators. An important application for neural operators is learning surrogate maps for the solution operators of partial differential equations (PDEs). We consider standard PDEs such as the Burgers, Darcy subsurface flow, and the Navier-Stokes equations, and show that the proposed neural operators have superior performance compared to existing machine learning based methodologies, while being several orders of magnitude faster than conventional PDE solvers.

Neural operators have shown promise in learning solution maps of partial differential equations (PDEs), but they often struggle to generalize when test inputs lie outside the training distribution, such as novel initial conditions, unseen PDE coefficients or unseen physics. Prior works address this limitation with large-scale multiple physics pretraining followed by fine-tuning, but this still requires examples from the new dynamics, falling short of true zero-shot generalization. In this work, we propose a method to enhance generalization at test time, i.e., without modifying pretrained weights. Building on DISCO, which provides a dictionary of neural operators trained across different dynamics, we introduce a neural operator splitting strategy that, at test time, searches over compositions of training operators to approximate unseen dynamics. On challenging out-of-distribution tasks including parameter extrapolation and novel combinations of physics phenomena, our approach achieves state-of-the-art zero-shot generalization results, while being able to recover the underlying PDE parameters. These results underscore test-time computation as a key avenue for building flexible, compositional, and generalizable neural operators.

The very challenging task of learning solution operators of PDEs on arbitrary domains accurately and efficiently is of vital importance to engineering and industrial simulations. Despite the existence of many operator learning algorithms to approximate such PDEs, we find that accurate models are not necessarily computationally efficient and vice versa. We address this issue by proposing a geometry aware operator transformer (GAOT) for learning PDEs on arbitrary domains. GAOT combines novel multiscale attentional graph neural operator encoders and decoders, together with geometry embeddings and (vision) transformer processors to accurately map information about the domain and the inputs into a robust approximation of the PDE solution. Multiple innovations in the implementation of GAOT also ensure computational efficiency and scalability. We demonstrate this significant gain in both accuracy and efficiency of GAOT over several baselines on a large number of learning tasks from a diverse set of PDEs, including achieving state of the art performance on a large scale three-dimensional industrial CFD dataset.

The neural operator has emerged as a powerful tool in learning mappings between function spaces in PDEs. However, when faced with real-world physical data, which are often highly non-uniformly distributed, it is challenging to use mesh-based techniques such as the FFT. To address this, we introduce the Non-Uniform Neural Operator (NUNO), a comprehensive framework designed for efficient operator learning with non-uniform data. Leveraging a K-D tree-based domain decomposition, we transform non-uniform data into uniform grids while effectively controlling interpolation error, thereby paralleling the speed and accuracy of learning from non-uniform data. We conduct extensive experiments on 2D elasticity, (2+1)D channel flow, and a 3D multi-physics heatsink, which, to our knowledge, marks a novel exploration into 3D PDE problems with complex geometries. Our framework has reduced error rates by up to 60% and enhanced training speeds by 2x to 30x. The code is now available at https://github.com/thu-ml/NUNO.

We frame the problem of unifying representations in neural models as one of sparse model recovery and introduce a framework that extends sparse autoencoders (SAEs) to lifted spaces and infinite-dimensional function spaces, enabling mechanistic interpretability of large neural operators (NO). While the Platonic Representation Hypothesis suggests that neural networks converge to similar representations across architectures, the representational properties of neural operators remain underexplored despite their growing importance in scientific computing. We compare the inference and training dynamics of SAEs, lifted-SAE, and SAE neural operators. We highlight how lifting and operator modules introduce beneficial inductive biases, enabling faster recovery, improved recovery of smooth concepts, and robust inference across varying resolutions, a property unique to neural operators.

In this paper, we construct a mixture of neural operators (MoNOs) between function spaces whose complexity is distributed over a network of expert neural operators (NOs), with each NO satisfying parameter scaling restrictions. Our main result is a distributed universal approximation theorem guaranteeing that any Lipschitz non-linear operator between L^2([0,1]^d) spaces can be approximated uniformly over the Sobolev unit ball therein, to any given varepsilon>0 accuracy, by an MoNO while satisfying the constraint that: each expert NO has a depth, width, and rank of O(varepsilon^{-1}). Naturally, our result implies that the required number of experts must be large, however, each NO is guaranteed to be small enough to be loadable into the active memory of most computers for reasonable accuracies varepsilon. During our analysis, we also obtain new quantitative expression rates for classical NOs approximating uniformly continuous non-linear operators uniformly on compact subsets of L^2([0,1]^d).

Deep Neural Networks (DNNs) are generated by sequentially performing linear and non-linear processes. Using a combination of linear and non-linear procedures is critical for generating a sufficiently deep feature space. The majority of non-linear operators are derivations of activation functions or pooling functions. Mathematical morphology is a branch of mathematics that provides non-linear operators for a variety of image processing problems. We investigate the utility of integrating these operations in an end-to-end deep learning framework in this paper. DNNs are designed to acquire a realistic representation for a particular job. Morphological operators give topological descriptors that convey salient information about the shapes of objects depicted in images. We propose a method based on meta-learning to incorporate morphological operators into DNNs. The learned architecture demonstrates how our novel morphological operations significantly increase DNN performance on various tasks, including picture classification and edge detection.

Recent advances in scientific machine learning (SciML) have enabled neural operators (NOs) to serve as powerful surrogates for modeling the dynamic evolution of physical systems governed by partial differential equations (PDEs). While existing approaches focus primarily on learning simulations from the target PDE, they often overlook more fundamental physical principles underlying these equations. Inspired by how numerical solvers are compatible with simulations of different settings of PDEs, we propose a multiphysics training framework that jointly learns from both the original PDEs and their simplified basic forms. Our framework enhances data efficiency, reduces predictive errors, and improves out-of-distribution (OOD) generalization, particularly in scenarios involving shifts of physical parameters and synthetic-to-real transfer. Our method is architecture-agnostic and demonstrates consistent improvements in normalized root mean square error (nRMSE) across a wide range of 1D/2D/3D PDE problems. Through extensive experiments, we show that explicit incorporation of fundamental physics knowledge significantly strengthens the generalization ability of neural operators. We will release models and codes at https://sites.google.com/view/sciml-fundemental-pde.

Time-dependent partial differential equations are ubiquitous in physics-based modeling, but they remain computationally intensive in many-query scenarios, such as real-time forecasting, optimal control, and uncertainty quantification. Reduced-order modeling (ROM) addresses these challenges by constructing a low-dimensional surrogate model but relies on a fixed discretization, which limits flexibility across varying meshes during evaluation. Operator learning approaches, such as neural operators, offer an alternative by parameterizing mappings between infinite-dimensional function spaces, enabling adaptation to data across different resolutions. Whereas ROM provides rigorous numerical error estimates, neural operator learning largely focuses on discretization convergence and invariance without quantifying the error between the infinite-dimensional and the discretized operators. This work introduces the reduced-order neural operator modeling (RONOM) framework, which bridges concepts from ROM and operator learning. We establish a discretization error bound analogous to those in ROM, and get insights into RONOM's discretization convergence and discretization robustness. Moreover, three numerical examples are presented that compare RONOM to existing neural operators for solving partial differential equations. The results demonstrate that RONOM using standard vector-to-vector neural networks can achieve comparable performance in input generalization and achieves superior performance in both spatial super-resolution and discretization robustness, while also offering novel insights into temporal super-resolution scenarios and ROM-based approaches for learning on time-dependent data.

We present a data-driven control framework for partial differential equations (PDEs). Our approach integrates Time-Integrated Deep Operator Networks (TI-DeepONets) as differentiable PDE surrogate models within the Differentiable Predictive Control (DPC)-a self-supervised learning framework for constrained neural control policies. The TI-DeepONet architecture learns temporal derivatives and couples them with numerical integrators, while the DPC algorithm uses automatic differentiation to compute policy gradients by backpropagating the expectations of the optimal control loss through the learned TI-DeepONet. This approach enables efficient offline optimization of neural policies without the need for online optimization or supervisory controllers. We empirically demonstrate the proposed method across diverse PDE systems, including the heat, the nonlinear Burgers', and the reaction-diffusion equations. The learned policies achieve target tracking, constraint satisfaction, and curvature minimization objectives, while generalizing across distributions of initial conditions and parameters. Moreover, we demonstrate four orders of magnitude acceleration at inference time compared to nonlinear model predictive control benchmarks. These results highlight the promise of operator learning for scalable model-based control of PDEs.

Autoregressive learning of time-stepping operators provides an effective approach to data-driven partial differential equation (PDE) simulation, yet for conservation laws, they face a fundamental challenge: learned updates may violate global conservation over long rollouts. For the important subclass of mass-conservation-type equations, the problem is compounded by inherent physical bounds (e.g., nonnegativity or concentrations in [0,1]) whose violation further destabilizes predictions. We introduce FluxNet, which learns cumulative transport amounts representing the total conserved quantity redistributed between each cell and a configurable neighborhood over the full surrogate interval. A conservative update guarantees exact discrete conservation by construction; modular capacity-constrained transport heads (L, U, and D) enforce lower bounds, upper bounds, or near-zero dual-bound violations through architectural design. Unlike flux-rate surrogates that require temporal integration and thus inherit CFL constraints, FluxNet involves no such integration; configurable transport neighborhoods enable large-timestep prediction at full spatial resolution. Ghost cells extend the framework to non-periodic boundaries. Experiments on four benchmarks (1D convection--diffusion, 2D shallow water, 1D traffic flow, 2D Cahn--Hilliard) demonstrate exact conservation, structural bound preservation, architecture modularity, and superior stability over flux-rate surrogates at large temporal strides. The code is publicly available at: https://github.com/Lan-zs/FluxNet.

