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+# Grigory Bartosh<sup>1</sup> Dmitry Vetrov<sup>2</sup> Christian A. Naesseth<sup>1</sup>
+
+# Abstract
+
+Diffusion models have shown remarkable performance on many generative tasks. Despite recent success, most diffusion models are restricted in that they only allow linear transformation of the data distribution. In contrast, broader family of transformations can help train generative distributions more efficiently, simplifying the reverse process and closing the gap between the true negative log-likelihood and the variational approximation. In this paper, we present Neural Diffusion Models (NDMs), a generalization of conventional diffusion models that enables defining and learning time-dependent non-linear transformations of data. We show how to optimise NDMs using a variational bound in a simulation-free setting. Moreover, we derive a time-continuous formulation of NDMs, which allows fast and reliable inference using off-the-shelf numerical ODE and SDE solvers. Finally, we demonstrate the utility of NDMs through experiments on many image generation benchmarks, including MNIST, CIFAR-10, downsampled versions of ImageNet and CelebA-HQ. NDMs outperform conventional diffusion models in terms of likelihood, achieving state-of-the-art results on ImageNet and CelebAHQ, and produces high-quality samples.
+
+# 1. Introduction
+
+Generative models are a powerful class of probabilistic machine learning models with a wide range of applications from e.g. art and music to medicine and physics (Tomczak, 2022; Creswell et al., 2018; Papamakarios et al., 2021; Yang et al., 2022). Generative models learn to mimic the underlying probability distribution of a given data set and can generate novel samples that are similar to the original data.
+
+<sup>1</sup>University of Amsterdam <sup>2</sup>Constructor University, Bremen. Correspondence to: Grigory Bartosh <g.bartosh@uva.nl>, Dmitry Vetrov <dvetrov@constructor.university>, Christian A. Naesseth <c.a.naeseth@uva.nl>.
+
+Proceedings of the  $41^{st}$  International Conference on Machine Learning, Vienna, Austria. PMLR 235, 2024. Copyright 2024 by the author(s).
+
+They can for example be used for data augmentation, as well as for unsupervised learning.
+
+Diffusion models have emerged as a family of generative models that excel at several generative tasks (Sohl-Dickstein et al., 2015; Ho et al., 2020). They parameterize the data model through an iterative refinement process, the reverse process, that builds up the data step-by-step from pure noise. For training purposes an auxiliary noising process, the forward process, is introduced that successively adds noise to data. The reverse process is then optimized to resemble the forward process. Despite success in various domains (Sohl-Dickstein et al., 2015; Ho et al., 2020; Sahara et al., 2021; Popov et al., 2021; Watson et al., 2022; Trippe et al., 2023), a key limitation of most existing diffusion models is that they rely on a fixed and pre-specified forward process that is unable to adapt to the specific task or data at hand. At the same time there are many works (Hoogeboom & Salimans, 2023; Rombach et al., 2022; Lipman et al., 2023) that improve performance of diffusion models by modifications of the forward processes.
+
+In this paper we develop Neural Diffusion Models (NDMs), a general framework that enables non-linear, time-dependent and learnable data transformations. We extend the approach by Song et al. (2021a) and construct the forward process as a non-Markovian sequence of latent variables; each latent variable is constructed through a transformation of the data to which we then inject noise. This is then leveraged in the corresponding reverse process. To train NDMs efficiently we generalize the diffusion objective while keeping it a simulation-free bound on the log-likelihood. Furthermore, we derive the time-continuous analogue of the objective function as well as the stochastic differential equation (SDE) and ordinary differential equation (ODE) corresponding to the reverse process.
+
+We demonstrate how NDMs generalize several existing diffusion models and then propose a new model with learnable transformations of data parameterized by a neural network. To illustrate the empirical properties of NDMs we provide experimental results on a synthetic data as well as on MNIST, CIFAR-10, downsampled ImageNet and CelebA-HQ image datasets. NDMs consistently outperforms baselines in terms of negative log-likelihood, achieving state-of-the-art results for diffusion models on ImageNet 32 and 64, as well as CelebA-HQ.
+
+The main motivation for NDMs is improved likelihood and density estimation, crucial for applications to compression (MacKay, 2003), semi-supervised learning (Dai et al., 2017), adversarial purification (Song et al., 2017), and many others. However, for completeness we also study the impact of NDMs on image generation quality. We find that for small to medium number of steps NDMs achieves better image generation quality than denoising diffusion probabilistic models (DDPMs) (Ho et al., 2020), and comparable results for a large number of steps.
+
+Finally, we demonstrate that NDMs allows learning simpler generative dynamics like dynamical optimal transport, which conventional diffusion models are incapable of learning.
+
+We summarize the contributions as follows:
+
+1. We propose neural diffusion models or NDMs, a new framework that generalizes conventional diffusion models in both discrete and continuous time settings.  
+2. We develop an objective function to optimize NDMs that upper bounds the negative log-likelihood and study its properties.  
+3. We demonstrate the utility of NDMs with learnable transformations in terms of consistently and significantly improved log-likelihood, as well as better or comparable generation quality.
+
+# 2. Background
+
+Diffusion models are generative models that make use of latent variables. Given a sample from the data distribution  $\mathbf{x} \sim q(\mathbf{x})$ , we define a forward noisung process that produces latent variables  $\mathbf{z}_0, \mathbf{z}_1, \ldots, \mathbf{z}_T$ . In contrast, the reverse generative process reverts the forward process, starting by first generating the same latent variables and then data  $\mathbf{x}$ .
+
+The standard approach to specify the forward process is as a linear Gaussian Markov chain (Sohl-Dickstein et al., 2015; Ho et al., 2020). However, we can also use an implicit definition of the forward process from Song et al. (2021a). This will turn out to be useful for our purposes and is what we focus on here. To construct the implicit forward process we first define the marginal distributions  $q(\mathbf{z}_t|\mathbf{x})$ . Using these marginal distributions we can define the joint distribution of all latent variables  $\mathbf{z}_0,\mathbf{z}_1,\dots ,\mathbf{z}_T$  as follows:
+
+$$
+q \left(\mathbf {z} _ {0: T} | \mathbf {x}\right) = q \left(\mathbf {z} _ {T} | \mathbf {x}\right) \prod_ {t = 1} ^ {T} q \left(\mathbf {z} _ {t - 1} | \mathbf {z} _ {t}, \mathbf {x}\right),
+$$
+
+with  $q(\mathbf{z}_{t - 1}|\mathbf{z}_t,\mathbf{x})$  such that
+
+$$
+q \left(\mathbf {z} _ {t - 1} \mid \mathbf {x}\right) = \int q \left(\mathbf {z} _ {t} \mid \mathbf {x}\right) q \left(\mathbf {z} _ {t - 1} \mid \mathbf {z} _ {t}, \mathbf {x}\right) d \mathbf {z} _ {t}. \tag {1}
+$$
+
+![](images/8b1837d58a59bd712f3e3afd93411761cfce8bee91eb182d9fe13584c6327334.jpg)
+
+![](images/073331fbbe3b0b6b70e8c69fce3a349dca109529cce1cf4aa30cb167699e3a28.jpg)  
+(a) DDIM  
+(b) NDM  
+Figure 1. The directed graphical models of DDIM and NDM.
+
+Here we make use of the posterior distribution  $q(\mathbf{z}_{t-1}|\mathbf{z}_t, \mathbf{x})$  instead of the regular forward distribution  $q(\mathbf{z}_t|\mathbf{z}_{t-1})$ . Due to the dependence also on the data  $\mathbf{x}$  it is a non-Markovian forward process (see Figure 1a). In general the forward process is considered fixed and has no trainable parameters. Moreover, it is specified in such a way that  $q(\mathbf{z}_0|\mathbf{x}) \approx \delta(\mathbf{z}_0 - \mathbf{x})$  and  $q(\mathbf{z}_T|\mathbf{x}) \approx \mathcal{N}(\mathbf{z}_T; 0, I)$ . So if  $q(\mathbf{z}_{t-1}|\mathbf{z}_t)$  was available we could sample  $\mathbf{z}_T \sim \mathcal{N}(\mathbf{z}_T; 0, I)$  and run the reverse process to get  $\mathbf{z}_0 \sim q(\mathbf{z}_0) \approx q(\mathbf{x})$ . However, the distribution  $q(\mathbf{z}_{t-1}|\mathbf{z}_t)$  depends implicitly on the data distribution  $q(\mathbf{x})$  and thus has a complicated form, so we instead approximate the reverse process using a Markov chain with distribution  $p_{\theta}(\mathbf{z}_{0:T})$ :
+
+$$
+p _ {\theta} \left(\mathbf {z} _ {0: T}\right) = p \left(\mathbf {z} _ {T}\right) \prod_ {t = 1} ^ {T} p _ {\theta} \left(\mathbf {z} _ {t - 1} \mid \mathbf {z} _ {t}\right), \tag {2}
+$$
+
+where  $p(\mathbf{z}_T) = \mathcal{N}(\mathbf{z}_T;0,I)$
+
+The combination of the forward process  $q$  and the reverse process  $p_{\theta}$  is a form of (hierarchical) variational autoencoder (Kingma & Welling, 2014; Rezende et al., 2014). Therefore, it can be trained by optimizing the usual variational bound on the negative log-likelihood. In the case of diffusion models, it can be written as follows (see Section A of Ho et al. (2020)):
+
+$$
+\begin{array}{l} \mathbb {E} _ {q} \left[ \underbrace {\mathrm {D} _ {\mathrm {K L}} \left(q (\mathbf {z} _ {T} | \mathbf {x}) | | p (\mathbf {z} _ {T})\right)} _ {\mathcal {L} _ {\mathrm {p r i o r}}} - \underbrace {\log p _ {\theta} (\mathbf {x} | \mathbf {z} _ {0})} _ {\mathcal {L} _ {\mathrm {r e c}}} \right. \\ \left. \right.\left. + \underbrace {\sum_ {t = 1} ^ {T} \mathrm {D} _ {\mathrm {K L}} \left(q \left(\mathbf {z} _ {t - 1} \mid \mathbf {z} _ {t} , \mathbf {x}\right) \mid p _ {\theta} \left(\mathbf {z} _ {t - 1} \mid \mathbf {z} _ {t}\right)\right)} _ {\mathcal {L} _ {\text {d i f f}}} \right] \quad (3) \\ \end{array}
+$$
+
+Since the process  $q$  and the distribution  $p_{\theta}(\mathbf{z}_T) = p(\mathbf{z}_T)$  are fixed, the prior term  $\mathcal{L}_{\mathrm{prior}}$  does not depend on the parameters  $\theta$ , so it can be omitted. The distribution  $p_{\theta}(\mathbf{x}|\mathbf{z}_0)$  is often taken to be a Gaussian distribution, with low variance,
+
+Table 1. Summary of existing diffusion models as instances of Neural Diffusion Models (NDM). See extended table in Appendix B.  
+
+<table><tr><td>Model</td><td>Distribution q(zt|x)</td><td>NDM&#x27;s F(x,t)</td><td>Comment</td></tr><tr><td>DDPM (Ho et al., 2020)/DDIM (Song et al., 2021a)</td><td>N(zt; αt x, σt2I)</td><td>x</td><td></td></tr><tr><td>Flow Matching OT(Lipman et al., 2023)</td><td>N(zt; αt x, σt2I)</td><td>x</td><td>αt = t,σt = 1 - (1 - σmin)t</td></tr><tr><td>VDM(Kingma et al., 2021)</td><td>N(zt; αt x, σt2I)</td><td>x</td><td>α2t = sigmoid(-γη(t)),σ2t = sigmoid(γη(t))</td></tr><tr><td>Soft Diffusion(Daras et al., 2022)</td><td>N(zt; Ct x, st2I)</td><td>Ct x</td><td>αt = 1, σt2 = st2</td></tr><tr><td>LSGM(Vahdat et al., 2021)</td><td>N(zt; αt E(x), σt2I)</td><td>E(x)</td><td>p(x|z0) = N(x; aD(z0), σ2)</td></tr></table>
+
+for continuous data and a dequantization distribution for discrete data. Thus, also the reconstruction term  $\mathcal{L}_{\mathrm{rec}}$  does not depend on the parameter  $\theta$ .
+
+This means that the only part that depends on the model parameters  $\theta$  is the diffusion term  $\mathcal{L}_{\mathrm{diff}}$ . It is a sum of Kullback-Leibler (KL) divergences between posterior distributions in the forward process  $q(\mathbf{z}_{t - 1}|\mathbf{z}_t,\mathbf{x})$  and the distributions  $p_{\theta}(\mathbf{z}_{t - 1}|\mathbf{z}_t)$  from the reverse process. In the general case this KL divergence is intractable, so the standard choice here is to set the marginal conditional distributions to be Gaussian, i.e.  $q(\mathbf{z}_t|\mathbf{x}) = \mathcal{N}(\mathbf{z}_t;\alpha_t\mathbf{x},\sigma_t^2 I)$ . The posterior distribution then takes the form:
+
+$$
+q \left(\mathbf {z} _ {s} \mid \mathbf {z} _ {t}, \mathbf {x}\right) = \mathcal {N} \left(\mathbf {z} _ {s}; \mu_ {s \mid t}, \tilde {\sigma} _ {s \mid t} ^ {2} I\right), \quad \text {f o r} \quad 0 \leq s \leq t \leq T,
+$$
+
+$$
+\mu_ {s \mid t} = \alpha_ {s} \mathbf {x} + \frac {\sqrt {\sigma_ {s} ^ {2} - \tilde {\sigma} _ {s \mid t} ^ {2}}}{\sigma_ {t}} (\mathbf {z} _ {t} - \alpha_ {t} \mathbf {x}). \tag {4}
+$$
+
+Note that here we allow for an arbitrary choice of time grid, i.e.  $s$  and  $t$ , whereas above it was equidistant. It is straightforward to check that such a posterior distribution satisfies (1) for any  $\tilde{\sigma}_{s|t}^2 \leq \sigma_s^2$ . The exact schedule of  $\tilde{\sigma}_{s|t}^2$  is a user design choice.
+
+Finally, the reverse distribution is set to  $p_{\theta}(\mathbf{z}_s|\mathbf{z}_t) = q(\mathbf{z}_s|\mathbf{z}_t,\hat{\mathbf{x}}_\theta (\mathbf{z}_t,t))$ , where  $\hat{\mathbf{x}}_{\theta}(\mathbf{z}_t,t)$  is the model's prediction of  $\mathbf{x}$ . Since  $q(\mathbf{z}_s|\mathbf{z}_t,\mathbf{x})$  and  $p_{\theta}(\mathbf{z}_s|\mathbf{z}_t)$  are both Gaussian distributions, we can compute the KL divergences in  $\mathcal{L}_{\mathrm{diff}}$  in closed form.
+
+This choice of forward and reverse processes, resulting in analytic expressions for the diffusion terms given data, is what makes diffusion models a simulation-free approach. Simulation-free means that we do not have to sample all latent variables for each optimization step. Rather than
+
+calculating all individual terms in  $\mathcal{L}_{\mathrm{diff}}$ , we can uniformly sample  $t$  and optimize only a subset of KL divergences using stochastic gradient descent.
+
+By choosing a specific value for  $\tilde{\sigma}_{s|t}^2$ , we can obtain equality between the processes of DDPM and DDIM (see section 4.1 of Song et al. (2021a)). Furthermore, as Song et al. (2021c) demonstrated, when the number of steps  $T$  in DDPM goes to infinity, we can transition to continuous time. In this scenario, the reverse process can be described using a Stochastic Differential Equation (SDE):
+
+$$
+d \mathbf {z} _ {t} = [ r (t) \mathbf {z} _ {t} - g ^ {2} (t) s _ {\theta} (\mathbf {z} _ {t}, t) ] d t + g (t) d \mathbf {w} _ {t},
+$$
+
+$$
+s _ {\theta} (\mathbf {z} _ {t}, t) = \frac {\alpha_ {t} \hat {\mathbf {x}} _ {\theta} (\mathbf {z} _ {t} , t) - \mathbf {z} _ {t}}{\sigma_ {t} ^ {2}}, \quad r (t) = \frac {d \log \alpha_ {t}}{d t},
+$$
+
+$$
+g ^ {2} (t) = \frac {d \sigma_ {t} ^ {2}}{d t} - 2 \frac {d \log \alpha_ {t}}{d t} \sigma_ {t} ^ {2}, \tag {5}
+$$
+
+with time running backwards from  $t = 1$  to  $t = 0$ . This formulation allows us to switch to the equivalent ODE and to use different SDE and ODE solvers for sampling and density estimation.
+
+# 3. Neural diffusion models
+
+In theory, we can view diffusion models as a special type of hierarchical variational autoencoders (VAE). From this perspective, the conventional diffusion model resembles a VAE with a fixed variational distribution, in which the latent variables are inferred using scaling of data points and injecting of Gaussian noise. Such a formulation limits diffusion models in terms of the flexibility of the latent space. Introducing a more flexible (and learnable) distribution of latent variables could effectively reduce the gap between the log-likelihood and the variational bound. In practical
+
+# Algorithm 1 Learning NDM
+
+Require:  $q(\mathbf{x}),F_{\varphi},\hat{\mathbf{x}}_{\theta}$
+
+for learning iterations do
+
+$$
+\mathbf {x} \sim q (\mathbf {x}), t \sim U [ 1, T ], \varepsilon \sim \mathcal {N} (0, I)
+$$
+
+$$
+\mathbf {z} _ {t} \sim q _ {\varphi} (\mathbf {z} _ {t} | \mathbf {x})
+$$
+
+$$
+\mathcal {L} = \mathcal {L} _ {\text {r e c}} + \mathcal {L} _ {\text {d i f f}} + \mathcal {L} _ {\text {p r i o r}}
+$$
+
+Gradient step on  $\theta$  and  $\varphi$  w.r.t.  $\mathcal{L}$
+
+# end for
+
+terms, a more flexible forward process might simplify the task of learning the reverse (generative) process, thereby enhancing model quality. To overcome this limitation of conventional diffusion models, we propose a general form of transformations of data that allows to define and learn distributions on the latent space.
+
+In this section, we introduce the Neural Diffusion Models (NDMs) - a simulation-free framework that generalises conventional diffusion models. The key idea in NDMs is to apply a time-dependent transformation  $F_{\varphi}(\mathbf{x},t)$  to the data  $\mathbf{x}$  at each step of forward process before injecting noise. Previous diffusion models arise as special cases when the data transformation is either linear, time-independent, or pre-defined non-linear (see Table 1). In contrast, the NDM can work with any time-dependent transformation of data and may be learned end-to-end. In Section 4 we provide experimental results with  $F_{\varphi}(\mathbf{x},t)$  parameterized by neural networks.
+
+# 3.1. Model definition and variational objective
+
+We introduce NDMs constructively. First, we define the desired marginal distributions:
+
+$$
+q _ {\varphi} (\mathbf {z} _ {t} | \mathbf {x}) = \mathcal {N} \left(\mathbf {z} _ {t}; \alpha_ {t} F _ {\varphi} (\mathbf {x}, t), \sigma_ {t} ^ {2} I\right), \tag {6}
+$$
+
+where  $F_{\varphi}(\mathbf{x},t):\mathbb{R}^d\times [0,T]\mapsto \mathbb{R}^d$  is a function parameterized by  $\varphi$  that applies a time-dependent transform to the data point  $\mathbf{x}$ . We adapt the approach from DDIM, as described in Section 2, and choose the following posterior distribution that satisfies (6) (we provide derivation and proof in Appendix A.1):
+
+$$
+q _ {\varphi} \left(\mathbf {z} _ {s} \mid \mathbf {z} _ {t}, \mathbf {x}\right) = \mathcal {N} \left(\mathbf {z} _ {s}; \mu_ {s | t} ^ {F _ {\varphi}}, \tilde {\sigma} _ {s | t} ^ {2} I\right), \tag {7}
+$$
+
+$$
+\mu_ {s | t} ^ {F _ {\varphi}} = \alpha_ {s} F _ {\varphi} (\mathbf {x}, s) + \frac {\sqrt {\sigma_ {s} ^ {2} - \tilde {\sigma} _ {s | t} ^ {2}}}{\sigma_ {t}} \Big (\mathbf {z} _ {t} - \alpha_ {t} F _ {\varphi} (\mathbf {x}, t) \Big),
+$$
+
+for  $0 \leq s \leq t \leq T$  where  $\tilde{\sigma}_{s|t}^2 \leq \sigma_s^2$  is a design choice. Using this posterior we can define an implicit forward process according to (1) (see Figure 1b). This forward process provides access to both marginal and posterior distributions just like in the DDIM framework (Song et al., 2021a). The
+
+# Algorithm 2 Sampling from NDM
+
+Require:  $F_{\varphi},\hat{\mathbf{x}}_{\theta}$
+
+$$
+\mathbf {z} _ {T} \sim \mathcal {N} (0, I)
+$$
+
+$$
+\mathbf {f o r} t = T, \dots , 1 \mathbf {d o}
+$$
+
+$$
+\hat {\mathbf {x}} = \hat {\mathbf {x}} _ {\theta} (\mathbf {z} _ {t}, t)
+$$
+
+$$
+\mathbf {z} _ {t - 1} \sim q _ {\varphi} (\mathbf {z} _ {t - 1} | \mathbf {z} _ {t}, \hat {\mathbf {x}})
+$$
+
+end for
+
+$$
+x \sim p (\mathbf {x} | \mathbf {z} _ {0})
+$$
+
+corresponding NDM variational objective has the following form:
+
+$$
+\begin{array}{l} \mathbb {E} _ {q _ {\varphi}} \left[ \underbrace {\mathrm {D} _ {\mathrm {K L}} \left(q _ {\varphi} (\mathbf {z} _ {T} | \mathbf {x}) | | p (\mathbf {z} _ {T})\right)} _ {\mathcal {L} _ {\text {p r i o r}}} - \underbrace {\log p _ {\theta} (\mathbf {x} | \mathbf {z} _ {0})} _ {\mathcal {L} _ {\text {r e c}}} \right. \\ \left. + \underbrace {\sum_ {t = 1} ^ {T} \mathrm {D} _ {\mathrm {K L}} \left(q _ {\varphi} \left(\mathbf {z} _ {t - 1} \mid \mathbf {z} _ {t} , \mathbf {x}\right) \mid p _ {\theta} \left(\mathbf {z} _ {t - 1} \mid \mathbf {z} _ {t}\right)\right)} _ {\mathcal {L} _ {\text {d i f f}}} \right]. \tag {8} \\ \end{array}
+$$
+
+While the objective has the same form as in DDIM (3), the individual terms are different. If the transformation  $F_{\varphi}(\mathbf{x},t)$  is actually parameterized by learnable parameters  $\varphi$ , the prior term  $\mathcal{L}_{\mathrm{prior}}$  and the reconstruction term  $\mathcal{L}_{\mathrm{rec}}$  depend on the parameter  $\varphi$  as well. Therefore, in that case these terms cannot be excluded from the optimization process.
+
+For the standard parameterization of the reverse process through approximate posteriors  $p_{\theta}(\mathbf{z}_s|\mathbf{z}_t) = q_{\varphi}(\mathbf{z}_s|\mathbf{z}_t,\hat{\mathbf{x}}_\theta (\mathbf{z}_t,t))$  the KL divergences in the diffusion term  $\mathcal{L}_{\mathrm{diff}}$  are (see Appendix A.2):
+
+$$
+\begin{array}{l} \mathrm {D} _ {\mathrm {K L}} \left(q _ {\varphi} \left(\mathbf {z} _ {s} \mid \mathbf {z} _ {t}, \mathbf {x}\right) | | p _ {\theta} \left(\mathbf {z} _ {s} \mid \mathbf {z} _ {t}\right)\right) = \\ \frac {1}{2 \tilde {\sigma} _ {s | t} ^ {2}} \left\| \alpha_ {s} \left(F _ {\varphi} (\mathbf {x}, s) - F _ {\varphi} (\hat {\mathbf {x}} _ {\theta} (\mathbf {z} _ {t}, t), s)\right) + \right. \\ \left. \frac {\sqrt {\sigma_ {s} ^ {2} - \tilde {\sigma} _ {s \mid t} ^ {2}}}{\sigma_ {t}} \alpha_ {t} \left(F _ {\varphi} \left(\hat {\mathbf {x}} _ {\theta} (\mathbf {z} _ {t}, t), t\right) - F _ {\varphi} (\mathbf {x}, t)\right) \right\| _ {2} ^ {2}. \tag {9} \\ \end{array}
+$$
+
+Note a distinction between the objectives of NDM and DDIM here. In the case of DDIM, the model tries to accurately predict the data point  $\mathbf{x}$ . In contrast, NDM aims to predict the transformed data point  $F_{\varphi}(\mathbf{x},t)$ . Despite this change, NDM's optimization remains simulation-free, so we can efficiently train the NDM by sampling time steps and calculating corresponding KL divergences. We summarise the training and sampling procedures in Algorithms 1 and 2.
+
+Given that NDM is a generalization of DDIM, we can leverage the same techniques for inference. Specifically, we can adjust the number of intermediate time steps, the schedule
+
+of  $\tilde{\sigma}_{s|t}^2$  as well as sampling with various dynamics, including a deterministic dynamic corresponding to  $\tilde{\sigma}_{s|t}^2 = 0$ .
+
+# 3.2. Continuous time NDMs
+
+We previously formulated NDMs in the discrete time setting with  $T$  steps. However, like conventional diffusion models, we can let the number of steps  $T$  go to infinity and switch to continuous time. In this case, the set of time steps  $\{0,1,\dots,T\}$  transforms to the range  $[0,1]$  and the diffusion term of the objective reduces to an expectation over time (see derivation in Appendix A.4):
+
+$$
+\begin{array}{l} \mathcal {L} _ {\mathrm {d i f f}} = \\ \mathbb {E} _ {q (\mathbf {x}) u (t) q (\mathbf {z} _ {t} | \mathbf {x})} \left[ \frac {1}{g ^ {2} (t)} \right\| \alpha_ {t} \left(\dot {F} _ {\varphi} (\mathbf {x}, t) - \dot {F} _ {\varphi} \left(\hat {\mathbf {x}} _ {\theta} (\mathbf {z} _ {t}, t), t\right)\right) \\ + \frac {1}{2} \left(\frac {\partial \sigma_ {t} ^ {2}}{\partial t} - 2 r (t) \sigma_ {t} ^ {2} + g ^ {2} (t)\right). \\ \left. \cdot \left(s (\mathbf {x}, \mathbf {z} _ {t}, t) - s \left(\hat {\mathbf {x}} _ {\theta} \left(\mathbf {z} _ {t}, t\right), \mathbf {z} _ {t}, t\right)\right) \right\| _ {2} ^ {2} \bigg ], \tag {10} \\ \end{array}
+$$
+
+where  $r(t) = \frac{\partial\log\alpha_t}{\partial t},g^2 (t) = \dot{\nu}_t\sigma_t^2,$  and  $s(\mathbf{x},\mathbf{z}_t,t) = \frac{\alpha_tF_\varphi(\mathbf{x},t) - \mathbf{z}_t}{\sigma_t^2}.$
+
+Similar to training a discrete time NDM, we can train a continuous time NDMs by sampling time. In our experiments we use importance sampling (Song et al., 2021b) and sample time from a distribution proportional to  $\frac{1}{g^2(t)}$ .
+
+Note, that we may not have access to the partial derivative of the transformation  $F_{\varphi}(\cdot ,t)$  with respect to  $t$  in closed form. However, for any differentiable  $F_{\varphi}(\cdot ,t)$  we can use Jacobian-Vector product (Smale & Hirsch, 1974) to obtain this derivative.
+
+The discrete time reverse process also becomes a continuous time process, described by a Stochastic Differential Equation (SDE). If we parameterize the noise injection in the posterior distribution as  $\tilde{\sigma}_{s|t}^2 = \sigma_s^2 (1 - e^{\nu_s - \nu_t})$ , we obtain the following SDE (see derivation in Appendix A.3):
+
+$$
+\begin{array}{l} d \mathbf {z} _ {t} = \left[ \alpha_ {t} \dot {F} _ {\varphi} \left(\hat {\mathbf {x}} _ {\theta} \left(\mathbf {z} _ {t}, t\right), t\right) + r (t) \mathbf {z} _ {t} \right. \\ \left. - \frac {1}{2} \left(g ^ {2} (t) - 2 r (t) \sigma_ {t} ^ {2}\right) s _ {\theta} (\mathbf {z} _ {t}, t) \right] d t + g (t) d \mathbf {w}, \tag {11} \\ \end{array}
+$$
+
+where  $s_{\theta}(\mathbf{z}_t,t) = \frac{\alpha_tF_\varphi(\hat{\mathbf{x}}_\theta(\mathbf{z}_t,t),t) - \mathbf{z}_t}{\sigma_t^2}$ .
+
+By changing the function  $\nu_{t}$ , we can obtain different dynamics. In the extreme case where  $\nu_{t}$  is equal to a constant we have deterministic dynamics described by an ODE. This enables the use of SDE or ODE solvers for inference. Moreover, we can estimate densities by considering the model as a continuous normalizing flow (Chen et al., 2018) in the deterministic case.
+
+# 4. Experiments
+
+We present empirical results for the proposed Neural Diffusion Models with learnable transformations on a synthetic datasets as well as multiple image datasets. Qualitatively, NDMs learn transformations that simplify the data distribution, leading to predictions of  $\mathbf{x}$  that are more aligned with the data. Quantitatively, NDMs consistently outperform the baseline in terms of likelihood, achieving state-of-the-art diffusion model results for ImageNet and CelebA-HQ. Moreover, for a small to medium number of steps, NDMs achieve better image generation quality than DDPM, while being comparable for a large number of steps. We also provide a proof of concept experiment that demonstrates that NDMs can learn simple generative trajectories, something conventional diffusion models are incapable of learning.
+
+# 4.1. Implementation details
+
+We demonstrate NDMs with learnable transformations on the MNIST (Deng, 2012), CIFAR-10 (Krizhevsky et al., 2009), downsampled ImageNet (Deng et al., 2009; Van Den Oord et al., 2016) and CelebA-HQ-256 (Karras et al., 2017) datasets. In all experiments we use same neural network architectures to parameterize both the generative process and the transformations  $F_{\varphi}$ . In experiments with images we use the U-Net architecture from Dhariwal & Nichol (2021). To ensure consistency with Song et al. (2021c;b), we apply horizontal flipping as a data augmentation technique for training models on CIFAR-10 and ImageNet. Unless otherwise stated, we utilize the DDPM variance-preserving schedule of noise injection for  $\alpha_{t}$  and  $\sigma_t^2$ . For density estimation of discrete data we use uniform dequantization.
+
+In the experiments we report negative log-likelihood (NLL) in bits per dimension (BPD), negative evidence lower bound (NELBO) (8), and sample quality as measured by the Frechet Inception Distance (FID) (Heusel et al., 2017). We calculate NLL by integrating the corresponding ODEs using the RK45 solver from Dormand & Prince (1980), and both NLL and NELBO are calculated on test data. For FID we report the score computed using  $50\mathrm{k}$  generated images.
+
+In Section 3 we parameterize the reverse process through  $\hat{\mathbf{x}}_{\theta}$  function. However, in practice we reparameterize the generative process in terms of prediction of injected noise. For a detailed description of parameterizations and other experimental details, please refer to Appendix C.
+
+# 4.2. Learned transformations
+
+Let us examine some of the transformations that NDM learns. Figure 2a-2c illustrates the transformations that NDM learns for the 2D checkerboard distribution, MNIST, and CIFAR-10 datasets. For the checkerboard, we observe that  $F_{\varphi}$  learns to transform the interleaved pattern into a
+
+![](images/5d3f9a00c9366c32c60b76566304ecd76e70ad4620d8ac6f2e01b6e583330d52.jpg)
+
+![](images/64edcf3c5e13e7a150f86a698d274d121085900e327333d191c4a9c7c962d8bd.jpg)
+
+![](images/4514ea3edcb9bdad3547e2f6d1e08520e8081778e11c771ee0d49e032213d848.jpg)  
+(a) Top: Data  $\mathbf{x}$ . Bottom:  $F_{\varphi}(\mathbf{x},T)$ , transformed data.
+
+![](images/1d672f188e323a905a4a14fcaedaa70fb1a5eb5ff7587e28d8d5c9d0c6a8783a.jpg)  
+(b) Top: CIFAR data samples. Bottom:  $F_{\varphi}(\mathbf{x}, T)$ , transformed data samples.
+
+![](images/761e2c3539d85769fd993b450b0ecec7c8176e51fc0f9132d83988257ae8905f.jpg)  
+(c) Top: MNIST data samples. Bottom:  $F_{\varphi}(\mathbf{x}, T)$ , transformed data samples.
+
+![](images/52ef072051ed99b028e15ea1fa77d98a9f65a634478e0e8799003ea7a2d79bcc.jpg)  
+(d) DDPM predictions  $\hat{\mathbf{x}}_{\theta}(\mathbf{z}_T,T)$  for  $\mathbf{z}_T\sim \mathcal{N}(\mathbf{z}_T;0,I)$  
+(e) NDM predictions  $\hat{\mathbf{x}}_{\theta}(\mathbf{z}_T,T)$  for  $\mathbf{z}_T\sim \mathcal{N}(\mathbf{z}_T;0,I)$  
+Figure 2. Learned transforms for the 2D checkerboard distribution (left). Learned transforms for CIFAR-10 and MNIST (top right), as well as predictions for MNIST (bottom right). NDM learns useful forward transformations and more accurately predicts the data from injected noise.
+
+Table 2. We report Negative Log-Likelihood (NLL) in Bits Per Dimension (BPD) on the test sets for CIFAR10, ImageNet 32x32, and ImageNet 64x64. Our results were obtained using the continuous-time formulation of our model, integrated via the corresponding Ordinary Differential Equation (ODE), as detailed in Section 3.2.  