Compressing a predefined deep neural network (DNN) into a compact sub-network with competitive performance is crucial in the efficient machine learning realm. This topic spans various techniques, from structured pruning to neural architecture search, encompassing both pruning and erasing operators perspectives. Despite advancements, existing methods suffers from complex, multi-stage processes that demand substantial engineering and domain knowledge, limiting their broader applications. We introduce the third-generation Only-Train-Once (OTOv3), which first automatically trains and compresses a general DNN through pruning and erasing operations, creating a compact and competitive sub-network without the need of fine-tuning. OTOv3 simplifies and automates the training and compression process, minimizes the engineering efforts required from users. It offers key technological advancements: (i) automatic search space construction for general DNNs based on dependency graph analysis; (ii) Dual Half-Space Projected Gradient (DHSPG) and its enhanced version with hierarchical search (H2SPG) to reliably solve (hierarchical) structured sparsity problems and ensure sub-network validity; and (iii) automated sub-network construction using solutions from DHSPG/H2SPG and dependency graphs. Our empirical results demonstrate the efficacy of OTOv3 across various benchmarks in structured pruning and neural architecture search. OTOv3 produces sub-networks that match or exceed the state-of-the-arts. The source code will be available at https://github.com/tianyic/only_train_once.

We consider the general class of time-homogeneous stochastic dynamical systems, both discrete and continuous, and study the problem of learning a representation of the state that faithfully captures its dynamics. This is instrumental to learning the transfer operator or the generator of the system, which in turn can be used for numerous tasks, such as forecasting and interpreting the system dynamics. We show that the search for a good representation can be cast as an optimization problem over neural networks. Our approach is supported by recent results in statistical learning theory, highlighting the role of approximation error and metric distortion in the learning problem. The objective function we propose is associated with projection operators from the representation space to the data space, overcomes metric distortion, and can be empirically estimated from data. In the discrete-time setting, we further derive a relaxed objective function that is differentiable and numerically well-conditioned. We compare our method against state-of-the-art approaches on different datasets, showing better performance across the board.

Time domain simulation, i.e., modeling the system's evolution over time, is a crucial tool for studying and enhancing power system stability and dynamic performance. However, these simulations become computationally intractable for renewable-penetrated grids, due to the small simulation time step required to capture renewable energy resources' ultra-fast dynamic phenomena in the range of 1-50 microseconds. This creates a critical need for solutions that are both fast and scalable, posing a major barrier for the stable integration of renewable energy resources and thus climate change mitigation. This paper explores operator learning, a family of machine learning methods that learn mappings between functions, as a surrogate model for these costly simulations. The paper investigates, for the first time, the fundamental concept of simulation time step-invariance, which enables models trained on coarse time steps to generalize to fine-resolution dynamics. Three operator learning methods are benchmarked on a simple test system that, while not incorporating practical complexities of renewable-penetrated grids, serves as a first proof-of-concept to demonstrate the viability of time step-invariance. Models are evaluated on (i) zero-shot super-resolution, where training is performed on a coarse simulation time step and inference is performed at super-resolution, and (ii) generalization between stable and unstable dynamic regimes. This work addresses a key challenge in the integration of renewable energy for the mitigation of climate change by benchmarking operator learning methods to model physical systems.

In this work, we propose marginalized operators, a new class of off-policy evaluation operators for reinforcement learning. Marginalized operators strictly generalize generic multi-step operators, such as Retrace, as special cases. Marginalized operators also suggest a form of sample-based estimates with potential variance reduction, compared to sample-based estimates of the original multi-step operators. We show that the estimates for marginalized operators can be computed in a scalable way, which also generalizes prior results on marginalized importance sampling as special cases. Finally, we empirically demonstrate that marginalized operators provide performance gains to off-policy evaluation and downstream policy optimization algorithms.

Currently, prominent Transformer architectures applied on graphs and meshes for shape analysis tasks employ traditional attention layers that heavily utilize spectral features requiring costly eigenvalue decomposition-based methods. To encode the mesh structure, these methods derive positional embeddings, that heavily rely on eigenvalue decomposition based operations, e.g. on the Laplacian matrix, or on heat-kernel signatures, which are then concatenated to the input features. This paper proposes a novel approach inspired by the explicit construction of the Hodge Laplacian operator in Discrete Exterior Calculus as a product of discrete Hodge operators and exterior derivatives, i.e. (L := star_0^{-1} d_0^T star_1 d_0). We adjust the Transformer architecture in a novel deep learning layer that utilizes the multi-head attention mechanism to approximate Hodge matrices star_0, star_1 and star_2 and learn families of discrete operators L that act on mesh vertices, edges and faces. Our approach results in a computationally-efficient architecture that achieves comparable performance in mesh segmentation and classification tasks, through a direct learning framework, while eliminating the need for costly eigenvalue decomposition operations or complex preprocessing operations.

Neural operators, as a powerful approximation to the non-linear operators between infinite-dimensional function spaces, have proved to be promising in accelerating the solution of partial differential equations (PDE). However, it requires a large amount of simulated data, which can be costly to collect. This can be avoided by learning physics from the physics-constrained loss, which we refer to it as mean squared residual (MSR) loss constructed by the discretized PDE. We investigate the physical information in the MSR loss, which we called long-range entanglements, and identify the challenge that the neural network requires the capacity to model the long-range entanglements in the spatial domain of the PDE, whose patterns vary in different PDEs. To tackle the challenge, we propose LordNet, a tunable and efficient neural network for modeling various entanglements. Inspired by the traditional solvers, LordNet models the long-range entanglements with a series of matrix multiplications, which can be seen as the low-rank approximation to the general fully-connected layers and extracts the dominant pattern with reduced computational cost. The experiments on solving Poisson's equation and (2D and 3D) Navier-Stokes equation demonstrate that the long-range entanglements from the MSR loss can be well modeled by the LordNet, yielding better accuracy and generalization ability than other neural networks. The results show that the Lordnet can be 40times faster than traditional PDE solvers. In addition, LordNet outperforms other modern neural network architectures in accuracy and efficiency with the smallest parameter size.

No interatomic potential has offered universality across chemistry, near-first-principles accuracy and the speed of empirical potentials at once. Here we introduce DPA4C, an equivariant potential whose architecture and compressed CUDA operators are co-designed under deployment constraints to pursue accuracy and efficiency together. Five variants spanning a 49-fold parameter range form the high-throughput end of the measured accuracy--throughput frontier. The largest variant approaches the accuracy of the MACE-Omat models at about two orders of magnitude higher measured throughput. The most compact reduces the energy, force and stress errors of the fastest existing universal MLIP by 61.4%, 48.1% and 34.3% at 1.92 times its saturated throughput. All five variants complete multimillion-atom simulations on a single GPU and run molecular dynamics for 2.048 billion atoms on 1,024 16-GB NVIDIA V100 GPUs at 83.3--91.2% weak-scaling efficiency. Compared with the MEAM empirical potential, DPA4C-Nano reaches 1.8 and 2.5 times the saturated throughput in single-GPU scans on the same V100 hardware for diamond carbon and FCC copper, respectively. DPA4C therefore brings quantum-trained universal accuracy into a regime of speed and system size previously associated with empirical potentials.

In this technical report, we present the Ring-linear model series, specifically including Ring-mini-linear-2.0 and Ring-flash-linear-2.0. Ring-mini-linear-2.0 comprises 16B parameters and 957M activations, while Ring-flash-linear-2.0 contains 104B parameters and 6.1B activations. Both models adopt a hybrid architecture that effectively integrates linear attention and softmax attention, significantly reducing I/O and computational overhead in long-context inference scenarios. Compared to a 32 billion parameter dense model, this series reduces inference cost to 1/10, and compared to the original Ring series, the cost is also reduced by over 50%. Furthermore, through systematic exploration of the ratio between different attention mechanisms in the hybrid architecture, we have identified the currently optimal model structure. Additionally, by leveraging our self-developed high-performance FP8 operator library-linghe, overall training efficiency has been improved by 50%. Benefiting from the high alignment between the training and inference engine operators, the models can undergo long-term, stable, and highly efficient optimization during the reinforcement learning phase, consistently maintaining SOTA performance across multiple challenging complex reasoning benchmarks.