+
+<table><tr><td>Model</td><td>CIFAR10</td><td>ImageNet 32</td><td>ImageNet 64</td></tr><tr><td>DDPM (Ho et al., 2020)</td><td>3.69</td><td></td><td></td></tr><tr><td>Improved DDPM (Nichol &amp; Dhariwal, 2021)</td><td>2.94</td><td></td><td>3.54</td></tr><tr><td>VDM (Kingma et al., 2021)</td><td>2.65</td><td>3.72</td><td>3.40</td></tr><tr><td>Score SDE (Song et al., 2021c)</td><td>2.99</td><td></td><td></td></tr><tr><td>Score Flow (Song et al., 2021b)</td><td>2.83</td><td>3.76</td><td></td></tr><tr><td>NDM (ours)</td><td>2.70</td><td>3.55</td><td>3.35</td></tr></table>
+
+non-interleaved one. In the case of the grayscale digits of the MNIST dataset,  $F_{\varphi}$  learns to highlight the distinctive features of the numbers. It thickens the lines and even creates bubbles at the corners. For the color images of CIFAR-10,  $F_{\varphi}$  learns to increase the image contrast. In all cases, our model learns a way to simplify the data distribution. These transformations may enable the reverse process to transition more smoothly from simple distributions to complex ones.
+
+Furthermore, we would like to emphasize the difference between the predictions of  $\mathbf{x}$  that NDM and DDPM makes. Figure 2d and Figure 2e shows the predictions  $\hat{\mathbf{x}}_{\theta}(\mathbf{z}_T,T)$  generated by NDM and DDPM models trained on the MNIST dataset. In each case, the model samples from a standard normal distribution  $\mathbf{z}_T\sim \mathcal{N}(\mathbf{z}_T;0,I)$  and based on this value tries to predict  $\mathbf{x}$ . Therefore, we do not expect these samples to be of high quality. However, as we can
+
+see, NDM's predictions of  $\mathbf{x}$  are much more similar to the data distribution than DDPM's predictions. We attribute this behavior to the fact that our model aims to predict not the datapoint  $\mathbf{x}$ , but the transformed datapoint  $F_{\varphi}(\mathbf{x}, t)$ . Thus, to make better predictions of the transformed datapoint, it may be critical to generate predictions of  $\mathbf{x}$  that resemble real data. Any deviation from the  $\mathbf{x}$ -distribution is exaggerated by the transformation and thus less likely to happen for NDM's predictions.
+
+In Appendix D we provide additional samples for terminal and intermediate timesteps.
+
+# 4.3. Image generation
+
+Next, we evaluate NDMs with learnable transformations quantitatively. We train continuous time NDM on MNIST,
+
+Table 3. Performance comparison of the DDPM and NDM on CIFAR-10 and ImageNet 32 datasets. We report FID scores for DDPM-style (FID) and DDIM-style  $(\mathrm{FID}^{*})$  sampling procedures.  
+
+<table><tr><td rowspan="2">Steps</td><td rowspan="2">Model</td><td colspan="4">CIFAR-10</td><td colspan="4">ImageNet 32</td></tr><tr><td>NLL ↓</td><td>NELBO ↓</td><td>FID ↓</td><td>FID* ↓</td><td>NLL ↓</td><td>NELBO ↓</td><td>FID ↓</td><td>FID* ↓</td></tr><tr><td rowspan="2">1000</td><td>DDPM</td><td>3.11</td><td>3.18</td><td>11.44</td><td>13.35</td><td>3.89</td><td>3.95</td><td>16.18</td><td>19.08</td></tr><tr><td>NDM</td><td>3.02</td><td>3.03</td><td>11.82</td><td>13.79</td><td>3.79</td><td>3.82</td><td>17.02</td><td>19.76</td></tr><tr><td rowspan="2">10</td><td>DDPM</td><td>5.02</td><td>5.13</td><td>37.83</td><td>19.89</td><td>6.28</td><td>6.42</td><td>53.51</td><td>26.47</td></tr><tr><td>NDM</td><td>4.63</td><td>4.74</td><td>31.56</td><td>22.20</td><td>5.81</td><td>5.94</td><td>45.38</td><td>29.95</td></tr><tr><td rowspan="2">1000 → 10</td><td>DDPM</td><td>8.78</td><td>8.98</td><td>43.85</td><td>17.73</td><td>10.99</td><td>11.23</td><td>58.35</td><td>25.53</td></tr><tr><td>NDM</td><td>8.58</td><td>8.81</td><td>48.41</td><td>16.96</td><td>10.78</td><td>11.06</td><td>62.12</td><td>23.77</td></tr></table>
+
+CIFAR-10, and downsampled ImageNet datasets. Table 2 summarizes our results, reporting NLL. NDMs demonstrate performance on CIFAR-10 that is comparable with the baselines and outperforms baselines on ImageNet.
+
+Then, we compare NDM with the DDPM baseline on MNIST, CIFAR-10, and ImageNet 32 datasets. To ensure a fair comparison, when implementing DDPM we use an NDM with fixed identity transformation  $F_{\varphi}(\mathbf{x},t) = \mathbf{x}$ . Therefore, we train both models with the same objective (8) and hyperparameters. The first part of Table 3 summarizes our results, reporting NLL, NELBO (8), and FID score. NDM demonstrates comparable sample quality with the baseline on all datasets and consistently outperforms the baseline on NLL and NELBO, especially for smaller numbers of steps. This improvement may be attributed to NDM's ability to fit distributions of the forward process and simplify the denoising task for the reverse process.
+
+We also compare NDM with DDPM in a setup where we train both models with  $T = 1000$  steps and then sample with fewer steps. The second part of Table 3 summarizes our results, which are consistent with the corresponding numbers of steps used during training. However, in absolute values, both models show worse performance when we decrease the number of steps, and NDM demonstrates a more severe degradation. This observation is especially noticeable for small numbers of steps, such as  $T = 10$ , where NDM has a better FID score than DDPM when trained with 10 steps, but a worse FID score when the number of steps is decreased from 1000 to 10. From this, we conclude that although NDM can in principle work with reduced number of steps it is less robust to such modifications compared to DDPM.
+
+Finally, we demonstrate that NDMs may be successfully combined with LSGM (Vahdat et al., 2021). For this experiment we replaced the linear diffusion in the LSGM baseline for CelebA-HQ-256 with NDMs featuring the learnable  $F_{\varphi}$ . We parameterise  $F_{\varphi}$  with the same neural network architecture as baseline's architecture for parameterisation
+
+Table 4. Generative results on CelebA-HQ-256 for LSGM and NDM with learnable transformations in the latent space of VAE.  
+
+<table><tr><td>Model</td><td>NLL ↓</td><td>FID ↓</td></tr><tr><td>LSGM (Vahdat et al., 2021)</td><td>≤ 0.70</td><td>7.22</td></tr><tr><td>Latent NDM (ours)</td><td>≤ 0.65</td><td>7.18</td></tr></table>
+
+of diffusion. Table 4 demonstrates that NDMs have better likelihood estimation and sample quality.
+
+In Appendix E we provide a proof of concept experiment, which demonstrates that we can learn simpler generative dynamics compared to conventional diffusion models. In this experiment we restrict the reverse process to learn dynamic optimal transport trajectories. It is not possible to match such a reverse process with a predefined forward process, but NDM allows to capture the data distribution with the simpler generative dynamics.
+
+See Appendix D for additional results and ablation studies.
+
+# 5. Related work
+
+NDMs build on diffusion probabilistic models originally proposed by Sohl-Dickstein et al. (2015), which can be considered as an instance of (hierarchical) variational autoencoders (VAEs) (Kingma & Welling, 2014; Rezende et al., 2014). Recently, the theory of diffusion models was extended to deterministic sampling (Song et al., 2021a) and continuous time (Song et al., 2021c). These results allowed to reach impressive performance in image generation tasks (Ho et al., 2020; Song et al., 2021c; Dhariwal & Nichol, 2021; Kingma et al., 2021). However, most existing diffusion models have a significant limitation in that they rely on a pre-specified and simple noise injection process that is unable to adapt to the specific task or data at hand. To overcome this, researchers have explored ways to generalize diffusion models.
+
+Various papers have since proposed ways to speed up sam
+
+pling from diffusion models. Tachibana et al. (2021) and Liu et al. (2022) proposed alternative SDE and ODE solvers. Xiao et al. (2021) proposed replacing simple Gaussian distributions at each generation step with distributions learned by GANs (Goodfellow et al., 2014). Some works proposed methods like iterative distillation with a reduction in the number of steps (Salimans & Ho, 2022) and iterative straightening of trajectories (Liu et al., 2023; Liu, 2022). While these methods change the generative process, they are compatible with NDMs.
+
+Several papers proposed constructing the process of data corruption not by noise injection, but rather by blurring (Rissanen et al., 2023; Daras et al., 2022; Hoogeboom & Salimans, 2023) or through another linear transformation (Singhal et al., 2023). Another line of work modifies directly the dynamics of diffusion models through mapping the data into the latent space of VAE (Vahdat et al., 2021; Rombach et al., 2022), hierarchical VAE (Gu et al., 2023) or normalizing flow (Kim et al., 2022) models and then runs standard linear diffusion. As we demonstrate in Tables 1, these arise as distinct special cases of NDMs for specific choices of the transformation  $F_{\varphi}$ .
+
+In another line of works (De Bortoli et al., 2021; Wang et al., 2021; Peluchetti), finite-time diffusion constructions were proposed using diffusion bridge theory to address the approximation error incurred by infinite-time denoising constructions. While such approaches allow learning forward transformations, they require inferring all latent variables for each optimization step. This limitation breaks the simulation-free paradigm and can make these models expensive to train. NDM in contrast allows learning forward transformations efficiently and simulation-free.
+
+Inspired by diffusion models, several works (Lipman et al., 2023; Neklyudov et al., 2022) have proposed simulation-free objectives for training continuous normalizing flows. These approaches are similar to diffusion models as they rely on the idea of reversing a predefined corruption process. Later, some works (Albergo & Vanden-Eijnden, 2023; Lee et al., 2023) extended these ideas and proposed to learn the forward process. However, although NDMs and these works are similar in spirit, they differ in that they optimize the forward process specifically to obtain straight generative trajectories, while in our approach we optimize learnable forward process to minimize variational bound on NLL, which not necessarily leads to straight generative trajectories.
+
+In concurrent and independent work, Nielsen et al. (2024) introduced DiffEnc, also adding time-dependent transformations in diffusion models. While the underlying idea is similar, there are some differences between DiffEnc and NDM. The two approaches utilize different parameterisations and noise injection schedules. In addition, DiffEnc approximates time derivatives of the data transformations
+
+leading to biased stochastic gradients, while NDM calculates exact time derivatives using JVP.
+
+In Appendix B, we provide further discussion, details and comparisons with related works.
+
+# 6. Limitations
+
+Compared to conventional diffusion models, NDMs with learnable transformations have twice as many parameters, which results in slower training. Specifically, in experiments on images, NDMs with learnable transformations take approximately 2.3 times longer than DDPM to train. However, no additional techniques where necessary to ensure stable training of NDMs. Additionally, in Appendix D, we provide an ablation study demonstrating that performance improvements are not achieved by increasing the number of parameters.
+
+Another distinction between NDMs and DDPM is the importance for NDMs in using the full objective (8) when training the model. A simplified objective, such as  $\mathcal{L}_{\mathrm{simple}}$  used in DDPM, which measures how well the model predicts injected noise and does not take into account the transformation  $F_{\varphi}$ , can cause the collapse of this transformation to 0. The reason for this is that it becomes trivial to identify the injected noise through  $\mathbf{z}_t$ .
+
+Finally, unlike conventional diffusion, the generative process of NDMs with learnable transformations depends on the parameters of the forward process. Therefore, in the case of learnable parameters, NDMs do not support conditional generation techniques with classifier guidance (Dhariwal & Nichol, 2021). However, we can utilize alternative approaches (Wu et al., 2023) to enable conditional generation from NDMs, but we will defer this to future research.
+
+# 7. Conclusion
+
+We introduced Neural Diffusion Models (NDMs), a new class of diffusion models that enables defining and learning the general forward noising process. First, we showed how to optimize NDMs using a variational bound in a simulation-free setting. Then, we derived a time-continuous formulation of NDMs allowing for fast and reliable inference and likelihood evaluation using off-the-shelf numerical ODE and SDE solvers. Next, we demonstrated how some existing diffusion models appear as a special cases of NDMs. For NDMs with learnable transformations we studied their utility on standard image generation benchmarks. NDMs significantly outperforms conventional diffusion models in terms of likelihood, achieving state-of-the-art results for ImageNet and CelebA-HQ, and produces samples of comparable or better quality.
+
+# Impact Statement
+
+This paper presents work whose goal is to advance the field of Machine Learning. There are many potential societal consequences of our work, none which we feel must be specifically highlighted here.
+
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+
+# A. Derivations and proofs
+
+# A.1.Forward posterior
+
+First, we rewrite the marginal distribution (6) in terms of standard normally distributed  $\varepsilon_{t},\varepsilon_{s}$  for  $s$  and  $t$  , where  $s <   t$  ..
+
+$$
+\mathbf {z} _ {t} = \alpha_ {t} F _ {\varphi} (\mathbf {x}, t) + \sigma_ {t} \varepsilon_ {t}, \tag {12}
+$$
+
+$$
+\mathbf {z} _ {s} = \alpha_ {s} F _ {\varphi} (\mathbf {x}, s) + \sigma_ {s} \varepsilon_ {s}. \tag {13}
+$$
+
+Next, we constructively introduce the posterior distribution  $q_{\varphi}(\mathbf{z}_s|\mathbf{z}_t,\mathbf{x})$ . To sample  $\mathbf{z}_s$  given  $\mathbf{z}_t$  and  $\mathbf{x}$  while preserving the correct marginal distribution  $q_{\varphi}(\mathbf{z}_s|\mathbf{x})$ , we can combine the noise  $\varepsilon_t$  with additional noise  $\tilde{\varepsilon}_{s|t}$  as follows:
+
+$$
+\mathbf {z} _ {s} = \alpha_ {s} F _ {\varphi} (\mathbf {x}, s) + \sqrt {\sigma_ {s} ^ {2} - \tilde {\sigma} _ {s | t} ^ {2}} \varepsilon_ {t} + \tilde {\sigma} _ {s | t} \tilde {\varepsilon} _ {s | t}. \tag {14}
+$$
+
+The samples  $\mathbf{z}_s$  follow a (conditional) normal distribution. By marginalizing  $\varepsilon_t$  and  $\tilde{\varepsilon}_{s|t}$ , we obtain a normal distribution with mean  $\alpha_s F\varphi(\mathbf{x}, s)$  and variance  $\sigma_s^2 - \tilde{\sigma}_{s|t}^2 + \tilde{\sigma}_{s|t}^2 = \sigma_s^2$ . Therefore, this sampling procedure satisfies  $q_{\varphi}(\mathbf{z}_s|\mathbf{x}) = \int q_{\varphi}(\mathbf{z}_t|\mathbf{x}) q_{\varphi}(\mathbf{z}_s|\mathbf{z}_t, \mathbf{x}) d\mathbf{z}_t$ .
+
+The equation (14) relies on  $\varepsilon_{t}$ , which we do not have explicit access to. However, once we know  $\mathbf{z}_t$  and  $\mathbf{x}$ , we can calculate it from (12) as  $\varepsilon_{t} = \frac{\mathbf{z}_{t} - \alpha_{t}F_{\varphi}(\mathbf{x},t)}{\sigma_{t}}$  and substitute it in (14):
+
+$$
+\mathbf {z} _ {s} = \alpha_ {s} F _ {\varphi} (\mathbf {x}, s) + \frac {\sqrt {\sigma_ {s} ^ {2} - \tilde {\sigma} _ {s \mid t} ^ {2}}}{\sigma_ {t}} \left(\mathbf {z} _ {t} - \alpha_ {t} F _ {\varphi} (\mathbf {x}, t)\right) + \tilde {\sigma} _ {s \mid t} \tilde {\varepsilon} _ {s \mid t}. \tag {15}
+$$
+
+Using this constructive definition, we obtain the posterior distribution (7).
+
+# A.2. Objective
+
+To calculate the diffusion term  $\mathcal{L}_{\mathrm{diff}}$  (9) of the objective, we need to compute the KL divergence between the forward posterior distribution  $q_{\varphi}(\mathbf{z}_s|\mathbf{z}_t,\mathbf{x})$  and the reverse distribution  $p_{\theta}(\mathbf{z}_s|\mathbf{z}_t)$ . Since we use parameterization  $p_{\theta}(\mathbf{z}_s|\mathbf{z}_t) = q_{\varphi}(\mathbf{z}_s|\mathbf{z}_t,\hat{\mathbf{x}}_\theta (\mathbf{z}_t,t))$ , both of these distributions are normal distributions with the same variance, so we can evaluate the KL divergence between them analytically as follows:
+
+$$
+\begin{array}{l} \mathrm {D} _ {\mathrm {K L}} \Big (q _ {\varphi} (\mathbf {z} _ {s} | \mathbf {z} _ {t}, \mathbf {x}) | | p _ {\theta} (\mathbf {z} _ {s} | \mathbf {z} _ {t}) \Big) = \\ = \frac {1}{2 \tilde {\sigma} _ {s | t} ^ {2}} \left\| \alpha_ {s} F _ {\varphi} (\mathbf {x}, s) + \frac {\sqrt {\sigma_ {s} ^ {2} - \tilde {\sigma} _ {s | t} ^ {2}}}{\sigma_ {t}} \left(\mathbf {z} / - \alpha_ {t} F _ {\varphi} (\mathbf {x}, t)\right) - \right. \\ \left. \alpha_ {s} F _ {\varphi} \left(\hat {\mathbf {x}} _ {\theta} \left(\mathbf {z} _ {t}, t\right), s\right) - \frac {\sqrt {\sigma_ {s} ^ {2} - \tilde {\sigma} _ {s \mid t} ^ {2}}}{\sigma_ {t}} \left(\mathbf {z} _ {t} - \alpha_ {t} F _ {\varphi} \left(\hat {\mathbf {x}} _ {\theta} \left(\mathbf {z} _ {t}, t\right), t\right)\right) \right\rVert_ {2} ^ {2} (16) \\ = \frac {1}{2 \tilde {\sigma} _ {s | t} ^ {2}} \Bigg \| \alpha_ {s} \Big (F _ {\varphi} (\mathbf {x}, s) - F _ {\varphi} (\hat {\mathbf {x}} _ {\theta} (\mathbf {z} _ {t}, t), s) \Big) + \\ \left. \frac {\sqrt {\sigma_ {s} ^ {2} - \tilde {\sigma} _ {s \mid t} ^ {2}}}{\sigma_ {t}} \alpha_ {t} \left(F _ {\varphi} \left(\hat {\mathbf {x}} _ {\theta} \left(\mathbf {z} _ {t}, t\right), t\right) - F _ {\varphi} (\mathbf {x}, t)\right) \right\| _ {2} ^ {2}. (17) \\ \end{array}
+$$
+
+With a learnable transformation  $F_{\varphi}$ , the term  $\mathcal{L}_{\mathrm{prior}}$  becomes dependent on the parameters  $\varphi$ , necessitating its optimization
+
+during training. We can compute the prior term as follows:
+
+$$
+\begin{array}{l} \left. \mathrm {D} _ {\mathrm {K L}} \left(q _ {\varphi} (\mathbf {z} _ {T} | \mathbf {x}) \| p (\mathbf {z} _ {T})\right) = \frac {1}{2} \left[ \log \frac {| I |}{| \sigma_ {T} ^ {2} I |} - d + T r \{I ^ {- 1} \sigma_ {t} ^ {2} I \} + \left\| 0 - \alpha_ {T} F _ {\varphi} (\mathbf {x}, T) \right\| _ {2} ^ {2} \right] \right. (18) \\ = \frac {1}{2} \left[ - d \log \sigma_ {T} ^ {2} - d + d \sigma_ {T} ^ {2} + \left\| \alpha_ {T} F _ {\varphi} (\mathbf {x}, T) \right\| _ {2} ^ {2} \right] (19) \\ = \frac {1}{2} \left[ d \left(\sigma_ {T} ^ {2} - \log \sigma_ {T} ^ {2} - 1\right) + \alpha_ {T} ^ {2} \| F _ {\varphi} (\mathbf {x}, T) \| _ {2} ^ {2} \right]. (20) \\ \end{array}
+$$
+
+Here,  $d$  represents the dimensionality of the data space.
+
+# A.3. Reverse SDE and ODE
+
+As discussed in Section 3.2, when the number of steps, denoted as  $T$ , tends to infinity for NDM, we can switch to continuous time.
+
+In the discrete time setting, we define the time step as  $t \in [0,1,\dots,T]$ . In the continuous time setting, we utilize the unit interval, denoting time as  $t \in [0,1]$ . Nevertheless, for the sake of notational simplicity in this and subsequent sections, we will consider the discrete time to also lie within the unit interval, with  $t \in [\frac{0}{T},\frac{1}{T},\dots,\frac{T}{T}]$ .
+
+To derive the stochastic differential equation (SDE) for the reverse process  $p_{\theta}(\mathbf{z}_s|\mathbf{z}_t)$  in NDM, we first obtain an SDE that depends on the data point  $\mathbf{x}$  and whose solution corresponds to the posterior distribution  $q_{\varphi}(\mathbf{z}_{t - \Delta t}|\mathbf{z}_t,\mathbf{x})$ . By defining  $p_{\theta}(\mathbf{z}_s|\mathbf{z}_t)$  through  $q_{\varphi}(\mathbf{z}_s|\mathbf{z}_t,\mathbf{x})$  with the prediction  $\hat{\mathbf{x}}_{\theta}(\mathbf{z}_t,t)$  instead of  $\mathbf{x}$ , we can subsequently replace the prediction  $\hat{\mathbf{x}}_{\theta}(\mathbf{z}_t,t)$  and derive the SDE for the reverse process.
+
+We constructively derive the SDE for the posteriors  $q_{\varphi}(\mathbf{z}_s|\mathbf{z}_t,\mathbf{x})$ . First, let us consider the following auxiliary SDE with backward time flow:
+
+$$
+d \varepsilon_ {t} = \frac {\dot {\nu} _ {t}}{2} \varepsilon_ {t} d t + \sqrt {\dot {\nu} _ {t}} d \mathbf {w}. \tag {21}
+$$
+
+It is straightforward to show that the solution to this SDE corresponds to the following distribution:
+
+$$
+q \left(\varepsilon_ {s} \mid \varepsilon_ {t}\right) = \mathcal {N} \left(\varepsilon_ {s}; \sqrt {1 - \bar {\sigma} _ {s \mid t} ^ {2}} \varepsilon_ {t}; \bar {\sigma} _ {s \mid t} ^ {2} I\right), \quad \text {w h e r e} \quad \bar {\sigma} _ {s \mid t} ^ {2} = 1 - e ^ {\nu_ {s} - \nu_ {t}}. \tag {22}
+$$
+
+To derive the SDE for the posteriors  $q_{\varphi}(\mathbf{z}_s|\mathbf{z}_t,\mathbf{x})$ , we can apply the following function to both the SDE (21) and the distribution (22):
+
+$$
+G (\mathbf {x}, \varepsilon_ {t}, t) = \alpha_ {t} F _ {\varphi} (\mathbf {x}, t) + \sigma_ {t} \varepsilon_ {t} \tag {23}
+$$
+
+Note that after applying the function  $G$ , the distribution (22) matches the posterior distribution  $q_{\varphi}(\mathbf{z}_s|\mathbf{z}_t,\mathbf{x})$ . Therefore, the desired SDE for  $q_{\varphi}(\mathbf{z}_s|\mathbf{z}_t,\mathbf{x})$  is obtained by transforming the SDE (21) using Ito's formula ( $\emptyset$ ksendal &  $\emptyset$ ksendal, 2003):
+
+$$
+d \mathbf {z} _ {t} = \left[ \frac {\partial G (\mathbf {x} , \varepsilon , t)}{\partial t} \right| _ {\varepsilon = \varepsilon_ {t}} + \frac {\dot {\nu} _ {t}}{2} \frac {\partial G (\mathbf {x} , \varepsilon_ {t} , t)}{\partial \varepsilon_ {t}} \varepsilon_ {t} - \frac {\dot {\nu} _ {t}}{2} \frac {\partial^ {2} G (\mathbf {x} , \varepsilon_ {t} , t)}{\partial \varepsilon_ {t} ^ {2}} \Bigg ] d t + \sqrt {\dot {\nu} _ {t}} \frac {\partial G (\mathbf {x} , \varepsilon_ {t} , t)}{\partial \varepsilon_ {t}} d \mathbf {w} \tag {24}
+$$
+
+$$
+\begin{array}{l} = \left[ \dot {\alpha} _ {t} F _ {\varphi} (\mathbf {x}, t) + \alpha_ {t} \dot {F} _ {\varphi} (\mathbf {x}, t) + \dot {\sigma} _ {t} \varepsilon_ {t} + \frac {\dot {\nu} _ {t}}{2} \sigma_ {t} \varepsilon_ {t} \right] d t + \sqrt {\dot {\nu} _ {t}} \sigma_ {t} d \mathbf {w} (25) \\ = \left[ \frac {\dot {\alpha} _ {t}}{\alpha_ {t}} \left(\mathbf {z} _ {t} - \sigma_ {t} \varepsilon_ {t}\right) + \alpha_ {t} \dot {F} _ {\varphi} (\mathbf {x}, t) + \dot {\sigma} _ {t} \varepsilon_ {t} + \frac {\dot {\nu} _ {t}}{2} \sigma_ {t} \varepsilon_ {t} \right] d t + \sqrt {\dot {\nu} _ {t}} \sigma_ {t} d \mathbf {w} (26) \\ = \left[ \alpha_ {t} \dot {F} _ {\varphi} (\mathbf {x}, t) + \frac {\partial \log \alpha_ {t}}{\partial t} \mathbf {z} _ {t} - \frac {1}{2} \left(\frac {\partial \sigma_ {t} ^ {2}}{\partial t} - 2 \frac {\partial \log \alpha_ {t}}{\partial t} \sigma_ {t} ^ {2} + \dot {\nu} _ {t} \sigma_ {t} ^ {2}\right) \left(- \frac {\varepsilon_ {t}}{\sigma_ {t}}\right) \right] d t + \sqrt {\dot {\nu} _ {t}} \sigma_ {t} d \mathbf {w} (27) \\ = \left[ \alpha_ {t} \dot {F} _ {\varphi} (\mathbf {x}, t) + r (t) \mathbf {z} _ {t} - \frac {1}{2} \left(\frac {\partial \sigma_ {t} ^ {2}}{\partial t} - 2 r (t) \sigma_ {t} ^ {2} + \dot {\nu} _ {t} \sigma_ {t} ^ {2}\right) s (\mathbf {x}, \mathbf {z} _ {t}, t) \right] d t + \sqrt {\dot {\nu} _ {t}} \sigma_ {t} d \mathbf {w}, (28) \\ \end{array}
+$$
+
+$$
+\text {w h e r e} \quad r (t) = \frac {\partial \log \alpha_ {t}}{\partial t} \quad \text {a n d} \quad s (\mathbf {x}, \mathbf {z} _ {t}, t) = \frac {\alpha_ {t} F _ {\varphi} (\mathbf {x} , t) - \mathbf {z} _ {t}}{\sigma_ {t} ^ {2}} \tag {29}
+$$
+
+To obtain the SDE for the reverse process, we can substitute the prediction  $\hat{\mathbf{x}}_{\theta}(\mathbf{z}_t,t)$  instead of  $\mathbf{x}$ . This substitution yields the SDE (11):
+
+$$
+d \mathbf {z} _ {t} = \left[ \alpha_ {t} \dot {F} _ {\varphi} \left(\hat {\mathbf {x}} _ {\theta} \left(\mathbf {z} _ {t}, t\right), t\right) + r (t) \mathbf {z} _ {t} - \frac {1}{2} \left(\frac {\partial \sigma_ {t} ^ {2}}{\partial t} - 2 r (t) \sigma_ {t} ^ {2} + \dot {\nu} _ {t} \sigma_ {t} ^ {2}\right) s _ {\theta} \left(\mathbf {z} _ {t}, t\right) \right] d t + \sqrt {\dot {\nu} _ {t}} \sigma_ {t} d \mathbf {w}, \tag {30}
+$$
+
+$$
+\text {w h e r e} \quad s _ {\theta} \left(\mathbf {z} _ {t}, t\right) = \frac {\alpha_ {t} F _ {\varphi} \left(\hat {\mathbf {x}} _ {\theta} \left(\mathbf {z} _ {t} , t\right) , t\right) - \mathbf {z} _ {t}}{\sigma_ {t} ^ {2}} \tag {31}
+$$
+
+As discussed earlier, we can leverage the Jacobian-Vector Product (JVP) trick (Smale & Hirsch, 1974) to calculate  $\dot{F}_{\varphi}$ .
+
+In the case where  $\nu_{t}$  is a constant, the dynamics become deterministic and can be described by ordinary differential equations (ODEs). In our experiments, we utilize these ODEs to model the generative process as a continuous normalizing flow (Chen et al., 2018; Grathwohl et al., 2019) and estimate densities.
+
+# A.4. Continuous time objective
+
+When we switch to continuous time, the discrete objective (9) transforms from finite sum of KL divergences into integral, which we can easily derive as soon as we have access to both stochastic differential equations associated with the forward process (28) and with the reverse process (30). In continuous time the diffusion term  $\mathcal{L}_{\mathrm{diff}}$  (10) is equal to:
+
+$$
+\begin{array}{l} \mathcal {L} _ {\mathrm {d i f f}} = \int_ {0} ^ {1} \frac {1}{g ^ {2} (t)} \left\| \alpha_ {t} \left(\dot {F} _ {\varphi} (\mathbf {x}, t) - \dot {F} _ {\varphi} (\hat {\mathbf {x}} _ {\theta} (\mathbf {z} _ {t}, t), t)\right) + \right. \\ \frac {1}{2} \left(\frac {\partial \sigma_ {t} ^ {2}}{\partial t} - 2 r (t) \sigma_ {t} ^ {2} + g ^ {2} (t)\right) \left(s (\mathbf {x}, \mathbf {z} _ {t}, t) - s (\hat {\mathbf {x}} _ {\theta} (\mathbf {z} _ {t}, t), \mathbf {z} _ {t}, t)\right) \Bigg \| _ {2} ^ {2} d t, \\ \end{array}
+$$
+
+$$
+\text {w h e r e} \quad r (t) = \frac {\partial \log \alpha_ {t}}{\partial t}, \quad g ^ {2} (t) = \dot {\nu} _ {t} \sigma_ {t} ^ {2} \quad \text {a n d} \quad s (\mathbf {x}, \mathbf {z} _ {t}, t) = \frac {\alpha_ {t} F _ {\varphi} (\mathbf {x} , t) - \mathbf {z} _ {t}}{\sigma_ {t} ^ {2}}. \tag {32}
+$$
+
+As we can see, these equations contain  $\dot{F}_{\varphi}$  as a component. In general, we do not have explicit access to the time derivative of the forward transformation  $F_{\varphi}$ . However, we will focus on cases where the forward transformation is differentiable. By utilizing automatic differentiation tools, we can calculate the time derivatives of  $F_{\varphi}$ . Nevertheless, when  $\mathbf{x}$  is fixed, the function  $F_{\varphi}(\mathbf{x}, \cdot)$  becomes a scalar-to-vector function. To compute its time derivative using simple backpropagation, we would need to execute it for all outputs of  $F_{\varphi}$ , resulting in quadratic computational complexity. Fortunately, there exists a more efficient method to obtain the time derivative, the Jacobian-Vector Product trick (Smale & Hirsch, 1974). The Jacobian of the transformation function with  $\mathbf{x}$  fixed is represented as a column matrix. Therefore, by computing the product of the Jacobian with a one-dimensional vector, we can obtain a vector of time derivatives.
+
+# B. Connections with other works
+
+We introduce NDMs as a comprehensive framework that generalises various existing approaches. Here we provide Table 5 which is an extended version of Table 1, that demonstrates how existing approaches appear as a spatial cases of NDMs.
+
+We also provide an extended discussion on the connection between NDM and other related works.
+
+# B.1. Diffusion in latent space
+
+The concept of a learnable forward process is not entirely new. In some sense models that run a diffusion process in the latent space of a VAE (Vahdat et al., 2021; Rombach et al., 2022), a hierarchical VAE (Gu et al., 2023), or a Flow model (Kim et al., 2022) can be viewed as diffusion models with a learnable forward process. These models optimize the mapping to the latent space. Consequently, projecting the diffusion generative dynamic from the latent to the data space introduces a novel, nonlinear, and learnable generative dynamic. However, these models still rely on conventional diffusion in the latent space.
+
+Additionally, these models can be viewed as spatial cases of NDM with a specific choice of the transformation  $F_{\varphi}(\mathbf{x},t)$
+
+Table 5. Summary of existing diffusion models as instances of Neural Diffusion Models (NDM).  