Neural operators generalize classical neural networks to maps between infinite-dimensional spaces, e.g., function spaces. Prior works on neural operators proposed a series of novel methods to learn such maps and demonstrated unprecedented success in learning solution operators of partial differential equations. Due to their close proximity to fully connected architectures, these models mainly suffer from high memory usage and are generally limited to shallow deep learning models. In this paper, we propose U-shaped Neural Operator (U-NO), a U-shaped memory enhanced architecture that allows for deeper neural operators. U-NOs exploit the problem structures in function predictions and demonstrate fast training, data efficiency, and robustness with respect to hyperparameters choices. We study the performance of U-NO on PDE benchmarks, namely, Darcy's flow law and the Navier-Stokes equations. We show that U-NO results in an average of 26% and 44% prediction improvement on Darcy's flow and turbulent Navier-Stokes equations, respectively, over the state of the art. On Navier-Stokes 3D spatiotemporal operator learning task, we show U-NO provides 37% improvement over the state of art methods.

In many reasoning problems, the premises are not observed as discrete symbols, but must be inferred from high-dimensional inputs. Further, the predicate vocabulary, argument structure, and trusted evidence are supplied by a Knowledge Graph (KG), or rule definitions. Classical neuro-symbolic pipelines have a discrete interface between perception and deduction. We present a neuro-soft-symbolic architecture for differentiable deductive reasoning over latent perceptual facts and knowledge-provided predicates. SoftReason removes the gradient gap by representing the deductive state as a local soft interpretation tensor over candidate constants and predicates. Perception proposes probabilistic base facts, KG triples enter as high-confidence soft evidence, and every query anchor, predicate choice, and closure update remains differentiable. Our core innovation is a learned differentiable lift of the immediate-consequence operator. It uses predicate-definition embeddings and latent composition channels to form soft body-predicate mixtures, aggregate over all possible witnesses, propose query-conditioned head facts, and update the interpretation through a monotone probabilistic OR. We instantiate the framework on Knowledge-aware Visual Question Answering (KVQA), and demonstrates how SoftReason supports end-to-end perceptual grounding, KG evidence injection, and differentiable deductive closure in one trainable architecture.

We present a feasibility-seeking approach to neural network training. This mathematical optimization framework is distinct from conventional gradient-based loss minimization and uses projection operators and iterative projection algorithms. We reformulate training as a large-scale feasibility problem: finding network parameters and states that satisfy local constraints derived from its elementary operations. Training then involves projecting onto these constraints, a local operation that can be parallelized across the network. We introduce PJAX, a JAX-based software framework that enables this paradigm. PJAX composes projection operators for elementary operations, automatically deriving the solution operators for the feasibility problems (akin to autodiff for derivatives). It inherently supports GPU/TPU acceleration, provides a familiar NumPy-like API, and is extensible. We train diverse architectures (MLPs, CNNs, RNNs) on standard benchmarks using PJAX, demonstrating its functionality and generality. Our results show that this approach is as a compelling alternative to gradient-based training, with clear advantages in parallelism and the ability to handle non-differentiable operations.

Algorithm learning is a core problem in artificial intelligence with significant implications on automation level that can be achieved by machines. Recently deep learning methods are emerging for synthesizing an algorithm from its input-output examples, the most successful being the Neural GPU, capable of learning multiplication. We present several improvements to the Neural GPU that substantially reduces training time and improves generalization. We introduce a new technique - hard nonlinearities with saturation costs- that has general applicability. We also introduce a technique of diagonal gates that can be applied to active-memory models. The proposed architecture is the first capable of learning decimal multiplication end-to-end.

Trainable layers such as convolutional building blocks are the standard network design choices by learning parameters to capture the global context through successive spatial operations. When designing an efficient network, trainable layers such as the depthwise convolution is the source of efficiency in the number of parameters and FLOPs, but there was little improvement to the model speed in practice. This paper argues that simple built-in parameter-free operations can be a favorable alternative to the efficient trainable layers replacing spatial operations in a network architecture. We aim to break the stereotype of organizing the spatial operations of building blocks into trainable layers. Extensive experimental analyses based on layer-level studies with fully-trained models and neural architecture searches are provided to investigate whether parameter-free operations such as the max-pool are functional. The studies eventually give us a simple yet effective idea for redesigning network architectures, where the parameter-free operations are heavily used as the main building block without sacrificing the model accuracy as much. Experimental results on the ImageNet dataset demonstrate that the network architectures with parameter-free operations could enjoy the advantages of further efficiency in terms of model speed, the number of the parameters, and FLOPs. Code and ImageNet pretrained models are available at https://github.com/naver-ai/PfLayer.

Neural operators have proven to be a promising approach for modeling spatiotemporal systems in the physical sciences. However, training these models for large systems can be quite challenging as they incur significant computational and memory expense -- these systems are often forced to rely on autoregressive time-stepping of the neural network to predict future temporal states. While this is effective in managing costs, it can lead to uncontrolled error growth over time and eventual instability. We analyze the sources of this autoregressive error growth using prototypical neural operator models for physical systems and explore ways to mitigate it. We introduce architectural and application-specific improvements that allow for careful control of instability-inducing operations within these models without inflating the compute/memory expense. We present results on several scientific systems that include Navier-Stokes fluid flow, rotating shallow water, and a high-resolution global weather forecasting system. We demonstrate that applying our design principles to neural operators leads to significantly lower errors for long-term forecasts as well as longer time horizons without qualitative signs of divergence compared to the original models for these systems. We open-source our https://github.com/mikemccabe210/stabilizing_neural_operators{code} for reproducibility.

We introduce ECHO, a transformer-operator framework for generating million-point PDE trajectories. While existing neural operators (NOs) have shown promise for solving partial differential equations, they remain limited in practice due to poor scalability on dense grids, error accumulation during dynamic unrolling, and task-specific design. ECHO addresses these challenges through three key innovations. (i) It employs a hierarchical convolutional encode-decode architecture that achieves a 100 times spatio-temporal compression while preserving fidelity on mesh points. (ii) It incorporates a training and adaptation strategy that enables high-resolution PDE solution generation from sparse input grids. (iii) It adopts a generative modeling paradigm that learns complete trajectory segments, mitigating long-horizon error drift. The training strategy decouples representation learning from downstream task supervision, allowing the model to tackle multiple tasks such as trajectory generation, forward and inverse problems, and interpolation. The generative model further supports both conditional and unconditional generation. We demonstrate state-of-the-art performance on million-point simulations across diverse PDE systems featuring complex geometries, high-frequency dynamics, and long-term horizons.

With massive advancements in sensor technologies and Internet-of-things, we now have access to terabytes of historical data; however, there is a lack of clarity in how to best exploit the data to predict future events. One possible alternative in this context is to utilize operator learning algorithm that directly learn nonlinear mapping between two functional spaces; this facilitates real-time prediction of naturally arising complex evolutionary dynamics. In this work, we introduce a novel operator learning algorithm referred to as the Wavelet Neural Operator (WNO) that blends integral kernel with wavelet transformation. WNO harnesses the superiority of the wavelets in time-frequency localization of the functions and enables accurate tracking of patterns in spatial domain and effective learning of the functional mappings. Since the wavelets are localized in both time/space and frequency, WNO can provide high spatial and frequency resolution. This offers learning of the finer details of the parametric dependencies in the solution for complex problems. The efficacy and robustness of the proposed WNO are illustrated on a wide array of problems involving Burger's equation, Darcy flow, Navier-Stokes equation, Allen-Cahn equation, and Wave advection equation. Comparative study with respect to existing operator learning frameworks are presented. Finally, the proposed approach is used to build a digital twin capable of predicting Earth's air temperature based on available historical data.

Production deployments in complex systems require ML architectures to be highly efficient and usable against multiple tasks. Particularly demanding are classification problems in which data arrives in a streaming fashion and each class is presented separately. Recent methods with stochastic gradient learning have been shown to struggle in such setups or have limitations like memory buffers, and being restricted to specific domains that disable its usage in real-world scenarios. For this reason, we present a fully differentiable architecture based on the Mixture of Experts model, that enables the training of high-performance classifiers when examples from each class are presented separately. We conducted exhaustive experiments that proved its applicability in various domains and ability to learn online in production environments. The proposed technique achieves SOTA results without a memory buffer and clearly outperforms the reference methods.

Memory complexity and data scarcity have so far prohibited learning solution operators of partial differential equations (PDEs) at high resolutions. We address these limitations by introducing a new data efficient and highly parallelizable operator learning approach with reduced memory requirement and better generalization, called multi-grid tensorized neural operator (MG-TFNO). MG-TFNO scales to large resolutions by leveraging local and global structures of full-scale, real-world phenomena, through a decomposition of both the input domain and the operator's parameter space. Our contributions are threefold: i) we enable parallelization over input samples with a novel multi-grid-based domain decomposition, ii) we represent the parameters of the model in a high-order latent subspace of the Fourier domain, through a global tensor factorization, resulting in an extreme reduction in the number of parameters and improved generalization, and iii) we propose architectural improvements to the backbone FNO. Our approach can be used in any operator learning setting. We demonstrate superior performance on the turbulent Navier-Stokes equations where we achieve less than half the error with over 150x compression. The tensorization combined with the domain decomposition, yields over 150x reduction in the number of parameters and 7x reduction in the domain size without losses in accuracy, while slightly enabling parallelism.

Neural networks are one tool for approximating non-linear differential equations used in scientific computing tasks such as surrogate modeling, real-time predictions, and optimal control. PDE foundation models utilize neural networks to train approximations to multiple differential equations simultaneously and are thus a general purpose solver that can be adapted to downstream tasks. Current PDE foundation models focus on either learning general solution operators and/or the governing system of equations, and thus only handle numerical or symbolic modalities. However, real-world applications may require more flexible data modalities, e.g. text analysis or descriptive outputs. To address this gap, we propose a novel multimodal deep learning approach that leverages a transformer-based architecture to approximate solution operators for a wide variety of ODEs and PDEs. Our method integrates numerical inputs, such as equation parameters and initial conditions, with text descriptions of physical processes or system dynamics. This enables our model to handle settings where symbolic representations may be incomplete or unavailable. In addition to providing accurate numerical predictions, our approach generates interpretable scientific text descriptions, offering deeper insights into the underlying dynamics and solution properties. The numerical experiments show that our model provides accurate solutions for in-distribution data (with average relative error less than 3.3%) and out-of-distribution data (average relative error less than 7.8%) together with precise text descriptions (with correct descriptions generated 100% of times). In certain tests, the model is also shown to be capable of extrapolating solutions in time.

We consider solving partial differential equations (PDEs) with Fourier neural operators (FNOs), which operate in the frequency domain. Since the laws of physics do not depend on the coordinate system used to describe them, it is desirable to encode such symmetries in the neural operator architecture for better performance and easier learning. While encoding symmetries in the physical domain using group theory has been studied extensively, how to capture symmetries in the frequency domain is under-explored. In this work, we extend group convolutions to the frequency domain and design Fourier layers that are equivariant to rotations, translations, and reflections by leveraging the equivariance property of the Fourier transform. The resulting G-FNO architecture generalizes well across input resolutions and performs well in settings with varying levels of symmetry. Our code is publicly available as part of the AIRS library (https://github.com/divelab/AIRS).

Automatic operator design (AOD) has increasingly adopted large language models (LLMs) to generate problem-specific search and variation operators for multi-objective combinatorial optimization (MOCO). Yet most methods evolve the operators while leaving the process that discovers them fixed. This limits adaptation as new Pareto trade-offs and operator interactions change where and how the search should proceed. We introduce HEMO, which makes the AOD harness itself a self-evolving optimization target. HEMO maintains a persistent harness state and updates one module at a time, revising the logic that determines which experience to retrieve, which operators to improve, which parents to select, and how to formulate the orchestration of search steps. Downstream solution quality and runtime guide these updates, while a discounted UCB scheduler directs adaptation toward promising modules as their utility changes. A persistent memory filesystem preserves candidate outcomes and diagnostic feedback across harness states, enabling successive states to build on accumulated optimization experience. Experiments across eight MOCO tasks and two structurally distinct evolutionary backbones demonstrate strong performance against conventional evolutionary, neural, and LLM-based AOD baselines, together with effective transfer to related tasks. HEMO extends automatic operator design beyond evolving individual operators to improving the search process that discovers them.

Deep neural networks with millions of parameters are at the heart of many state of the art machine learning models today. However, recent works have shown that models with much smaller number of parameters can also perform just as well. In this work, we introduce the problem of architecture-learning, i.e; learning the architecture of a neural network along with weights. We introduce a new trainable parameter called tri-state ReLU, which helps in eliminating unnecessary neurons. We also propose a smooth regularizer which encourages the total number of neurons after elimination to be small. The resulting objective is differentiable and simple to optimize. We experimentally validate our method on both small and large networks, and show that it can learn models with a considerably small number of parameters without affecting prediction accuracy.

Low-Rank Adaptive Residual Connections (LARC) give a frozen model a compact numerical state that can learn from feedback. The map h+BAh adds a low-rank correction to a hidden representation. A slow state ρ learns starting factors across tasks; a private fast state Φ copies them, changes with feedback, and resets to the trained initialization. This report specifies an input-side realization of the numerical policy carrier in Memory-Mediated Learning Architecture and examines its factor-space dynamics and learning lifetime. We study a rank-4 input residual with 12,288 trainable parameters on a frozen MiniCPM5-1B-SFT substrate. In a four-candidate program-selection task, two feedback-gradient steps reduce expected query execution error by 24.65 and 36.65 percentage points relative to resetting to the respective trained static and post-adaptation initializations. These development results cover 16 parameter groups and three paired training seeds. A direct support-loss selection rule is much more accurate, reaching 0.78125% error. In a repository-balanced chronological replay of public continuous-integration jobs, retaining online updates raises half-Brier loss from 0.1274 to 0.1808. A fixed follow-up intervention records same-batch non-descent and inconsistent future benefit from shrinking updates. Together, the algebra and measurements distinguish residual capacity, adaptation relative to a starting point, and usefulness on later decisions.

We develop a unified theoretical framework for sparse knowledge distillation based on probability-domain softening operators. While the equivalence p^{1/T} propto softmax(z/T) is well known, our contribution is an operator-level analytical framework built on this foundation rather than the equivalence itself. The framework comprises four core components: (i) operator-agnostic bias--variance decompositions that characterize when sparse students outperform dense teachers, (ii) a homotopy path formalization of multi-stage pruning in function space explaining why iterative compression succeeds where one-shot pruning fails, (iii) convergence guarantees establishing O(1/n) rates for n-stage distillation with explicit parameter dependence, and (iv) equivalence class characterizations identifying distinct probability-domain operators that yield identical student models under capacity constraints. We introduce an axiomatic definition of probability-domain softening operators based on ranking preservation, continuity, entropy monotonicity, identity, and boundary behavior, and show that multiple non-equivalent operator families satisfy these axioms. All learning-theoretic guarantees are shown to hold uniformly across this operator class, independent of implementation details. These results provide theoretical grounding for black-box teacher distillation, partial-access settings such as top-k truncation and text-only outputs, and privacy-preserving model compression.

Neural operators provide deep neural networks for learning mappings between function spaces. Among them, the Fourier Neural Operator (FNO) is particularly effective: its spectral convolution relies on low-dimensional Fourier-domain representations and can handle inputs at different resolutions. This design aligns well with settings where the Fourier basis diagonalizes the underlying operator, such as linear, constant-coefficient PDEs on periodic domains, in which Fourier modes evolve independently. However, nonlinear PDEs may benefit from an additional inductive bias, as they exhibit structured interactions between modes, governed by polynomial nonlinearities. To capture this inductive bias, we introduce the Higher-Order Spectral Convolution, a spectral mixer that extends FNO from diagonal modulation to explicit n-linear mode mixing, aligned with the dynamics of nonlinear PDEs. Our experiments on standard benchmarks show that the proposed Higher-Order FNO (HO-FNO) retains the efficiency of FNO-based architectures and consistently improves over other spectral neural operators. HO-FNO also performs on par with or better than state-of-the-art transformers and state-space models on several datasets, with stronger gains in highly nonlinear regimes, such as the Poisson equation with polynomial forcing, where a single HO-FNO layer outperforms FNO models with up to 16 layers. We open-source our code for reproducibility at: https://github.com/AlexColagrande/HO-FNO.

Pre-training neural operators on diverse partial differential equation (PDE) datasets has emerged as a promising direction for building general-purpose surrogate models in scientific machine learning. However, the inherent complexity and structural diversity of PDE solution operators make multi-PDE pre-training fundamentally challenging. Existing methods mainly address this by increasing model capacity, while leaving the target solution operators unchanged. Inspired by classical numerical analysis, we instead propose to transform complex and diverse solution operators into simpler, better-aligned forms that are easier to model jointly. Since the optimal transformation varies across PDE types, it must be adaptive and input-dependent, allowing a single neural operator to approximate an entire family of operators. We instantiate this idea as AOT-POT (adaptive operator-transformation for pre-training operator transformer), which expands hidden representations into multiple parallel streams, adaptively aggregates and redistributes them before and after each sub-layer, and mixes streams through Sinkhorn-projected doubly stochastic matrices for stable training. These mechanisms together reshape diverse solution operators into a unified form that can be effectively modeled by a single architecture. Empirically, AOT-POT achieves state-of-the-art performance on 12 PDE benchmarks with only 3\% additional parameters, reducing relative L2 error by up to 77.6\% (40.9\% on average). Fine-tuning AOT-POT further reduces L2 error by up to 92\% on in-domain PDEs and 89\% on out-of-domain PDEs (unseen types during pre-training), demonstrating that adaptive operator transformation is an effective and complementary direction for advancing PDE foundation models beyond simply scaling model capacity.

Vector Symbolic Architectures (VSAs) have emerged as a novel framework for enabling interpretable machine learning algorithms equipped with the ability to reason and explain their decision processes. The basic idea is to represent discrete information through high dimensional random vectors. Complex data structures can be built up with operations over vectors such as the "binding" operation involving element-wise vector multiplication, which associates data together. The reverse task of decomposing the associated elements is a combinatorially hard task, with an exponentially large search space. The main algorithm for performing this search is the resonator network, inspired by Hopfield network-based memory search operations. In this work, we introduce a new variant of the resonator network, based on self-attention based update rules in the iterative search problem. This update rule, based on the Hopfield network with log-sum-exp energy function and norm-bounded states, is shown to substantially improve the performance and rate of convergence. As a result, our algorithm enables a larger capacity for associative memory, enabling applications in many tasks like perception based pattern recognition, scene decomposition, and object reasoning. We substantiate our algorithm with a thorough evaluation and comparisons to baselines.