+
+<table><tr><td>Model</td><td>Distribution q(zt|x)</td><td>NDM&#x27;s F(x,t)</td><td>Comment</td></tr><tr><td>DDPM (Ho et al., 2020)/DDIM (Song et al., 2021a)</td><td>N(zt; αt x, σt2I)</td><td>x</td><td></td></tr><tr><td>Flow Matching OT(Lipman et al., 2023)</td><td>N(zt; αt x, σt2I)</td><td>x</td><td>αt=t,σt=1-(1-σmin)t</td></tr><tr><td>VDM (Kingma et al., 2021)</td><td>N(zt; αt x, σt2I)</td><td>x</td><td>α2t=sigmoid(-γη(t)),σ2t=sigmoid(γη(t))</td></tr><tr><td>IHDM (Rissanen et al., 2023)</td><td>N(zt; Ve-At VTx, σ2I)</td><td>Ve-At VTx</td><td>αt=1,σt=σ,σis fixed</td></tr><tr><td>Blurring Diffusion(Hoogeboom &amp; Salimans, 2023)</td><td>N(zt; αt e-At VTx, σt2I)</td><td>e-At VTx</td><td>p(x|z0)=N(x;aVz0,σ)</td></tr><tr><td>Soft Diffusion(Daras et al., 2022)</td><td>N(zt; Ct x, st2I)</td><td>Ct x</td><td>αt=1,σt2=st2</td></tr><tr><td>LSGM (Vahdat et al., 2021)</td><td>N(zt; αt E(x), σt2I)</td><td>E(x)</td><td>p(x|z0)=N(x;aD(z0),σ2)</td></tr><tr><td rowspan="2">f-DM (Gu et al., 2023)</td><td rowspan="2">N(zt; αt x t, σt2I)</td><td rowspan="2">xt=(t-τk)̂x^k+(τk+1-t)x^k/τk+1-τk, where τk≤t&lt;τk+1</td><td>x^k=f0:k(x)</td></tr><tr><td>x^k=\left\{g_k(f_{k+1}(x^k)), \begin{array}{l} if k&lt;K, \\ x^k, \end{array}\right.</td></tr></table>
+
+For example,  $F_{\varphi}$  might be selected as the VAE's time independent encoder in the case of (Vahdat et al., 2021) or the time independent Flow model in the case of (Kim et al., 2022).
+
+# B.2. Schrödinger Bridges
+
+Another line of works (De Bortoli et al., 2021; Wang et al., 2021; Peluchetti; Chen et al., 2021) are approaches based on Schrödinger Bridge theory. While such approaches allow learning forward transformations, in contrast to NDM, these approaches are not simulation-free. In Schrödinger Bridge models, we typically lack direct access to the distribution  $q(\mathbf{z}_t|\mathbf{x})$ . Consequently, to sample the latent variable  $\mathbf{z}_t$  in training time, we must simulate the full stochastic process, such as the stochastic differential equations. This characteristic makes Schrödinger Bridge models expensive in training and not simulation-free.
+
+In contrast, NDM framework, by design, has access to  $q(\mathbf{z}_t|\mathbf{x})$ . Thus, with NDM, when training a model with  $T$  time steps, there is no need to propagate  $F_{\varphi}$  for  $T$  times at each step of the training procedure. Instead, the NDM framework enables sampling of the intermediate latent variables  $\mathbf{z}_t$  directly from the distribution  $q(\mathbf{z}_t|\mathbf{x})$ . Therefore, we can maintain the training paradigm outlined in Section 2. Instead of computing all  $T$  KL divergences for each time step, we can approximate the objective using the Monte Carlo method by calculating just one KL divergence for a uniformly sampled time step  $t \in [1;T]$ , as described in Algorithm 1.
+
+This approach allows us to train the model with batches of shape  $[batch\_size, d]$  rather than  $[batch\_size, T, d]$ . Consequently, NDM can leverage larger batch sizes and use just one call of  $F_{\varphi}$  for inferring latent variables  $\mathbf{z}_t$ .
+
+# B.3. Stochastic Interpolants
+
+Albergo & Vanden-Eijnden (2023) proposed a Stochastic Interpolant approach, which provide more flexibility than conventional diffusion models in defining and even learning of the forward process. While we find stochastic interpolants intriguing and promising, as well as related to our work, these methods differ significantly.
+
+Firstly, stochastic interpolants represent an approach to learning continuous-time deterministic generative dynamics, whereas NDM learns stochastic dynamics in either discrete or continuous time, which can subsequently may be transformed into a deterministic process.
+
+Secondly, in NDM, the model is trained by optimizing the variational bound on the likelihood, while Stochastic Interpolants are trained by optimizing the generalization of the Flow Matching objective (Lipman et al., 2023).
+
+Lastly, NDM joint learns both the forward and reverse processes by optimizing the likelihood, whereas stochastic interpolants
+
+learn the generative process with a fixed forward process. Albergo & Vanden-Eijnden (2023) demonstrate the possibility of constructing an optimization procedure for the forward process through a max-min game to solve a dynamic optimal transport problem. However, the purpose of this optimization differs from that of NDM.
+
+Moreover, max-min optimization, as employed in Stochastic Interpolants, is notably less stable compared to min-min optimization in NDM. Additionally, Stochastic Interpolants do not present experimental results for the optimization of the forward process.
+
+# B.4. DiffEnc
+
+In concurrent work, Nielsen et al. (2024) introduced DiffEnc. DiffEnc also proposes to add a time-dependent transformation to the data in the diffusion model. However, there are some distinctions between these two methods. Firstly, in NDM, we parameterize the reverse process by predicting the data point  $\mathbf{x}$ , while in DiffEnc, they predict the transformed data point  $F_{\varphi}(\mathbf{x},t)$ .
+
+Secondly, in NDM, we employ a Signal-to-Noise Ratio (SNR) schedule for noise injection from DDPM (Ho et al., 2020) and a straightforward parameterization of the model  $\hat{\mathbf{x}}_{\theta}(\mathbf{z}_t,t)$  through predicting the injected epsilon, as detailed in Appendix C.2. Simultaneously, in DiffEnc, the authors use a learnable SNR schedule (Kingma et al., 2021) and a v-parameterization (Salimans & Ho, 2022) of  $\hat{\mathbf{x}}_{\theta}(\mathbf{z}_t,t)$ .
+
+Finally, DiffEnc utilizes approximations of the time derivatives of data transformations  $F_{\varphi}$ , while in the NDM framework, we propose calculating exact time derivatives using Jacobian-Vector Products.
+
+# C. Implementation details
+
+All our experiments were conducted using synthetic 2D datasets and image datasets: MNIST (Deng, 2012), CIFAR-10 (Krizhevsky et al., 2009), downsampled ImageNet (Deng et al., 2009; Van Den Oord et al., 2016) and CelebA-HQ-256 (Karras et al., 2017). For CIFAR-10 and ImageNet datasets we applied center cropping and resizing. For synthetic data, we employed a 5-layer MLP with 512 neurons in each layer, while for the images, we utilized the U-Net architecture from Dhariwal & Nichol (2021). In our experiments both the DDPM and NDM approaches were trained on identical architectures, with the same hyper-parameters and for the same number of epochs. The hyper-parameters are presented in Table 6. In experiment where we report results for the continuous time models we use importance sampling of time (Song et al., 2021b) instead of uniform sampling.
+
+We trained models using the Adam optimizer, setting the following parameters:  $\beta_{1} = 0.9$ ,  $\beta_{2} = 0.999$ , weight decay of 0.0, and  $\varepsilon = 10^{-8}$ . To facilitate the training process, we employed a polynomial decay learning rate schedule, which includes a warm-up phase for a specified number of training steps. During the warm-up phase, the learning rate is linearly increased from  $10^{-8}$  to the peak learning rate. Once the peak learning rate is reached, the learning rate is linearly decayed to  $10^{-8}$  until the final training step. The training was performed using Tesla V100 GPUs.
+
+# C.1. Dequantization
+
+When reporting negative log-likelihood, we dequantize using the standard uniform dequantization. We report an importance-weighted estimate using
+
+$$
+\log \frac {1}{K} \sum_ {k = 1} ^ {K} p _ {\theta} (\mathbf {x} + u _ {k}), \quad \text {w h e r e} \quad u _ {k} \sim \mathcal {U} (0, 1), \tag {33}
+$$
+
+with  $\mathbf{x}\in [0,\dots ,255]$
+
+# C.2. Parameterization
+
+In order to simplify the derivations above, we have utilized the notation  $\hat{\mathbf{x}}_{\theta}(\mathbf{z}_t,t)$  to represent the prediction of the reverse process. However, prior research has shown that predicting the injected noise  $\varepsilon_{t}$  can lead to improved results (Ho et al., 2020; Nichol & Dhariwal, 2021; Dhariwal & Nichol, 2021). Therefore, in all the experiments, we opt for the following
+
+Table 6. Training hyper-parameters.  
+
+<table><tr><td></td><td>CIFAR-10</td><td>ImageNet 32</td><td>ImageNet 64</td></tr><tr><td>Channels</td><td>256</td><td>256</td><td>192</td></tr><tr><td>Depth</td><td>2</td><td>3</td><td>3</td></tr><tr><td>Channels multipliers</td><td>1,2,2,2</td><td>1,2,2,2</td><td>1,2,3,4</td></tr><tr><td>Heads</td><td>4</td><td>4</td><td>4</td></tr><tr><td>Heads Channels</td><td>64</td><td>64</td><td>64</td></tr><tr><td>Attention resolution</td><td>16</td><td>16,8</td><td>32,16,8</td></tr><tr><td>Dropout</td><td>0.0</td><td>0.0</td><td>0.0</td></tr><tr><td>Effective Batch size</td><td>256</td><td>1024</td><td>2048</td></tr><tr><td>GPUs</td><td>2</td><td>4</td><td>16</td></tr><tr><td>Epochs</td><td>1000</td><td>200</td><td>250</td></tr><tr><td>Iterations</td><td>391k</td><td>250k</td><td>157k</td></tr><tr><td>Learning Rate</td><td>4e-4</td><td>1e-4</td><td>1e-4</td></tr><tr><td>Learning Rate Scheduler</td><td>Polynomial</td><td>Polynomial</td><td>Constant</td></tr><tr><td>Warmup Steps</td><td>45k</td><td>20k</td><td>-</td></tr></table>
+
+parameterization:
+
+$$
+\hat {\mathbf {x}} _ {\theta} \left(\mathbf {z} _ {t}, t\right) = \frac {\mathbf {z} _ {t} - \sigma_ {t} \hat {\varepsilon} _ {\theta} \left(\mathbf {z} _ {t} , t\right)}{\alpha_ {t}}. \tag {34}
+$$
+
+It is worth noting that with this parameterization,  $\hat{\varepsilon}_{\theta}(\mathbf{z}_t,t)$  does not necessarily approximate the true injected noise  $\varepsilon_{t}$ , since this reparameterization does not account for the transformation  $F_{\varphi}$ . We believe that better parameterizations may exist for NDM, but we leave this for future research.
+
+Furthermore, we restrict the transformation  $F_{\varphi}$  to an identity transformation for  $t = 0$  through the following construction:
+
+$$
+F _ {\varphi} (\mathbf {x}, t) = (1 - t) \mathbf {x} + t \bar {F} _ {\varphi} (\mathbf {x}, t). \tag {35}
+$$
+
+This ensures that  $q(\mathbf{z}_0|\mathbf{x}) \approx \delta (\mathbf{z}_0 - \mathbf{x})$ , and thus also removes the need to optimize the reconstruction term  $\mathcal{L}_{\mathrm{rec}}$ .
+
+Finally, to ensure consistency with Ho et al. (2020) we use  $\tilde{\sigma}_{s|t}^2 = \left(\sigma_t^2 -\frac{\alpha_t^2}{\alpha_s^2}\sigma_s^2\right)\frac{\sigma_s^2}{\sigma_t^2}$  for the forward process (7). This choice of  $\tilde{\sigma}_{s|t}^2$  guarantees consistency between the NDM and DDPM forward processes. For  $\alpha_{t}$  and  $\sigma_t^2$  we use the DDPM schedule of noise injection.
+
+# C.3. Diffusion in latent space
+
+For experiment with diffusion in the latent space of VAE on CelebA-HQ-256, we followed LSGM (Vahdat et al., 2021) experiment setup. The only difference between LSGM baseline and our model is that we utilize learnable transformations  $F_{\varphi}$  according to NDMs framework. We apply the same hyperparameters, as LSGM.
+
+# D. Additional results
+
+# D.1. Additional evaluation
+
+Here we provide Table 7 which contains additional results to Table 3. This table compare DDPM and NDMs with learnable transformations on CIFAR-10 and ImageNet  $32 \times 32$  datasets with different numbers of steps.
+
+# D.2. Additional samples
+
+In this section, we present additional illustrations showcasing the properties of NDMs.
+
+Figure 4 provides a comparison between DDPM and NDM on a synthetic 2D data distribution. For this experiment, both models utilize  $T = 10$  discrete time steps. From Figure 4c, it is evident that NDM learns to transform the data distribution.
+
+Table 7. Performance comparison the DDPM and NDM on CIFAR-10 and ImageNet 32 datasets with different numbers of steps. We report the performance with same hyperparameters and neural networks on both models to quantify the effect of learnable transformation in fair setting. We provide likelihood (bits/dim) and negative ELBO. Additionally for CIFAR-10 and ImageNet 32 we provide FID score. Boldface numbers represent the best performance. NDM consistently outperforms in terms of NLL and NELBO with comparable sample quality to DDPM on all datasets.  
+
+<table><tr><td rowspan="2">Steps</td><td rowspan="2">Model</td><td colspan="3">CIFAR-10</td><td colspan="3">ImageNet 32</td></tr><tr><td>NLL ↓</td><td>NELBO ↓</td><td>FID ↓</td><td>NLL ↓</td><td>NELBO ↓</td><td>FID ↓</td></tr><tr><td rowspan="2">1000</td><td>DDPM</td><td>3.11</td><td>3.18</td><td>11.44</td><td>3.89</td><td>3.95</td><td>16.18</td></tr><tr><td>NDM</td><td>3.02</td><td>3.03</td><td>11.82</td><td>3.79</td><td>3.82</td><td>17.02</td></tr><tr><td rowspan="2">100</td><td>DDPM</td><td>3.31</td><td>3.38</td><td>11.78</td><td>4.14</td><td>4.23</td><td>16.66</td></tr><tr><td>NDM</td><td>3.05</td><td>3.12</td><td>11.98</td><td>3.83</td><td>3.92</td><td>17.74</td></tr><tr><td rowspan="2">50</td><td>DDPM</td><td>3.49</td><td>3.57</td><td>13.22</td><td>4.37</td><td>4.47</td><td>18.70</td></tr><tr><td>NDM</td><td>3.22</td><td>3.30</td><td>13.15</td><td>4.05</td><td>4.14</td><td>18.93</td></tr><tr><td rowspan="2">10</td><td>DDPM</td><td>5.02</td><td>5.13</td><td>37.83</td><td>6.28</td><td>6.42</td><td>53.51</td></tr><tr><td>NDM</td><td>4.63</td><td>4.74</td><td>31.56</td><td>5.81</td><td>5.94</td><td>45.38</td></tr><tr><td rowspan="2">1000 → 100</td><td>DDPM</td><td>3.38</td><td>3.45</td><td>12.29</td><td>4.23</td><td>4.32</td><td>17.49</td></tr><tr><td>NDM</td><td>3.30</td><td>3.37</td><td>12.70</td><td>4.15</td><td>4.23</td><td>18.48</td></tr><tr><td rowspan="2">1000 → 50</td><td>DDPM</td><td>4.08</td><td>4.17</td><td>15.24</td><td>5.10</td><td>5.21</td><td>20.09</td></tr><tr><td>NDM</td><td>3.98</td><td>4.07</td><td>16.83</td><td>5.00</td><td>5.10</td><td>21.11</td></tr><tr><td rowspan="2">1000 → 10</td><td>DDPM</td><td>8.78</td><td>8.98</td><td>43.85</td><td>10.99</td><td>11.23</td><td>58.35</td></tr><tr><td>NDM</td><td>8.58</td><td>8.81</td><td>48.41</td><td>10.78</td><td>11.06</td><td>62.12</td></tr></table>
+
+Additionally, after injecting noise (Figures 4a and 4d), the distributions of samples  $\mathbf{z}_t$  show minimal differences between DDPM and NDM. However, when examining the predictions of data points  $\hat{\mathbf{x}}_{\theta}(\mathbf{z}_t,t)$  (Figures 4b and 4e), NDM produces predictions that more closely resemble the true data distribution compared to DDPM.
+
+A similar pattern emerges when applying these models to the MNIST dataset, as depicted in Figure 5. For this experiment we also use  $T = 10$  discrete time steps. DDPM generates blurry predictions  $\hat{\mathbf{x}}_{\theta}(\mathbf{z}_t,t)$  for  $t$  close to  $T$ , which bear little resemblance to real MNIST samples. Conversely, NDM produces predictions that are more similar to the true MNIST distribution, despite both models generating similar-looking noisy samples.
+
+Finally, we include samples from both DDPM and NDM models with  $T = 1000$  steps on the CIFAR-10 dataset in Figure 6. As outlined in Table 1, NDM exhibits lower sample quality based on FID measurements; however, visually there is no drop in quality.
+
+# D.3. Ablation studies
+
+Finally, we address the question of whether the improved performance of NDM is due to the proposed method or merely the result of increasing the number of model parameters. To investigate this issue, we provide additional experiments where we double the number of DDPM parameters in two ways. The first way is to simply stack two U-Net architectures, which is the closest form to NDM. The second way is to increase the width of the U-Net architecture. Specifically, for the second way we use 384 channels instead of 256. Importantly, we left all other hyper-parameters (see Table 6), such as the learning rate and number of iterations, unchanged. As shown in Table 8, neither of these approaches yields the same results as NDM with learnable transformations. This means that the improved performance is not simply a result of the increased number of parameters.