Existing neural operator architectures face challenges when solving multiphysics problems with coupled partial differential equations (PDEs) due to complex geometries, interactions between physical variables, and the limited amounts of high-resolution training data. To address these issues, we propose Codomain Attention Neural Operator (CoDA-NO), which tokenizes functions along the codomain or channel space, enabling self-supervised learning or pretraining of multiple PDE systems. Specifically, we extend positional encoding, self-attention, and normalization layers to function spaces. CoDA-NO can learn representations of different PDE systems with a single model. We evaluate CoDA-NO's potential as a backbone for learning multiphysics PDEs over multiple systems by considering few-shot learning settings. On complex downstream tasks with limited data, such as fluid flow simulations, fluid-structure interactions, and Rayleigh-B\'enard convection, we found CoDA-NO to outperform existing methods by over 36%.

Quantizing the activation, weight, and gradient to 4-bit is promising to accelerate neural network training. However, existing 4-bit training methods require custom numerical formats which are not supported by contemporary hardware. In this work, we propose a training method for transformers with all matrix multiplications implemented with the INT4 arithmetic. Training with an ultra-low INT4 precision is challenging. To achieve this, we carefully analyze the specific structures of activation and gradients in transformers to propose dedicated quantizers for them. For forward propagation, we identify the challenge of outliers and propose a Hadamard quantizer to suppress the outliers. For backpropagation, we leverage the structural sparsity of gradients by proposing bit splitting and leverage score sampling techniques to quantize gradients accurately. Our algorithm achieves competitive accuracy on a wide range of tasks including natural language understanding, machine translation, and image classification. Unlike previous 4-bit training methods, our algorithm can be implemented on the current generation of GPUs. Our prototypical linear operator implementation is up to 2.2 times faster than the FP16 counterparts and speeds up the training by up to 35.1%.

Learning in weight spaces, where neural networks process the weights of other deep neural networks, has emerged as a promising research direction with applications in various fields, from analyzing and editing neural fields and implicit neural representations, to network pruning and quantization. Recent works designed architectures for effective learning in that space, which takes into account its unique, permutation-equivariant, structure. Unfortunately, so far these architectures suffer from severe overfitting and were shown to benefit from large datasets. This poses a significant challenge because generating data for this learning setup is laborious and time-consuming since each data sample is a full set of network weights that has to be trained. In this paper, we address this difficulty by investigating data augmentations for weight spaces, a set of techniques that enable generating new data examples on the fly without having to train additional input weight space elements. We first review several recently proposed data augmentation schemes %that were proposed recently and divide them into categories. We then introduce a novel augmentation scheme based on the Mixup method. We evaluate the performance of these techniques on existing benchmarks as well as new benchmarks we generate, which can be valuable for future studies.

Partial differential equations (PDEs) are widely used to model complex physical systems, but solving them efficiently remains a significant challenge. Recently, Transformers have emerged as the preferred architecture for PDEs due to their ability to capture intricate dependencies. However, they struggle with representing continuous dynamics and long-range interactions. To overcome these limitations, we introduce the Mamba Neural Operator (MNO), a novel framework that enhances neural operator-based techniques for solving PDEs. MNO establishes a formal theoretical connection between structured state-space models (SSMs) and neural operators, offering a unified structure that can adapt to diverse architectures, including Transformer-based models. By leveraging the structured design of SSMs, MNO captures long-range dependencies and continuous dynamics more effectively than traditional Transformers. Through extensive analysis, we show that MNO significantly boosts the expressive power and accuracy of neural operators, making it not just a complement but a superior framework for PDE-related tasks, bridging the gap between efficient representation and accurate solution approximation. Our code is available on https://github.com/Math-ML-X/Mamba-Neural-Operator

Building on the probability-domain distillation framework of Sparse-KD, we develop an axiomatic, operator-theoretic framework for multi-teacher ensemble knowledge distillation. Rather than prescribing a specific aggregation formula, we define five core axioms governing valid knowledge aggregation operators, encompassing convexity, positivity, continuity, weight monotonicity, and temperature coherence. We prove the existence and non-uniqueness of operator families satisfying these axioms, establishing that multiple distinct aggregation mechanisms conform to the same foundational principles. Within this framework, we establish operator-agnostic guarantees showing that multi-teacher aggregation reduces both stochastic variance and systematic supervisory bias under heterogeneous teachers, while providing Jensen-type bounds, log-loss guarantees, and safety attenuation properties. For aggregation operators linear in teacher weights, we further establish classical ensemble variance-reduction results under standard independence assumptions, with extensions to correlated-error regimes. The framework provides theoretical grounding for multi-teacher distillation from diverse frontier models while admitting multiple valid implementation strategies.

Neural architecture search (NAS) has achieved remarkable results in deep neural network design. Differentiable architecture search converts the search over discrete architectures into a hyperparameter optimization problem which can be solved by gradient descent. However, questions have been raised regarding the effectiveness and generalizability of gradient methods for solving non-convex architecture hyperparameter optimization problems. In this paper, we propose L^{2}NAS, which learns to intelligently optimize and update architecture hyperparameters via an actor neural network based on the distribution of high-performing architectures in the search history. We introduce a quantile-driven training procedure which efficiently trains L^{2}NAS in an actor-critic framework via continuous-action reinforcement learning. Experiments show that L^{2}NAS achieves state-of-the-art results on NAS-Bench-201 benchmark as well as DARTS search space and Once-for-All MobileNetV3 search space. We also show that search policies generated by L^{2}NAS are generalizable and transferable across different training datasets with minimal fine-tuning.

The Koopman operator provides a powerful framework for modeling dynamical systems and has attracted growing interest from the machine learning community. However, its infinite-dimensional nature makes identifying suitable finite-dimensional subspaces challenging, especially for deep architectures. We argue that these difficulties come from suboptimal representation learning, where latent variables fail to balance expressivity and simplicity. This tension is closely related to the information bottleneck (IB) dilemma: constructing compressed representations that are both compact and predictive. Rethinking Koopman learning through this lens, we demonstrate that latent mutual information promotes simplicity, yet an overemphasis on simplicity may cause latent space to collapse onto a few dominant modes. In contrast, expressiveness is sustained by the von Neumann entropy, which prevents such collapse and encourages mode diversity. This insight leads us to propose an information-theoretic Lagrangian formulation that explicitly balances this tradeoff. Furthermore, we propose a new algorithm based on the Lagrangian formulation that encourages both simplicity and expressiveness, leading to a stable and interpretable Koopman representation. Beyond quantitative evaluations, we further visualize the learned manifolds under our representations, observing empirical results consistent with our theoretical predictions. Finally, we validate our approach across a diverse range of dynamical systems, demonstrating improved performance over existing Koopman learning methods. The implementation is publicly available at https://github.com/Wenxuan52/InformationKoopman.

In this paper, we propose a highly parameter-efficient approach to scaling pre-trained language models (PLMs) to a deeper model depth. Unlike prior work that shares all parameters or uses extra blocks, we design a more capable parameter-sharing architecture based on matrix product operator (MPO). MPO decomposition can reorganize and factorize the information of a parameter matrix into two parts: the major part that contains the major information (central tensor) and the supplementary part that only has a small proportion of parameters (auxiliary tensors). Based on such a decomposition, our architecture shares the central tensor across all layers for reducing the model size and meanwhile keeps layer-specific auxiliary tensors (also using adapters) for enhancing the adaptation flexibility. To improve the model training, we further propose a stable initialization algorithm tailored for the MPO-based architecture. Extensive experiments have demonstrated the effectiveness of our proposed model in reducing the model size and achieving highly competitive performance.

Foundation models are transitioning from offline predictors to deployed systems expected to operate over long time horizons. In real deployments, objectives are not fixed: domains drift, user preferences evolve, and new tasks appear after the model has shipped. This elevates continual learning and instant personalization from optional features to core architectural requirements. Yet most adaptation pipelines still follow a static weight paradigm: after training (or after any adaptation step), inference executes a single parameter vector regardless of user intent, domain, or instance-specific constraints. This treats the trained or adapted model as a single point in parameter space. In heterogeneous and continually evolving regimes, distinct objectives can induce separated feasible regions over parameters, forcing any single shared update into compromise, interference, or overspecialization. As a result, continual learning and personalization are often implemented as repeated overwriting of shared weights, risking degradation of previously learned behaviors. We propose HY-WU (Weight Unleashing), a memory-first adaptation framework that shifts adaptation pressure away from overwriting a single shared parameter point. HY-WU implements functional (operator-level) memory as a neural module: a generator that synthesizes weight updates on-the-fly from the instance condition, yielding instance-specific operators without test-time optimization.