+
+# E. Dynamic optimal transport
+
+In this section, we present a proof-of-concept experiment demonstrating that the NDMs framework enables the learning of simpler generative trajectories. Specifically, we conduct experiments involving a 1D mixture of Gaussian distribution and dynamic optimal transport (OT).
+
+Table 8. Comparison of NDM and DDPM with doubled number of parameters on CIFAR-10 for 10 and 1000 steps. The performance of DDPM stays the same while doubling the number of parameters, and NDM still achieves the best NLL and NELBO despite comparable number of parameters.  
+
+<table><tr><td rowspan="2">Model</td><td colspan="3">10 steps</td><td colspan="3">1000 steps</td></tr><tr><td>NLL ↓</td><td>NELBO ↓</td><td>FID ↓</td><td>NLL ↓</td><td>NELBO ↓</td><td>FID ↓</td></tr><tr><td>DDPM</td><td>5.02</td><td>5.13</td><td>37.83</td><td>3.11</td><td>3.18</td><td>11.44</td></tr><tr><td>DDPM (stack)</td><td>5.02</td><td>5.13</td><td>38.05</td><td>3.10</td><td>3.18</td><td>11.42</td></tr><tr><td>DDPM (wide)</td><td>5.01</td><td>5.11</td><td>37.88</td><td>3.11</td><td>3.17</td><td>11.39</td></tr><tr><td>NDM</td><td>4.63</td><td>4.74</td><td>31.56</td><td>3.02</td><td>3.03</td><td>11.82</td></tr></table>
+
+While NDMs don't inherently have a direct connection with OT, we can establish a connection given the presence of infinitely many pairs of matched forward and reverse processes. This connection is facilitated by the NDMs' ability to learn the forward process. Therefore, we can consider the following setup.
+
+We consider NDMs with a learnable function  $F_{\varphi}$ . Then, we constrain the reverse process to exclusively learn dynamic OT mappings. Finally, we train both the forward and reverse processes jointly, following the NDMs framework. In such a setup we can expect the forward process to learn such a transition from data distribution to Gaussian distribution, that aligns with the limitations imposed on the reverse process.
+
+# E.1. Restricted reverse process
+
+To restrict the reverse process we parameterise the reverse deterministic process to have linear trajectories:
+
+$$
+\mathbf {z} _ {t} = h _ {\theta} (t, \varepsilon) = (1 - t) \hat {\mathbf {x}} _ {\theta} (\varepsilon) + t \varepsilon , \tag {36}
+$$
+
+where  $\varepsilon$  is a sample drawn from a unit Gaussian distribution. Since we are working with smooth 1D distributions, it is enough for  $\hat{\mathbf{x}}_{\theta}$  to be monotonically increasing, so the trajectories  $\mathbf{z}_t$  correspond to dynamic OT. Which means that for any parameters  $\theta$  the reverse process describes dynamic OT between the standard Gaussian distribution and another distribution (not necessarily exactly the target data distribution). In practice, we parameterize  $\hat{\mathbf{x}}_{\theta}$  using the neural network proposed by Kingma et al. (2021) for the parameterization of the Signal-to-Noise Ratio (SNR) function.
+
+Then, we can derive an ordinary differential equation (ODE) for the reverse process:
+
+$$
+d \mathbf {z} _ {t} = \underbrace {\varepsilon - \hat {\mathbf {x}} _ {\theta} (\varepsilon) \Big | _ {\varepsilon = h _ {\theta} ^ {- 1} (t , \mathbf {z} _ {t})}} _ {f _ {\theta} (t, \mathbf {z} _ {t})} d t. \tag {37}
+$$
+
+Next, we may switch to a stochastic differential equation (SDE) according to Song et al. (2021c):
+
+$$
+d \mathbf {z} _ {t} = \underbrace {\left[ f _ {\theta} (t , \mathbf {z} _ {t}) - \frac {g ^ {2} (t)}{2} \nabla_ {\mathbf {z} _ {t}} \log p _ {\theta} (\mathbf {z} _ {t}) \right]} _ {f _ {\theta} ^ {r} (t, \mathbf {z} _ {t})} d t + g (t) d \bar {\mathbf {w}}. \tag {38}
+$$
+
+As soon as we have access to  $h_{\theta}^{-1}$ , we may find:
+
+$$
+\begin{array}{l} \nabla_ {\mathbf {z} _ {t}} \log p _ {\theta} (\mathbf {z} _ {t}) = \nabla_ {\mathbf {z} _ {t}} \left[ \log p (\varepsilon) - \log \left| \frac {\partial \mathbf {z} _ {t}}{\partial \varepsilon} \right| \right] \Big | _ {\varepsilon = h _ {\theta} ^ {- 1} (t, \mathbf {z} _ {t})} (39) \\ = \nabla_ {\mathbf {z} _ {t}} \left[ \log p (\varepsilon) - \log \left| (1 - t) \frac {\partial \mathbf {x} _ {t}}{\partial \varepsilon} + t \right| \right] \Big | _ {\varepsilon = h _ {\theta} ^ {- 1} (t, \mathbf {z} _ {t})}. (40) \\ \end{array}
+$$
+
+# E.2. Objective function
+
+To train a model with such a specific reverse process, we can utilize a slightly modified NDMs framework. The only component of the NDMs' objective that is unclear is the diffusion term  $\mathcal{L}_{\mathrm{diff}}$ . NDMs provide a conditional reverse SDE
+
+![](images/c146d12f6e6eb31a3cca1990c879ab85fc232fa49ff9879ddd3f1988d14f97fa.jpg)  
+(a) DDPM with regular reverse process.
+
+![](images/751c1e794ed06afc86f1d4e7e00d9cb7034e07b32b573e0c6c1c67f66cdd710d.jpg)  
+(b) NDM with restricted (OT) reverse process.  
+Figure 3. Comparison of DDPM and NDM with restricted reverse process to be optimal transport, 1D distribution.
+
+associated with the forward process (28) in the following form:
+
+$$
+d \mathbf {z} _ {t} = f _ {\varphi} ^ {f} (x, t, \mathbf {z} _ {t}) d t + g (t) d \bar {\mathbf {w}}. \tag {41}
+$$
+
+Also, here we have the reverse SDE (38). Therefore, we may find diffusion term  $\mathcal{L}_{\mathrm{diff}}$  of objective as follows:
+
+$$
+\mathcal {L} _ {\text {d i f f}} = \mathbb {E} _ {q (\mathbf {x})} \mathbb {E} _ {u (t)} \mathbb {E} _ {q \left(\mathbf {z} _ {t} \mid \mathbf {x}\right)} \frac {1}{g ^ {2} (t)} \left\| f _ {\varphi} ^ {f} (x, t, \mathbf {z} _ {t}) - f _ {\theta} ^ {r} (t, \mathbf {z} _ {t}) \right\| _ {2} ^ {2}. \tag {42}
+$$
+
+# E.3. Results and discussion
+
+Figures 3a and 3b illustrate trajectories learned by DDPM and NDM with learnable  $F_{\varphi}$  and restricted reverse process. As expected, DDPM learns curved trajectories predetermined by fixed forward process. At the same time NDM effectively learns dynamic OT. It worths noting that DDPM with the restricted reverse process is by design not able to learn the data distribution, since it's impossible to match the fixed forward process (with curved trajectories) with the reverse process (with straight trajectories).
+
+The proposed approach is limited to 1D data, monotonically increasing  $\hat{\mathbf{x}}_{\theta}$ , and a nontrivial  $h_\theta^{-1}$  function, which we resolve using 5 iterations of Newton's method. Nevertheless, this experiment clearly demonstrates that NDMs may be utilised for learning OT as well as other (e.g. computationally efficient ones) dynamics by restricting the reverse process. Establishing rigorous theoretical connections with OT, developing specific techniques for efficient parameterisation of the reverse process and generalising to higher dimensions are interesting avenue for future work.
+
+![](images/00e85b808ef3f26afed9c222cd0532a9b9854cb23e026812f626b56b1fa51e2e.jpg)
+
+![](images/c2177e12f37a25d0d9448c42b56ab2d563dbad0614c9a9723678a64e7bb73a77.jpg)  
+(a) DDPM, samples  $\mathbf{z}_t$  from forward process.
+
+![](images/c62b22ffd22c0d136907159e21c36557ca8eae7f7b98ce55ec034ddd7d57f841.jpg)
+
+![](images/18d5b50d1d53e35a3537a6a5bc6b2fcadada123c2ea0bc58887af27e57dd5295.jpg)
+
+![](images/c9668a8b160c592bf074eb09aa922651767e50af70bdf895e9cb1fef20db2d59.jpg)
+
+![](images/c0c68b59dacc7601d9606e466ee4bfeb74174d97499a7726ad54876d9555ac3a.jpg)
+
+![](images/bfee79170b1e11a74f7a80bdf9c5130fb78f1fdcdecd6f53fdd32e9e496d3ad3.jpg)
+
+![](images/a08a908715452e63b004dec232cb4bacc532c6398332242f4b1ccac67e21db9d.jpg)  
+(b) DDPM, predictions  $\hat{\mathbf{x}}_{\theta}(\mathbf{z}_t,t)$  for different time steps.
+
+![](images/5ce750a6a81e9adfb30cf15d976b3bec7969d4cb26b6aeac853c1eb9a6a70b4c.jpg)
+
+![](images/f35f18341364dd8ed8d49534b8bd43c273d06e04942ae52a9b42aa4ffc6e013a.jpg)
+
+![](images/1c7e9b1a1b434ef41ae317f4e306a7aa815ec07d103640568137d048c0fb1d50.jpg)
+
+![](images/1ec54041c0daf5741418efeb9937da050a8531fbf9fa4c9b813293184ff797a6.jpg)
+
+![](images/01fed67831d2ebd246669df0041f400e5ee838d239ca2cb04a69e0cf6ee3ae03.jpg)
+
+![](images/a345507578ba24dca4ceba16e800f4816fb5b01115087ec46266139b64e5cd89.jpg)
+
+![](images/2e88599dc6d7f2f1bee9e266b7d811fb943debf8b31568cd2dfab5b00d7aae38.jpg)
+
+![](images/9a100151f3f1b7129e01d79933ddc5004ff727a3b276bfd2ceac313bf32c4972.jpg)
+
+![](images/ea8de29e668ce1d5e13ba6d1ccbfba9d52a3e63a5f7275f0b8797e431b4c2bfe.jpg)
+
+![](images/38b56ee85d9e27c7e0081782e9b96b4d64f7c29985eff2123843168c996956d7.jpg)
+
+![](images/d6e8db23d4feff33260dfd6e9816c803ae6d3c990a6bb725f2e0d7ff41c5c887.jpg)
+
+![](images/8da885998c7e1579e38c6cb361e295953cd2e9d5ceeaf133a10547051fa0c17a.jpg)  
+(c) NDM, forward transformations  $F_{\varphi}(\mathbf{x},t)$
+
+![](images/529c08288ee1252bf574c5eed2ca1a77cbd42ddee525a854d67181027ca2c80a.jpg)
+
+![](images/49b17f0396cce24ae5189cd1c23aefbe2d8ff0bac7bc096025769b735180fc4f.jpg)
+
+![](images/99be39388e3d25b9a76c3cb13b20ccbb71d62c182f3d250d4835d4b0786c8014.jpg)
+
+![](images/1689fa16518fe2859bc3c64c1676568d575cb9c89da57e27f3db7c4cf3a564fb.jpg)
+
+![](images/0139e82ee8d0a582ea11e2fd8611340a6f9ad2cf385afe891c48c888f9a64fab.jpg)  
+(e) NDM, predictions  $\hat{\mathbf{x}}_{\theta}(\mathbf{z}_t,t)$  for different time steps.
+
+![](images/3fb544b6bce2c1b9a1a591b2f1bfb16eb173c690eedb710d01135d16645414a2.jpg)  
+(d) NDM, samples  $\mathbf{z}_t$  from forward process.
+
+![](images/0d1add10c35b75b4f055e646ccda2669d8834a53f5f9fb2601b1a9935eca92c6.jpg)  
+Figure 4. Comparison of DDPM and NDM on 2D distribution.
+
+![](images/599e44cb1fce048aa06403a8a2aba0f20d42699e5052af23b527a42c66d0837f.jpg)
+
+![](images/1bf54b9be439716a9e22a1cb3bb4bf291c850af6eaf4052f53b157ca0592eaf8.jpg)
+
+![](images/587dc6ae68b2b12ca6e70aa949f3fa1848ab0fd91e1bc6ce4f50d63fd2d3830c.jpg)
+
+![](images/5caef49653bb9e40ef82c1529badb7b2a9907bfaf9339eb8e81495c1c6540422.jpg)
+
+![](images/23d11600d92261c0b90fc403016b9d39a2847d99a3ba883296cb5fcde5110279.jpg)
+
+![](images/f33d3b2dd48651115624af1e3bc495876ec5eac0e93c72814432c69b188885cc.jpg)
+
+![](images/c3ec8997e1607263d878d7441ca0304372e4e235af3689d7625c68d9672ed58d.jpg)
+
+![](images/b60c29d9f160b41646cbbe85cc2042eb483025801fb91f43021cfc119dbbbd21.jpg)
+
+![](images/81272052c4075b7427648c38a7012021ea6a1488e36f47438c3de152993ea706.jpg)
+
+![](images/a8122dac66e4238f3f5a78751b5628601a4a98c60de547b8863a0bc429bbb24d.jpg)
+
+![](images/9323a16362aefc9cb406b40cfbac44e4826ed14b7381f020baf38eae294c3d84.jpg)
+
+![](images/747899fd7f7bdbb19b2ca4428000f952c77f065e4026da64891f795ca1cf1b59.jpg)
+
+![](images/eba6d0ebc3b6671bf1b4fb924a0da6a7341311cff3f0933b3e331b83d62616d4.jpg)  
+(a) DDPM
+
+![](images/d081bd3917997680c48e2d967bd66d9a19152f2925b78a1843605da90b41d670.jpg)
+
+![](images/b232512e31a0bf52d8bf80e36ae37e030e46362ab0a4452a05bc02d1d2527c56.jpg)
+
+![](images/c387a06dca70cbec666cc8e0c389de90fd2e7a85338484e05f1407c8c0fbdafc.jpg)
+
+![](images/5f215846e10e9f51e22f4ad4aa8bcd8b836307fb4d2f7c45cd5005a9c54de42d.jpg)
+
+![](images/5d757aedc6fe64f7d79e637fa38bd2fc0a3b0d9ffd9b03b2a1c3a408d66b8278.jpg)
+
+![](images/ddef1d5a4c0ab57fea05e36dc92b470792d0d807c82a428ab49fae332e962174.jpg)
+
+![](images/de7006bf04054f4dfa4ff08f63924c29559034238617a8c8e914e6010735f33a.jpg)
+
+![](images/326c1464d43e58dff856a4ced0b3b1d1420437f17722c1075706e78f4f14fdc1.jpg)
+
+![](images/a99914f34b25051e442cf2eef28bbef27a12befcea30c0585d5323aad2620c63.jpg)
+
+![](images/c5399491d039787d8efb5e50767bf81b7fe2aefc317834a82fe63563f97334df.jpg)  
+(b) NDM  
+Figure 5. Samples  $\mathbf{z}_t$  from forward process and predicted data points  $\hat{\mathbf{x}}_{\theta}(\mathbf{z}_t,t)$  on MNIST. (a) Samples from DDPM. (b) Samples from NDM. In each group, Left: data sample, Top: noised samples  $\mathbf{z}_t$ , Bottom: predicted data points  $\hat{\mathbf{x}}_{\theta}(\mathbf{z}_t,t)$ .
+
+![](images/d778e086fbed4e7e6abedfd7fca802bfad365f345f235be36e4e7d4145c2ae67.jpg)
+
+![](images/dd6e26d1a7703978f31b60326abb8c70d5a138d5cc88df5aaf1ccc626ae707db.jpg)  
+(a) DDPM, FID = 11.44  
+(b) NDM, FID = 11.82  
+Figure 6. Samples on CIFAR-10. (a) Samples from DDPM. (b) Samples from NDM. Samples of both models are generated with the same random seed.

2024/Neural Diffusion Models/images.zip

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2024/Neural Image Compression with Text-guided Encoding for both Pixel-level and Perceptual Fidelity/00d62faa-ad68-4c8a-920d-e9cfc279879d_origin.pdf

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2024/Neural Image Compression with Text-guided Encoding for both Pixel-level and Perceptual Fidelity/full.md

@@ -0,0 +1,534 @@
+# Neural Image Compression with Text-guided Encoding for both Pixel-level and Perceptual Fidelity
+
+Hagyeong Lee  $^{*1}$  Minkyu Kim  $^{*1}$  Jun-Hyuk Kim  $^{2}$  Seungeon Kim  $^{2}$  Dokwan Oh  $^{2}$  Jaeho Lee
+
+# Abstract
+
+Recent advances in text-guided image compression have shown great potential to enhance the perceptual quality of reconstructed images. These methods, however, tend to have significantly degraded pixel-wise fidelity, limiting their practicality. To fill this gap, we develop a new text-guided image compression algorithm that achieves both high perceptual and pixel-wise fidelity. In particular, we propose a compression framework that leverages text information mainly by text-adaptive encoding and training with joint image-text loss. By doing so, we avoid decoding based on text-guided generative models—known for high generative diversity—and effectively utilize the semantic information of text at a global level. Experimental results on various datasets show that our method can achieve high pixel-level and perceptual quality, with either human- or machine-generated captions. In particular, our method outperforms all baselines in terms of LPIPS, with some room for even more improvements when we use more carefully generated captions.
+
+Project Page:
+
+taco-nic.github.io
+
+Code:
+
+github.com/effl-lab/TACO
+
+# 1. Introduction
+
+How can text improve machine vision? This question has been asked repeatedly in various domains of visual computing, giving birth to a number of vision-language models with tremendous multi-modal reasoning and generative capabilities (Radford et al., 2021; Liu et al., 2023a).
+
+The value of text has gained much attention in image compression as well, motivated by the astonishing ability of
+
+*Equal contribution ¹POSTECH ²Samsung Advanced Institute of Technology. Correspondence to: Jaeho Lee <jaeho.lee@postech.ac.kr>.
+
+Proceedings of the  $41^{st}$  International Conference on Machine Learning, Vienna, Austria. PMLR 235, 2024. Copyright 2024 by the author(s).
+
+![](images/bbdaa0ab06d00b9530d969c08c399c1cb8dafd4f5ae275b85350b1ff692cc6ae.jpg)  
+Figure 1. Pixel-wise fidelity vs. perceptual fidelity, at 0.40 bpp. We compare pixel-wise and perceptual fidelity of image compression codes on MS-COCO 30k. The proposed TACO achieves competitive results in both metrics. The reported figures for Qin 2023 has been measured on MS-COCO 40k, and some figures have been interpolated from nearest bpp models (●: PSNR-focused, ■: perception-focused, ▲: text-guided, ×: handcrafted).
+
+modern text-guided generative models to synthesize realistic images from the given text prompt (Rombach et al., 2022). The conventional way to leverage these models is via text-guided decoding (or textual transform coding; Weissman (2023)): We use text-guided generative model as our decoder, and the encoding is done by finding a good textual code. The code may consist of a text prompt plus any additional latent prompt vectors, that can generate the reconstruction that looks close to the original image. As the text can be represented with a very small number of bits, we expect such an approach to be very bit-efficient (Shannon, 1951; Bown et al., 2018; Weissman, 2023).
+
+Following this intuition, recent works have developed text-guided neural image compression codes that can reconstruct images with high perceptual quality, i.e., either highly realistic (e.g., low FID) or perceptually similar to the original image (e.g., low LPIPS). One line of work shows that using pretrained text-guided generative models as decoders
+
+"A woman wearing a red hat and a red dress"
+
+![](images/1c6a076463f491f34d8859739318d463494437caaa5e2b8500456bfce7435a0e.jpg)  
+Figure 2. Qualitative results. We compare TACO against MS-ILLM and LIC-TCM, which focuses on perception and PSNR, respectively. TACO uses slightly less bpp than baselines. Comparing with MS-ILLM, TACO tends to suffer less from hallucinated artifacts (see teeth). Comparing with LIC-TCM, TACO can reconstruct sharper details (see lips). See Appendix D for more examples.
+
+![](images/dfafee32d551690f00bb40454f98a975009c5dfc6583cc2fcf37fcdbe0dcc876.jpg)  
+Original
+
+![](images/61d939c5eb201c93ed8b64d09b26c2ac889a7b328b3b788d52e38d259261aba5.jpg)  
+TACO
+
+![](images/9214d1065f0f1fff1674438154f7f9097846e9f81bdf48b645e808866a0f337f.jpg)  
+MS-ILLM
+
+![](images/360832d9af0decf9fa2d5471c6b82adec17e26969e6a1520eb9512fc53bbcddf.jpg)  
+LIC-TCM
+
+can give neural CODECs that achieve high realism at extremely low bitrates, e.g., less than 0.01 bits per pixel (Pan et al., 2022; Lei et al., 2023; Careil et al., 2024). Another line of work targets conventional bitrates (over 0.1 bpp) and shows that one can significantly improve the perceptual similarity of reconstructed images to the original image by using text captions as additional side information, where the text information is inserted to the decoder via some plug-in modules (Jiang et al., 2023; Qin et al., 2023).
+
+These approaches, however, commonly suffer from substantial degradations in pixel-wise fidelity, i.e., PSNR. For example, diffusion-based compression CODEs tend to achieve 3-5 dB less PSNR than standard neural image compression CODEs (Careil et al., 2024). Despite the practical importance of the pixel-wise fidelity (Liu et al., 2023b), it remains unknown whether or how such PSNR degradation can be mitigated for text-guided image compression schemes.
+
+Contribution. In this paper, we fill this gap by developing a text-guided image compression scheme that can achieve high pixel-level and perceptual quality simultaneously (we focus more on the perceptual fidelity, e.g., LPIPS, than on realism). Our key hypothesis is that text-guided decoding may not be an effective strategy for PSNR. For high PSNR, we must be able to decode with very small generative diversity, and effectively utilize the semantic information at the global level (i.e., not affecting only local regions). Diffusion-like decoders tend to have too much diversity (Pan et al., 2022), and plug-in approaches tend to have weak control on the global semantics of the reconstruction (Lei et al., 2023).
+
+Thus, we establish an alternative strategy of utilizing text only for encoding. Here, the main role of text is to provide additional supervision on how human perceives the image; using this information, the encoder can better preserve perceptually meaningful information without discarding it dur
+
+ing (lossy) compression. By using text only for encoder, we can leverage existing decoder architectures that can generate accurate reconstructions with minimal generative diversity. Furthermore, as we encode the textual information into the code itself, the injected semantic information can affect the whole pixels globally without any extra effort.
+
+Based on this intuition, we propose a simple yet effective encoder-centric text-guided image compression algorithm, coined TACO (Text-Adaptive COmpression). TACO transforms a popular PSNR-oriented neural codec architecture (ELIC) into a text-guided one by augmenting the encoder with a text adapter. The text adapter utilizes a pre-trained CLIP encoder (Radford et al., 2021) and bi-directional attention mechanism to inject textual information into the latent code. Then, TACO trains the model from scratch with a joint image-text loss, which encourages the reconstruction to be semantically aligned with the given text.
+
+In our experiments, we find that TACO achieves excellent performance in both pixel-level and perceptual quality at standard bpp levels (i.e.,  $\geq 0.1$ ). In particular, TACO outperforms all image compression baselines in terms of LPIPS, while achieving competitive performance in pixel-level fidelity and realism when compared to state-of-the-art methods in each metric (Figure 1). This trend holds true for either datasets that come with human-generated image descriptions (e.g., MS-COCO (Lin et al., 2014)), or image-only datasets paired with machine-generated captions (e.g., CLIC (Toderici et al., 2020) captioned by OFA (Wang et al., 2022)). TACO also outperforms conventional LPIPS-focused text-guided baselines (Jiang et al., 2023; Qin et al., 2023).
+
+Our findings suggest that the core value of text in image compression—at least in standard bpp range—may hinge more on its relationship with human perception, than it being an efficient way to store information (Weissman, 2023).
+
+# 2. Related Work
+
+Measuring Reconstruction Quality. The most popular metric for evaluating the quality of reconstructions is the pixel-wise distortion from the original image. Typically, the quantity is measured in mean squared error (MSE) or peak signal-to-noise ratio (PSNR). However, the pixel-wise distortion is often found ill-aligned with the human-perceived image quality (Eskicioglu & Fisher, 1995).
+
+To address this, various perceptual quality metrics have been proposed. Roughly, the metrics fall into two categories: perceptual distortion and realism. Perceptual distortion quantifies how different two images are for human perception. Multi-scale structural similarity (Wang et al., 2003, MS-SSIM) measures the discrepancy of patch-level statistics of both images in various scales. Learned perceptual image patch similarity (Zhang et al., 2018, LPIPS) measures the distortion in the feature space of pretrained neural net classifiers, which are fine-tuned to account for human perception. Similarly, PieAPP trains a model to approximate the human judgement on perceptual similarity (Prashnani et al., 2018).
+
+Realism, on the other hand, attempts to quantify how realistic an image is, without necessarily comparing it with the original image. For example, Fréchet inception distance (Heusel et al., 2017, FID) measures the distance of (Inception) feature distribution of reconstructed images, from that of natural images. Similarly, KID measures the maximum mean discrepancy between the feature distributions (Binkowski et al., 2018). A recent work by Jayasumana et al. (2023) argues that FID/KID are still misaligned with human perception, and proposes CMMD, an alternative metric that utilizes pre-trained multimodal embeddings.
+
+Neural Image Compression. An early work by Balle et al. (2017) proposes an autoencoder-like image compression pipeline which achieves competitive pixel-level fidelity and substantially stronger perceptual quality than handcrafted codecs, e.g., JPEG. Subsequent works have developed more advanced hyperpriors on the latent space codes and utilized more advanced autoencoder-like architectures (Balle et al., 2018; Minnen et al., 2018; Cheng et al., 2020; He et al., 2022; Liu et al., 2023b), which led to even smaller pixelwise and perceptual distortions on the reproduced image.
+
+Another line of work focuses on developing neural compression CODEs for improved realism. Mentzer et al. (2020) propose a GAN-based method to generate highly realistic images at low bitrates. More recent approaches use diffusion model to generate even more realistic images (Hoogeboom et al., 2023; Yang & Mandt, 2023). However, these methods tend to have greater pixel-level or perceptual distortions than autoencoder-based models. MS-ILLM shows that the tradeoff between the distortion and realism can be more alleviated by using a more tailored GAN discriminator (Muckley et al.,
+
+![](images/46bda8dcacc284c317963a00b76a7ddec967fe1ce1eec4a92ae91d9c0f61aac7.jpg)  
+Figure 3. Text-guided decoding strategies vs. TACO. (Top) Text-guided decoding with diffusion-based decoders (Careil et al., 2024). (Middle) Text-guided decoder utilizing GAN (Qin et al., 2023). (Bottom) TACO is a much simpler yet effective strategy.
+
+2023). The fundamental limit of this rate-distortion-realism tradeoff has been theoretically studied by Blau & Michaeli (2018), suggesting that both goals may not be achieved simultaneously given a fixed rate.
+
+Text-guided Image Compression. Following the great success of vision-language models, text-guided image compression algorithms have emerged (Bhown et al., 2018; Weissman, 2023). One line of work aims to leverage pre-trained text-guided generative models as decoders. Here, the key is to introduce an appropriate means to control the generation diversity of these models. Pan et al. (2022) utilize the stable diffusion (Rombach et al., 2022), and control the diversity by guiding the reverse step with a JPEG-compressed version of the image. Lei et al. (2023) uses ControlNet as the decoder (Zhang et al., 2023), where the additional control is done by the 'sketch' version of the image, extracted by an edge detector. A concurrent work by Careil et al. (2024) also uses the latent diffusion model, and proposes a more sophisticated local-global encoding for better reconstruction. These models show superior performance in terms of realism (e.g., FID), especially at extremely low bpp.
+
+Another line of work proposes to use decoders that are trained from scratch, using a dataset with image-text pairs such as MS-COCO (Lin et al., 2014). The network architecture typically follows GAN-based image compression codes (Mentzer et al., 2020), with additional components to insert textual information into the architectures. Jiang et al. (2023) proposes to insert textual information to both encoder and decoder through uni-directional attention-like modules. Qin et al. (2023) insert text only to the decoder through semantic-spatial aware blocks. These works have their main strength on perceptual distortion (e.g., LPIPS) at conventional bpp.
+
+In both lines of work, it has been discovered that one can achieve high realism or perceptual similarity, at the expense of a substantial increase in pixel-level distortion. It remains unanswered whether one can attain similar improvements in the perceptual quality of images from the text, with only minimal degradation in PSNR.
+
+# 3. Method
+
+We introduce TACO (Text-Adaptive COmpression), a simple yet effective text-guided image compression algorithm that can achieve both high pixel-level and perceptual fidelity.
+
+At a high level, TACO is a simple framework that transforms a conventional PSNR-oriented image compression backbone into a text-guided one. TACO first augments the encoder of the backbone with a text adapter module, which inserts the textual information into the encoder. Then we train the overall network from scratch using a joint image-text loss.
+
+# 3.1. Background: Autoencoder-based Codec
+
+As the backbone architecture, we use ELIC (He et al., 2022), a PSNR-oriented neural image compression codec. We use ELIC as a base model due to its popularity and performance, and TACO does not rely on any of its structural properties. In principle, we expect TACO to be able to be combined with other base models to enhance their perceptual quality.
+
+Similar to most autoencoder-based CODECs, ELIC roughly consists of four components: encoder, quantizer, entropy model, and decoder. An image is processed sequentially by the components as follows. First, the encoder maps an image  $x$  to a latent feature  $y$ . The feature is then quantized into  $\hat{y}$ . The quantized feature  $\hat{y}$  is (losslessly) compressed into the code and decompressed with the entropy model. Finally, the decoder synthesizes the reconstruction  $\hat{x}$  from the decompressed feature  $\hat{y}$ .
+
+$$
+x \xrightarrow {\text {e n c o d e}} y \xrightarrow {\text {q u a n t i z e} \& \text {e n t r o p y c o d i n g}} \hat {y} \xrightarrow {\text {d e c o d e}} \hat {x} \tag {1}
+$$
+
+The encoder, where TACO injects the text information, is a stack of three residual bottleneck blocks (RBs; He et al. (2016)). The residual blocks are interleaved with five  $5 \times 5$  convolutional layers (with stride 2) and two attention modules (Cheng et al., 2020). For a more detailed description, see the original paper of He et al. (2022).
+
+# 3.2. TACO: Text-Adaptive Compression
+
+To enable an effective utilization of the textual information, TACO transforms the backbone image compression model in two ways. First, TACO injects the text information to the encoder of the backbone through text adapter. Second, TACO trains the whole model (i.e., backbone model and the text adapter) with the combined loss function of rate,
+
+![](images/650d81c618d8f81a75e81f0a3b341f5fc2783cb69960b8c6e8bfd95ba6bdc23e.jpg)  
+Figure 4. Text adapter of TACO. The text adapter first extracts features from the image caption using the CLIP text encoder. The textual features are then injected into the ELIC encoder through multiple cross-attention layers, interleaved with linear layers.
+
+distortion, and the joint image-text loss.
+
+Text Adapter. The text adapter takes in the text segment  $c$  associated with the original image  $x$  (e.g., image caption) as an input, and injects the text information into the backbone encoder to generate the joint image-text latent feature  $y$ .
+
+$$
+x \xrightarrow {\text {e n c o d e}} y \quad \Rightarrow \quad (x, c) \xrightarrow {\text {e n c o d e} + \text {T A C O}} y \tag {2}
+$$
+
+The text adapter operates in two steps (Figure 4). (1) Embedding: Maps the given text to the joint image-text feature space. (2) Injection: Gradually processes and injects text features into the backbone encoder through multiple layers.
+
+Adapter: Embedding. We leverage a pre-trained CLIP encoder (Radford et al., 2021), which contains rich semantic information on image-text correspondences. By using CLIP features, we expect reductions in the amount of extra computation and data for further reducing the domain gap between features. We use CLIP as frozen without fine-tuning.
+
+More concretely, this step generates a length- $m$  sequence of  $d$ -dimensional text tokens. In our experiments, we use  $m = 38$ , the maximum token length of MS-COCO captions, and  $d = 512$ , the dimensionality of CLIP encoder outputs.
+
+$$
+c \xrightarrow {\text {C L I P e n c o d e r}} \mathbf {e} := \left(e _ {1}, \dots , e _ {m}\right), \quad e _ {i} \in \mathbb {R} ^ {d}. \tag {3}
+$$
+
+Adapter: Injection. We use a six-layer network consisting of three linear layers and three cross-attention (CA) layers. Here, CAs play a crucial role injecting the text knowledge to images, or vice versa. This use of CA for knowledge injection is inspired by Chen et al. (2023), who uses CA for adapting ViTs to another vision task (dense prediction). In this work, we repurpose CAs for injection across different modalities. Each CA plays the following role:
+
+![](images/4ec2ed7f25d1cd4420a861abd2e03524c5a70896e77471524823b964affec0a6.jpg)
+
+![](images/a01618bf47c0491f39fa102005fe8a1417ba6703f61fb76f9b54e945e13c5c34.jpg)  
+Figure 5. Compression results: MS-COCO 30k. TACO achieves the best or competitive results in all metrics. In particular, TACO achieves the best LPIPS among all methods considered, while achieving only  $\sim 1$  dB less PSNR than LIC-TCM. PerCo achieves an impressive FID, but TACO outperforms in terms of both PSNR and LPIPS. We also note a strange under-performance of HiFiC in FID under this setup. This may be due to the resizing operation when measuring FID.