We study the problem of teaching multiple learners simultaneously in the nonparametric iterative teaching setting, where the teacher iteratively provides examples to the learner for accelerating the acquisition of a target concept. This problem is motivated by the gap between current single-learner teaching setting and the real-world scenario of human instruction where a teacher typically imparts knowledge to multiple students. Under the new problem formulation, we introduce a novel framework -- Multi-learner Nonparametric Teaching (MINT). In MINT, the teacher aims to instruct multiple learners, with each learner focusing on learning a scalar-valued target model. To achieve this, we frame the problem as teaching a vector-valued target model and extend the target model space from a scalar-valued reproducing kernel Hilbert space used in single-learner scenarios to a vector-valued space. Furthermore, we demonstrate that MINT offers significant teaching speed-up over repeated single-learner teaching, particularly when the multiple learners can communicate with each other. Lastly, we conduct extensive experiments to validate the practicality and efficiency of MINT.

In-Context Operator Networks (ICONs) have demonstrated the ability to learn operators across diverse partial differential equations using few-shot, in-context learning. However, existing ICONs process each spatial point as an individual token, severely limiting computational efficiency when handling dense data in higher spatial dimensions. We propose Vision In-Context Operator Networks (VICON), which integrates vision transformer architectures to efficiently process 2D data through patch-wise operations while preserving ICON's adaptability to multiphysics systems and varying timesteps. Evaluated across three fluid dynamics benchmarks, VICON significantly outperforms state-of-the-art baselines: DPOT and MPP, reducing the averaged last-step rollout error by 37.9% compared to DPOT and 44.7% compared to MPP, while requiring only 72.5% and 34.8% of their respective inference times. VICON naturally supports flexible rollout strategies with varying timestep strides, enabling immediate deployment in imperfect measurement systems where sampling frequencies may differ or frames might be dropped - common challenges in real-world settings - without requiring retraining or interpolation. In these realistic scenarios, VICON exhibits remarkable robustness, experiencing only 24.41% relative performance degradation compared to 71.37%-74.49% degradation in baseline methods, demonstrating its versatility for deploying in realistic applications. Our scripts for processing datasets and code are publicly available at https://github.com/Eydcao/VICON.

We introduce convolutional multi-hybrid architectures, with a design grounded on two simple observations. First, operators in hybrid models can be tailored to token manipulation tasks such as in-context recall, multi-token recall, and compression, with input-dependent convolutions and attention offering complementary performance. Second, co-designing convolution operators and hardware-aware algorithms enables efficiency gains in regimes where previous alternative architectures struggle to surpass Transformers. At the 40 billion parameter scale, we train end-to-end 1.2 to 2.9 times faster than optimized Transformers, and 1.1 to 1.4 times faster than previous generation hybrids. On H100 GPUs and model width 4096, individual operators in the proposed multi-hybrid StripedHyena 2 architecture achieve two-fold throughput improvement over linear attention and state-space models. Multi-hybrids excel at sequence modeling over byte-tokenized data, as demonstrated by the Evo 2 line of models. We discuss the foundations that enable these results, including architecture design, overlap-add blocked kernels for tensor cores, and dedicated all-to-all and point-to-point context parallelism strategies.

Solving parametric partial differential equations (PDEs) and associated PDE-based, inverse problems is a central task in engineering and physics, yet existing neural operator methods struggle with high-dimensional, discontinuous inputs and require large amounts of {\em labeled} training data. We propose the Deep Generative Neural Operator (DGNO), a physics-aware framework that addresses these challenges by leveraging a deep, generative, probabilistic model in combination with a set of lower-dimensional, latent variables that simultaneously encode PDE-inputs and PDE-outputs. This formulation can make use of unlabeled data and significantly improves inverse problem-solving, particularly for discontinuous or discrete-valued input functions. DGNO enforces physics constraints without labeled data by incorporating as virtual observables, weak-form residuals based on compactly supported radial basis functions (CSRBFs). These relax regularity constraints and eliminate higher-order derivatives from the objective function. We also introduce MultiONet, a novel neural operator architecture, which is a more expressive generalization of the popular DeepONet that significantly enhances the approximating power of the proposed model. These innovations make DGNO particularly effective for challenging forward and inverse, PDE-based problems, such as those involving multi-phase media. Numerical experiments demonstrate that DGNO achieves higher accuracy across multiple benchmarks while exhibiting robustness to noise and strong generalization to out-of-distribution cases. Its adaptability, and the ability to handle sparse, noisy data while providing probabilistic estimates, make DGNO a powerful tool for scientific and engineering applications.

Multivariable time series forecasting methods can integrate information from exogenous variables, leading to significant prediction accuracy gains. Transformer architecture has been widely applied in various time series forecasting models due to its ability to capture long-range sequential dependencies. However, a na\"ive application of transformers often struggles to effectively model complex relationships among variables over time. To mitigate against this, we propose a novel architecture, namely the Spectral Operator Neural Network (Sonnet). Sonnet applies learnable wavelet transformations to the input and incorporates spectral analysis using the Koopman operator. Its predictive skill relies on the Multivariable Coherence Attention (MVCA), an operation that leverages spectral coherence to model variable dependencies. Our empirical analysis shows that Sonnet yields the best performance on 34 out of 47 forecasting tasks with an average mean absolute error (MAE) reduction of 1.1% against the most competitive baseline (different per task). We further show that MVCA -- when put in place of the na\"ive attention used in various deep learning models -- can remedy its deficiencies, reducing MAE by 10.7% on average in the most challenging forecasting tasks.

Learning dexterous manipulation skills presents significant challenges due to complex nonlinear dynamics that underlie the interactions between objects and multi-fingered hands. Koopman operators have emerged as a robust method for modeling such nonlinear dynamics within a linear framework. However, current methods rely on runtime access to ground-truth (GT) object states, making them unsuitable for vision-based practical applications. Unlike image-to-action policies that implicitly learn visual features for control, we use a dynamics model, specifically the Koopman operator, to learn visually interpretable object features critical for robotic manipulation within a scene. We construct a Koopman operator using object features predicted by a feature extractor and utilize it to auto-regressively advance system states. We train the feature extractor to embed scene information into object features, thereby enabling the accurate propagation of robot trajectories. We evaluate our approach on simulated and real-world robot tasks, with results showing that it outperformed the model-based imitation learning NDP by 1.08times and the image-to-action Diffusion Policy by 1.16times. The results suggest that our method maintains task success rates with learned features and extends applicability to real-world manipulation without GT object states.

Modern machine learning requires system designers to specify aspects of the learning pipeline, such as losses, architectures, and optimizers. Meta-learning, or learning-to-learn, instead aims to learn those aspects, and promises to unlock greater capabilities with less manual effort. One particularly ambitious goal of meta-learning is to train general-purpose in-context learning algorithms from scratch, using only black-box models with minimal inductive bias. Such a model takes in training data, and produces test-set predictions across a wide range of problems, without any explicit definition of an inference model, training loss, or optimization algorithm. In this paper we show that Transformers and other black-box models can be meta-trained to act as general-purpose in-context learners. We characterize transitions between algorithms that generalize, algorithms that memorize, and algorithms that fail to meta-train at all, induced by changes in model size, number of tasks, and meta-optimization. We further show that the capabilities of meta-trained algorithms are bottlenecked by the accessible state size (memory) determining the next prediction, unlike standard models which are thought to be bottlenecked by parameter count. Finally, we propose practical interventions such as biasing the training distribution that improve the meta-training and meta-generalization of general-purpose in-context learning algorithms.

Learning to Optimize (L2O), a technique that utilizes machine learning to learn an optimization algorithm automatically from data, has gained arising attention in recent years. A generic L2O approach parameterizes the iterative update rule and learns the update direction as a black-box network. While the generic approach is widely applicable, the learned model can overfit and may not generalize well to out-of-distribution test sets. In this paper, we derive the basic mathematical conditions that successful update rules commonly satisfy. Consequently, we propose a novel L2O model with a mathematics-inspired structure that is broadly applicable and generalized well to out-of-distribution problems. Numerical simulations validate our theoretical findings and demonstrate the superior empirical performance of the proposed L2O model.

We introduce a unified theoretical framework for the rigorous analysis and systematic construction of deep neural networks (DNNs). This framework addresses a gap in existing theory by explicitly modeling the structure of tensor operations -- lower level information that is often abstracted. Our framework enables two novel objectives: (1) analysis of the evolution of architectural complexity over deep learning history, and (2) automatic construction of novel architectures based on new types of tensor operations. Our study of DNNs introduced over the past 40 years reveals a connection between groundbreaking architectures and increases in different types of architectural complexity. Moreover, we identify several large classes of higher complexity architectures that have not yet been explored. We then collect a dataset of 3,000+ higher complexity architectures, which we publicly release at: https://github.com/combinatoriallabs/ArchitecturalComplexity.

We propose Super-resolution Neural Operator (SRNO), a deep operator learning framework that can resolve high-resolution (HR) images at arbitrary scales from the low-resolution (LR) counterparts. Treating the LR-HR image pairs as continuous functions approximated with different grid sizes, SRNO learns the mapping between the corresponding function spaces. From the perspective of approximation theory, SRNO first embeds the LR input into a higher-dimensional latent representation space, trying to capture sufficient basis functions, and then iteratively approximates the implicit image function with a kernel integral mechanism, followed by a final dimensionality reduction step to generate the RGB representation at the target coordinates. The key characteristics distinguishing SRNO from prior continuous SR works are: 1) the kernel integral in each layer is efficiently implemented via the Galerkin-type attention, which possesses non-local properties in the spatial domain and therefore benefits the grid-free continuum; and 2) the multilayer attention architecture allows for the dynamic latent basis update, which is crucial for SR problems to "hallucinate" high-frequency information from the LR image. Experiments show that SRNO outperforms existing continuous SR methods in terms of both accuracy and running time. Our code is at https://github.com/2y7c3/Super-Resolution-Neural-Operator

Dynamical systems provide a comprehensive way to study complex and changing behaviors across various sciences. Many modern systems are too complicated to analyze directly or we do not have access to models, driving significant interest in learning methods. Koopman operators have emerged as a dominant approach because they allow the study of nonlinear dynamics using linear techniques by solving an infinite-dimensional spectral problem. However, current algorithms face challenges such as lack of convergence, hindering practical progress. This paper addresses a fundamental open question: When can we robustly learn the spectral properties of Koopman operators from trajectory data of dynamical systems, and when can we not? Understanding these boundaries is crucial for analysis, applications, and designing algorithms. We establish a foundational approach that combines computational analysis and ergodic theory, revealing the first fundamental barriers -- universal for any algorithm -- associated with system geometry and complexity, regardless of data quality and quantity. For instance, we demonstrate well-behaved smooth dynamical systems on tori where non-trivial eigenfunctions of the Koopman operator cannot be determined by any sequence of (even randomized) algorithms, even with unlimited training data. Additionally, we identify when learning is possible and introduce optimal algorithms with verification that overcome issues in standard methods. These results pave the way for a sharp classification theory of data-driven dynamical systems based on how many limits are needed to solve a problem. These limits characterize all previous methods, presenting a unified view. Our framework systematically determines when and how Koopman spectral properties can be learned.

Large Language Models (LLMs) have demonstrated remarkable capabilities across a wide range of natural language processing and reasoning tasks. However, their performance in the foundational domain of arithmetic remains unsatisfactory. When dealing with arithmetic tasks, LLMs often memorize specific examples rather than learning the underlying computational logic, limiting their ability to generalize to new problems. In this paper, we propose a Composable Arithmetic Execution Framework (CAEF) that enables LLMs to learn to execute step-by-step computations by emulating Turing Machines, thereby gaining a genuine understanding of computational logic. Moreover, the proposed framework is highly scalable, allowing composing learned operators to significantly reduce the difficulty of learning complex operators. In our evaluation, CAEF achieves nearly 100% accuracy across seven common mathematical operations on the LLaMA 3.1-8B model, effectively supporting computations involving operands with up to 100 digits, a level where GPT-4o falls short noticeably in some settings.

Neural operators provide fast surrogates for time-dependent partial differential equations (PDEs) by applying a learned evolution operator recursively to its own predictions, but this autoregressive rollout feeds every prediction error back as input, so local errors accumulate. Existing rollout-training strategies reduce the mismatch between training inputs and self-generated states, yet their supervision still measures only the absolute discrepancy from the ground-truth trajectory. Such supervision is therefore uninformative about whether the operator has overcome the long-horizon failure behaviors it exhibited earlier during optimization. We propose history-enriched rollout training (HERO), which augments conventional absolute trajectory supervision with relative supervision derived from the model's optimization history. HERO ranks detached candidate rollouts from a periodically refreshed lagged operator, the current model, and a perturbed input by rollout error, spectral discrepancy, energy drift, and error growth, and selects the strongest failure trajectory as reference. This reference enters a margin-based objective as a fixed comparison baseline, inducing a bounded, sample-dependent reweighting of the ground-truth rollout gradient rather than an independent gradient direction, which we further analyze theoretically. Experiments on nine PDE benchmarks with spectral and attention-based backbones show that HERO consistently improves long-horizon accuracy, stable rollout length, and out-of-distribution robustness at no inference-time cost. These results indicate that history-enriched relative supervision is effective for stabilizing long-horizon autoregressive prediction.

Implicit Neural Representations (INRs) based on vanilla Multi-Layer Perceptrons (MLPs) are widely believed to be incapable of representing high-frequency content. This has directed research efforts towards architectural interventions, such as coordinate embeddings or specialized activation functions, to represent high-frequency signals. In this paper, we challenge the notion that the low-frequency bias of vanilla MLPs is an intrinsic, architectural limitation to learn high-frequency content, but instead a symptom of stable rank degradation during training. We empirically demonstrate that regulating the network's rank during training substantially improves the fidelity of the learned signal, rendering even simple MLP architectures expressive. Extensive experiments show that using optimizers like Muon, with high-rank, near-orthogonal updates, consistently enhances INR architectures even beyond simple ReLU MLPs. These substantial improvements hold across a diverse range of domains, including natural and medical images and novel view synthesis, with up to +9 dB PSNR over the same architecture. Code is available at (https://rank-inrs.github.io).

Do LLM scientific agents rely only on their initial task context, or do they adapt their decisions in response to experimental feedback? We study this question in neural operator adaptation, where a large language model (LLM) selects fine-tuning configurations under a limited trial budget. Across transfers within and between partial differential equation (PDE) families, the LLM achieves lower held-out test nRMSE than random search and Bayesian optimisation in nearly every matched comparison. Endpoint performance alone cannot distinguish what happens, so we verify each attribution with controlled interventions. Before observing any validation score, the LLM's first configuration already ranks near the top of the corresponding random-search pool, indicating a useful initial bias. A complementary cold-start intervention shows that the selected base learning rate shifts with the PDE description. Once feedback becomes available, reassigning validation scores among evaluated configurations changes the next proposal in every case tested, whereas a value-preserving rewrite produces no comparable aggregate effect. These interventions establish that the LLM's decision-level actions respond to the given task and observed outcomes, showing that it combines a task-dependent prior with sensitivity to experimental feedback.

We introduce and train distributed neural architectures (DNA) in vision and language domains. DNAs are initialized with a proto-architecture that consists of (transformer, MLP, attention, etc.) modules and routers. Any token (or patch) can traverse any series of modules in any order. DNAs are a natural generalization of the sparse methods such as Mixture-of-Experts, Mixture-of-Depths, parameter sharing, etc. Computation and communication patterns of DNA modules are learnt end-to-end during training and depend on the content and context of each token (or patch). These patterns can be shaped by further requirements added to the optimization objective such as compute/memory efficiency or load balancing. We empirically show that (i) trained DNAs are competitive with the dense baselines in both domains and (ii) compute efficiency/parameter sharing can be learnt from data. Next, we analyze the emergent connectivity and computation patterns in the trained DNAs. We find that the paths that tokens take through the models are themselves distributed according to a power-law. We show that some paths (or, equivalently, groups of modules) show emergent specialization. Finally, we demonstrate that models learn to allocate compute and active parameters in an interpretable way.

Learned optimizers are powerful alternatives to hand-designed update rules like Adam, yet they have seen limited practical adoption since they often fail to meta-generalize beyond their training distribution and incur high meta-training cost. For instance, prior work, VeLO, scaled meta-training to 4,000 TPU months (sim10times GPT-3 compute) to meta-train a general-purpose optimizer but it failed to generalize beyond 600M parameters tasks. In this work, we present a surprising finding: by crafting a simple normalized optimizer architecture and augmenting meta-training, it becomes feasible to meta-train a performant general-purpose learned update rule on a tiny fraction of VeLO compute, 4.5 GPU hours to be precise. Our learned update rule scales stably to a billion-scale pretraining task (GPT-3 XL 1.3B) which is six orders of magnitude larger than its meta-training distribution. Furthermore, it shows strong performance across diverse out-of-distribution tasks and is compatible with modern optimization harness that includes orthogonalization, distinct update rules for input-output and hidden weights, and decoupled weight decay. In all, this work paves the way for practically applicable learnable optimization algorithms, unlocking exploration of richer meta-training and data curation recipes to further improve performance.

We study the learning dynamics of self-predictive learning for reinforcement learning, a family of algorithms that learn representations by minimizing the prediction error of their own future latent representations. Despite its recent empirical success, such algorithms have an apparent defect: trivial representations (such as constants) minimize the prediction error, yet it is obviously undesirable to converge to such solutions. Our central insight is that careful designs of the optimization dynamics are critical to learning meaningful representations. We identify that a faster paced optimization of the predictor and semi-gradient updates on the representation, are crucial to preventing the representation collapse. Then in an idealized setup, we show self-predictive learning dynamics carries out spectral decomposition on the state transition matrix, effectively capturing information of the transition dynamics. Building on the theoretical insights, we propose bidirectional self-predictive learning, a novel self-predictive algorithm that learns two representations simultaneously. We examine the robustness of our theoretical insights with a number of small-scale experiments and showcase the promise of the novel representation learning algorithm with large-scale experiments.

Most of the dominant approaches to continual learning are based on either memory replay, parameter isolation, or regularization techniques that require task boundaries to calculate task statistics. We propose a static architecture-based method that doesn't use any of these. We show that we can improve the continual learning performance by replacing the final layer of our networks with our pairwise interaction layer. The pairwise interaction layer uses sparse representations from a Winner-take-all style activation function to find the relevant correlations in the hidden layer representations. The networks using this architecture show competitive performance in MNIST and FashionMNIST-based continual image classification experiments. We demonstrate this in an online streaming continual learning setup where the learning system cannot access task labels or boundaries.

Designing machine learning architectures for processing neural networks in their raw weight matrix form is a newly introduced research direction. Unfortunately, the unique symmetry structure of deep weight spaces makes this design very challenging. If successful, such architectures would be capable of performing a wide range of intriguing tasks, from adapting a pre-trained network to a new domain to editing objects represented as functions (INRs or NeRFs). As a first step towards this goal, we present here a novel network architecture for learning in deep weight spaces. It takes as input a concatenation of weights and biases of a pre-trained MLP and processes it using a composition of layers that are equivariant to the natural permutation symmetry of the MLP's weights: Changing the order of neurons in intermediate layers of the MLP does not affect the function it represents. We provide a full characterization of all affine equivariant and invariant layers for these symmetries and show how these layers can be implemented using three basic operations: pooling, broadcasting, and fully connected layers applied to the input in an appropriate manner. We demonstrate the effectiveness of our architecture and its advantages over natural baselines in a variety of learning tasks.

While the use of bottom-up local operators in convolutional neural networks (CNNs) matches well some of the statistics of natural images, it may also prevent such models from capturing contextual long-range feature interactions. In this work, we propose a simple, lightweight approach for better context exploitation in CNNs. We do so by introducing a pair of operators: gather, which efficiently aggregates feature responses from a large spatial extent, and excite, which redistributes the pooled information to local features. The operators are cheap, both in terms of number of added parameters and computational complexity, and can be integrated directly in existing architectures to improve their performance. Experiments on several datasets show that gather-excite can bring benefits comparable to increasing the depth of a CNN at a fraction of the cost. For example, we find ResNet-50 with gather-excite operators is able to outperform its 101-layer counterpart on ImageNet with no additional learnable parameters. We also propose a parametric gather-excite operator pair which yields further performance gains, relate it to the recently-introduced Squeeze-and-Excitation Networks, and analyse the effects of these changes to the CNN feature activation statistics.

Training large models from scratch usually costs a substantial amount of resources. Towards this problem, recent studies such as bert2BERT and LiGO have reused small pretrained models to initialize a large model (termed the ``target model''), leading to a considerable acceleration in training. Despite the successes of these previous studies, they grew pretrained models by mapping partial weights only, ignoring potential correlations across the entire model. As we show in this paper, there are inter- and intra-interactions among the weights of both the pretrained and the target models. As a result, the partial mapping may not capture the complete information and lead to inadequate growth. In this paper, we propose a method that linearly correlates each weight of the target model to all the weights of the pretrained model to further enhance acceleration ability. We utilize multi-linear operators to reduce computational and spacial complexity, enabling acceptable resource requirements. Experiments demonstrate that our method can save 76\% computational costs on DeiT-base transferred from DeiT-small, which outperforms bert2BERT by +12.0\% and LiGO by +20.7\%, respectively.

Numerical simulations in climate, chemistry, or astrophysics are computationally too expensive for uncertainty quantification or parameter-exploration at high-resolution. Reduced-order or surrogate models are multiple orders of magnitude faster, but traditional surrogates are inflexible or inaccurate and pure machine learning (ML)-based surrogates too data-hungry. We propose a hybrid, flexible surrogate model that exploits known physics for simulating large-scale dynamics and limits learning to the hard-to-model term, which is called parametrization or closure and captures the effect of fine- onto large-scale dynamics. Leveraging neural operators, we are the first to learn grid-independent, non-local, and flexible parametrizations. Our multiscale neural operator is motivated by a rich literature in multiscale modeling, has quasilinear runtime complexity, is more accurate or flexible than state-of-the-art parametrizations and demonstrated on the chaotic equation multiscale Lorenz96.

While bigger and deeper neural network architectures continue to advance the state-of-the-art for many computer vision tasks, real-world adoption of these networks is impeded by hardware and speed constraints. Conventional model compression methods attempt to address this problem by modifying the architecture manually or using pre-defined heuristics. Since the space of all reduced architectures is very large, modifying the architecture of a deep neural network in this way is a difficult task. In this paper, we tackle this issue by introducing a principled method for learning reduced network architectures in a data-driven way using reinforcement learning. Our approach takes a larger `teacher' network as input and outputs a compressed `student' network derived from the `teacher' network. In the first stage of our method, a recurrent policy network aggressively removes layers from the large `teacher' model. In the second stage, another recurrent policy network carefully reduces the size of each remaining layer. The resulting network is then evaluated to obtain a reward -- a score based on the accuracy and compression of the network. Our approach uses this reward signal with policy gradients to train the policies to find a locally optimal student network. Our experiments show that we can achieve compression rates of more than 10x for models such as ResNet-34 while maintaining similar performance to the input `teacher' network. We also present a valuable transfer learning result which shows that policies which are pre-trained on smaller `teacher' networks can be used to rapidly speed up training on larger `teacher' networks.

Proximal operators are ubiquitous in inverse problems, commonly appearing as part of algorithmic strategies to regularize problems that are otherwise ill-posed. Modern deep learning models have been brought to bear for these tasks too, as in the framework of plug-and-play or deep unrolling, where they loosely resemble proximal operators. Yet, something essential is lost in employing these purely data-driven approaches: there is no guarantee that a general deep network represents the proximal operator of any function, nor is there any characterization of the function for which the network might provide some approximate proximal. This not only makes guaranteeing convergence of iterative schemes challenging but, more fundamentally, complicates the analysis of what has been learned by these networks about their training data. Herein we provide a framework to develop learned proximal networks (LPN), prove that they provide exact proximal operators for a data-driven nonconvex regularizer, and show how a new training strategy, dubbed proximal matching, provably promotes the recovery of the log-prior of the true data distribution. Such LPN provide general, unsupervised, expressive proximal operators that can be used for general inverse problems with convergence guarantees. We illustrate our results in a series of cases of increasing complexity, demonstrating that these models not only result in state-of-the-art performance, but provide a window into the resulting priors learned from data.

The instruction learning paradigm -- where a model learns to perform new tasks from task descriptions alone -- has become popular in general-purpose model research. The capabilities of large transformer models as instruction learners, however, remain poorly understood. We use a controlled synthetic environment to characterize such capabilities. Specifically, we use the task of deciding whether a given string matches a regular expression (viewed as an instruction) to identify properties of tasks, instructions, and instances that make instruction learning challenging. For instance, we find that our model, a fine-tuned T5-based text2text transformer, struggles with large regular languages, suggesting that less precise instructions are challenging for models. Additionally, instruction executions that require tracking longer contexts of prior steps are also more difficult. We use our findings to systematically construct a challenging instruction learning dataset, which we call Hard RegSet. Fine-tuning on Hard RegSet, our large transformer learns to correctly interpret only 65.6% of test instructions (with at least 90% accuracy), and 11%-24% of the instructions in out-of-distribution generalization settings. We propose Hard RegSet as a challenging instruction learning task, and a controlled environment for studying instruction learning.

This book develops an effective theory approach to understanding deep neural networks of practical relevance. Beginning from a first-principles component-level picture of networks, we explain how to determine an accurate description of the output of trained networks by solving layer-to-layer iteration equations and nonlinear learning dynamics. A main result is that the predictions of networks are described by nearly-Gaussian distributions, with the depth-to-width aspect ratio of the network controlling the deviations from the infinite-width Gaussian description. We explain how these effectively-deep networks learn nontrivial representations from training and more broadly analyze the mechanism of representation learning for nonlinear models. From a nearly-kernel-methods perspective, we find that the dependence of such models' predictions on the underlying learning algorithm can be expressed in a simple and universal way. To obtain these results, we develop the notion of representation group flow (RG flow) to characterize the propagation of signals through the network. By tuning networks to criticality, we give a practical solution to the exploding and vanishing gradient problem. We further explain how RG flow leads to near-universal behavior and lets us categorize networks built from different activation functions into universality classes. Altogether, we show that the depth-to-width ratio governs the effective model complexity of the ensemble of trained networks. By using information-theoretic techniques, we estimate the optimal aspect ratio at which we expect the network to be practically most useful and show how residual connections can be used to push this scale to arbitrary depths. With these tools, we can learn in detail about the inductive bias of architectures, hyperparameters, and optimizers.

This paper considers the learning of logical (Boolean) functions with focus on the generalization on the unseen (GOTU) setting, a strong case of out-of-distribution generalization. This is motivated by the fact that the rich combinatorial nature of data in certain reasoning tasks (e.g., arithmetic/logic) makes representative data sampling challenging, and learning successfully under GOTU gives a first vignette of an 'extrapolating' or 'reasoning' learner. We then study how different network architectures trained by (S)GD perform under GOTU and provide both theoretical and experimental evidence that for a class of network models including instances of Transformers, random features models, and diagonal linear networks, a min-degree-interpolator (MDI) is learned on the unseen. We also provide evidence that other instances with larger learning rates or mean-field networks reach leaky MDIs. These findings lead to two implications: (1) we provide an explanation to the length generalization problem (e.g., Anil et al. 2022); (2) we introduce a curriculum learning algorithm called Degree-Curriculum that learns monomials more efficiently by incrementing supports.