+
+![](images/fd1140aea563f6a873fe98740927666ff53429a5c36f2a18c798a5f3f63fa47b.jpg)
+
+![](images/49c0b4ec1941a034c80ab1f131acaffbf7ae3da40efcfba2ff0e4801cda903cb.jpg)
+
+- CA1: Text  $\rightarrow$  Image. Updates image features based on text tokens. Given a  $(h, w, k)$ -dimensional image feature, we treat each spatial location as an image token. That is, the set of image tokens can be written as:
+
+$$
+\mathbf {v} = \left(v _ {1}, \dots , v _ {h \times w}\right), \quad v _ {i} \in \mathbb {R} ^ {k}. \tag {4}
+$$
+
+CA1 computes query from each  $\nu \in \mathbf{v}$  and keys and values from  $e \in \mathbf{e}$ . To match the dimensionality of  $e$  and  $\nu$ , we put a linear layer before CA1. The output of the layer is
+
+$$
+\tilde {v} _ {i} = \mathrm {L N} (v _ {i}) + \gamma \odot \text {A t t e n t i o n} (\mathrm {L N} (v _ {i}), \mathrm {L N} (\operatorname {L i n} _ {1} (\mathbf {e}))), \tag {5}
+$$
+
+where  $\text{Attention}(\cdot)$ ,  $\text{LN}(\cdot)$ , and  $\text{Lin}(\cdot)$  denotes the attention, LayerNorm, and linear layer, respectively. Here,  $\gamma$  is a learnable weight hyperparameter.
+
+- CA2: Image  $\rightarrow$  Text. Updates text tokens based on image features which are further processed by the backbone encoder. The operation is similar to CA1, but we compute queries from the text and keys/values from the image.  
+- CA3: Text  $\rightarrow$  Image (downsampled). Same operation as CA1, but with two changes: First, the text tokens have been updated by image features (by CA2). Second, image features have been downsampled, so that text affects more global image features than in CA1.
+
+Joint Image-Text Loss. To train the model, we use a mixture of four different loss functions. Given the original image  $x$ , the text description  $c$ , the reconstructed image  $\hat{x}$  and its quantized latent feature  $\hat{y}$ , we use the loss
+
+$$
+r (\hat {y}) + \lambda \cdot d (x, \hat {x}) + k _ {\mathrm {p}} \cdot \operatorname {L P I P S} (x, \hat {x}) + k _ {\mathrm {j}} \cdot L _ {\mathrm {j}} (x, \hat {x}, c) \tag {6}
+$$
+
+The first three losses are standard:  $r(\cdot)$  denotes the rate,  $d(\cdot, \cdot)$  denotes the MSE, and LPIPS( $\cdot, \cdot$ ) denotes the LPIPS loss. The fourth loss  $L_{\mathrm{j}}(\cdot, \cdot, \cdot)$  is what we call the joint image-text loss, which encourages the reconstructed image to be semantically close to the given text and the original image.
+
+Formally, the loss is defined as the following:
+
+$$
+L _ {\mathrm {j}} (x, \hat {x}, c) = L _ {\text {c o n}} \left(f _ {\mathrm {I}} (\hat {x}), f _ {\mathrm {T}} (c)\right) + \beta \cdot \| f _ {\mathrm {I}} (x) - f _ {\mathrm {I}} (\hat {x}) \| _ {2}. \tag {7}
+$$
+
+Here, the functions  $f_{\mathrm{I}}(\cdot), f_{\mathrm{T}}(\cdot)$  denote the CLIP image and text embeddings, and  $L_{\mathrm{con}}(\cdot, \cdot)$  denotes the contrastive loss used in CLIP (Radford et al., 2021). In other words, the joint image-text loss is a mixture of (1) the relevance between the text and reconstructed image and (2) the CLIP embedding distance between the original and reconstructed image. We note that Jiang et al. (2023) also uses a loss similar to eq (7), but based on the AttnGAN embedding (Xu et al., 2018).
+
+For better PNSR, we do not use the adversarial loss. Such practice is common in perception-oriented CODECs, such as MS-ILLM or HiFiC, but not in PSNR-oriented CODECs. TACO works by improving the perceptual quality of PSNR-oriented methods with texts instead of further improving the perceptual quality of already perceptually good models.
+
+# 4. Experiments
+
+This section is organized as follows. Section 4.1 describes the basic experimental setup. Section 4.2 compares TACO with image compression CODECs that do not utilize text. Section 4.3 compares with text-guided CODECs; these works do not release code and/or model checkpoints and evaluation setups vary significantly from paper to paper, necessitating a more fine-grained comparison. Finally, Section 4.4 provides various analyses and ablations on TACO.
+
+# 4.1. Experimental Setup
+
+Datasets. For training, we use the training split of the MS-COCO dataset (Lin et al., 2014), which consists of 82,783 images with five human-annotated captions for each image; we use all five captions for training. We randomly crop each
+
+![](images/106551da02f34bbd290224b3454e119c373e92a10d75b18f42a1a170024d247e.jpg)
+
+![](images/c7b8487b642cfb857796bbd4d1776273b1c008c95c0cdbedd0b1ee99b2ca6a8f.jpg)  
+Figure 6. Compression results: CLIC. TACO outperforms all other baselines in LPIPS, and achieves close-to-best results in PSNR and FID. In particular, TACO achieves the PSNR very close to LIC-TCM and ELIC, vastly outperforming realism-oriented codeccs.
+
+![](images/fb54f2d636acd0b3a48ff9b2c7eb4a51ad1a2cce125984d34311e546050c7b4d.jpg)
+
+![](images/65e14a0a8168f16691939b38e4439ec7bee3e63b25f7132e457122186ec23049.jpg)
+
+![](images/73d8dd651ce1a3cc7c41fff27af2c4549611908d074390290a7ca9ca4b4fc2e8.jpg)  
+Figure 7. Compression results: Kodak. Similar to MS-COCO and CLIC, TACO outperforms baselines with a substantial margin in terms of LIPPs, and closely matches the PSNR of LIC-TCM and ELIC. TACO also performs best in terms of CMMD.
+
+![](images/77a166afff14b79bdfe5d0f67265b32be0bd403da96a1fd19501bc34af197520.jpg)
+
+![](images/2b0586c85c670198020128ebefb10b855e7ecae74c68c1e789bd788717c03ab7.jpg)
+
+image to  $256\times 256$  resolution for training.
+
+For evaluation, we use three different datasets. (1) MS-COCO 30k: We use the "restval" subset of the MS-COCO validation split, which consists of total 30,504 images. For evaluation, we center-crop the images to the  $256 \times 256$  resolution; we use center drop to more faithfully preserve the semantic information in the image caption. (2) CLIC (Toderici et al., 2020): The test set of the Challenge on Learned Image Compression 2020, consisting of 428 images. We do not crop these images. (3) Kodak dataset (Franzen, 1999): A natural image dataset that consists of 24 images. We also evaluate on these images as-is. We note that CLIC and Kodak do not have any associated captions; we use the image captions generated by OFA (Wang et al., 2022).
+
+Metrics. We focus on three image quality metrics:
+
+-  $LPIPS$  is a perceptual fidelity metric that measures the distance between the original and reconstructed image in the deep feature space; we use AlexNet features.  
+- PSNR is a pixel-wise fidelity metric, measured in dB.  
+-  $FID$  is a realism metric that measures the statistical fi
+
+dely between two image distributions (original and reconstructed) of inception features. For measuring FID, we follow Muckley et al. (2023) to resize the images to  $299 \times 299$  resolution before measuring.
+
+As a quantitative realism metric for Kodak dataset, we use a recently proposed CMMD (Jayasumana et al., 2023) instead of FID. This is because FID is known to be unstable in small datasets. For CMMD, we use the publicly available version of CLIP (w.o. projection). We additionally report results on MS-SSIM, PieAPP, and CMMD in the Appendix B.
+
+Optimization & Hyperparameters. We use Adam with batch size 4, and train for 50 epochs. The initial learning rate is set to  $10^{-4}$ , and we decay the learning rate by  $1/10$  at epochs 45 and 48. For hyperparameters, we simply use a fixed set  $(k_{\mathrm{p}}, k_{\mathrm{j}}, \beta) = (3.5, 0.0025, 40)$  throughout all bpps, instead of conducting an extensive hyperparameter tuning for each bpp. To get models of various bitrates, we train models with  $\lambda \in \{4, 8, 16, 40, 90, 150\} \times 10^{-4}$ .
+
+# 4.2. Results: vs. Image Compression Codes
+
+We now compare the compression performance of TACO against various image compression codecs, including:
+
+![](images/75cd314407eea47621a479641ea353a02c8c20bea16fa6fed456666705bc577e.jpg)  
+Figure 8. Comparison with text-guided decoding baselines. We compare with the MS-COCO 40k results reported by Qin et al. (2023), to the performance of TACO on the same dataset. We also compare with the MS-COCO 1k result reported by TGIC (Jiang et al., 2023).
+
+![](images/05ad55c607f76deced2d3a3f60df5d3da64b69e108f4d5c83310462ab1bd3343.jpg)
+
+![](images/a699a1e62349ee2ac737b187afcc34f5b333398d24b22ce208c40922283695d6.jpg)
+
+![](images/aeab15cc0addfb9af1126ee356495ff71d00a75dc938fb47a3541f753b874cec.jpg)  
+Figure 9. Text adapter vs. TGIC module. We compare the effectiveness of the proposed text adapter with the image-text attention module, used by Jiang et al. (2023), on CLIC. Our text adapter performance better in all metrics, when attached on ELIC encoder.
+
+![](images/71fa07ddbf5d0b1c7024fdb5b63105f0d60a78a549561d04385a90df2d398260.jpg)
+
+![](images/04fffae727d6313c73a0987ef1144481b1d693cca3609470403aa8f2ee774226.jpg)
+
+- Handcrafted image compression codecs: VTM, BPG.  
+- PSNR-oriented neural image compression codes: ELIC (He et al., 2022), and LIC-TCM (Liu et al., 2023b).  
+- Perception-oriented neural codes: HiFiC (Mentzer et al., 2020) and MS-ILLM (Muckley et al., 2023).
+
+We have used the official checkpoints for evaluation, except for ELIC (for which no official checkpoint is available).2 We provide quantitative comparisons on MS-COCO 30k, CLIC, and Kodak on Figures 5 to 7, respectively.
+
+We observe that, in terms of LPIPS (perceptual fidelity), TACO outperforms all baselines in all datasets. For example, on MS-COCO 30k, TACO achieves 0.025 LPIPS at 0.226 bpp, which is less than half achieved by the best baseline (HiFiC), achieving 0.053 at 0.233 bpp. In terms of PSNR, TACO closely achieves the performance of ELIC and LIC-TCM, falling within the 1dB range from these baselines. In realism, TACO outperforms baselines in MS-COCO and Kodak, but not in CLIC; the gap In CLIC, however, tends to be very small, especially at a high bpp regime.
+
+# 4.3. Results: vs. Text-guided Decoding
+
+We now compare the performance of TACO with text-guided codecs. In particular, we compare with codecs whose goal is to achieve high LPIPS at non-extreme bpp (i.e., over 0.1 bpp), such as TGIC (Jiang et al., 2023) and "Qin 2023" (Qin et al., 2023). Two things make a direct comparison difficult: (1) The baselines do not release official code or have model checkpoints publicly available. (2) The baselines report performance on non-unified setups: TGIC reports performance on MS-COCO 1k (along with CUB and Oxford-102), consisting of 1k randomly drawn test set samples. Qin et al. (2023) reports performance on MS-COCO 40k, i.e., the whole validation set, without any cropping.
+
+We focus on comparing with (more reproducible) Qin 2023 baseline. We evaluate TACO on MS-COCO 40k and compare it with the results reported in Qin et al. (2023). From Figure 8, we observe that TACO achieves much better performance than the previous text-guided baseline, based on the text-guided decoding strategy. This is somewhat surprising, as the original intention of using text-guided encoding has been on preventing the degradations in PSNR; instead, we also achieve better LPIPS and FID. This improvement can be attributed to the effectiveness of the text-adapter-based
+
+![](images/cacef711ece4450b30a2de082b203d698a494fa417e1215fc1b6c2b6244894bb.jpg)  
+Figure 10. Without text adapter. We observe both CMMD and LPIPS substantially degrade, while there is a tiny gain in PSNR.
+
+![](images/b428ff301a6b88c2297f5f354aaa1e75470c0328e8c9bbb6d4a6f19fa24a75ff.jpg)
+
+![](images/1229976d43e1e284254015a58ffae14562e8913b6b3a251b215f1d1d1854dea9.jpg)
+
+![](images/636c3310c0f88c08425676ff117b81f499d214ea68276281986daf2be1b8984c.jpg)  
+Figure 11. Without joint image text loss. We observe LPIPS severely degrades, while PSNR remains similar, with a tiny gain in CMMD.
+
+![](images/fac11956ebb003451eda8110820809d55f44b4643680197602c73ab4fc9d466b.jpg)
+
+![](images/21031dc2d2e9651df9abba63a5ce7260acdd585d2d165d67aaf5915fd35c4255.jpg)
+
+knowledge injection strategy.
+
+# 4.4. Ablation Studies
+
+To validate that both our proposed text adapter and joint image-text loss improve the performance, we conduct an ablation study. Through experiments on the Kodak dataset, we confirm that both components play essential roles.
+
+Without text adapter. From Figure 10, we observe that perceptual fidelity greatly decreases without a text adapter (more than  $4 \times$  in LPIPS). This gap is well-aligned with our intuition that the textual information plays an important role in how human perceives visual signals.
+
+Without joint image-text loss. From Figure 11, we observe that the joint image-text loss contributes in improving not only the perceptual fidelity, but also realism (with a tiny degradation in PSNR). We hypothesize that the manifold-level information transferred from the CLIP embedding, which is trained with a large-scale dataset, helps the reconstructed image to remain realistic.
+
+# 4.5. Further Analyses
+
+We now ask ourselves several questions regarding the validity and practicality of the proposed method, TACO.
+
+# Q1. Is the adapter architecture (Section 3.2) effective?
+
+A. Yes. It outperforms the baseline method for text injection. We compare it to another text injection mechanism introduced in TGIC (Jiang et al., 2023), which utilizes several convolutional layers to inject text information into the image encoder (coined "image-text attention module"). We compare the performance of these two modules for injecting text information to the encoder, in Figure 9. From the figure, we observe that the proposed text adapter is indeed much more effective than what TGIC uses.
+
+# Q2. Does TACO preserve textual information better?
+
+A. Yes, and No. TACO-compressed images tend to preserve textual information much better than PSNR-focused methods, similar to LIC-TCM. We measure various text similarity scores between machine-generated captions for the original and reconstructed image (Table 1); we use OFA for this purpose. We find that the images compressed by TACO preserve text semantics better than LIC-TCM and ELIC. On the other hand, the comparison with MS-ILLM is inconclusive; TACO performs slightly worse, but with slightly less bpp. In general, we find that models with high perceptual quality tend to preserve textual information better than PSNR-oriented compression CODECs.
+
+# Q3. How much computation does TACO add?
+
+A.  $\sim 5 - 10\%$  , depending on the image resolution. We
+
+Table 1. Comparing text similarity. We compare the text similarity scores for the CLIC images compressed by TACO and other baseline CODECs. We find that TACO achieves much higher text similarity score than LIC-TCM, but is slightly worse than MS-ILLM that uses slightly more bits per pixels.  
+
+<table><tr><td></td><td>BLEU-4 (↑) 
+(Papineni et al., 2002)</td><td>CIDEr-D (↑) 
+(Vedantam et al., 2015)</td><td>SPICE (↑) 
+(Anderson et al., 2016)</td></tr><tr><td>LIC-TCM (0.2810)</td><td>0.76</td><td>7.44</td><td>0.80</td></tr><tr><td>MS-ILLM (0.2314)</td><td>0.85</td><td>8.39</td><td>0.87</td></tr><tr><td>ELIC (0.3055)</td><td>0.78</td><td>7.58</td><td>0.81</td></tr><tr><td>TACO (0.2146)</td><td>0.84</td><td>8.21</td><td>0.86</td></tr></table>
+
+Table 2. Computational cost of TACO. We compare the wall-clock encoding and decoding latency of TACO with baseline image compression codecs. We average over 100 MS-COCO images  $(256\times 256)$ , randomly drawn from the validation split (0.0148 bpp). We also compare the computational cost when encoding a high-resolution image ('High' in table), observing that the increasing cost of attention remains relatively small  $(+10.2\% \rightarrow +4.8\%)$ .  
+
+<table><tr><td></td><td>Enc. (ms)</td><td>Enc.@High (ms)</td><td>Dec. (ms)</td><td>Total (ms)</td></tr><tr><td>LIC-TCM</td><td>112.07</td><td>960.99</td><td>125.26</td><td>237.33</td></tr><tr><td>MS-ILLM</td><td>70.39</td><td>800.45</td><td>53.74</td><td>124.14</td></tr><tr><td>ELIC</td><td>71.35</td><td>793.41</td><td>102.07</td><td>173.42</td></tr><tr><td>TACO</td><td>78.60 (+10.2%)</td><td>832.07 (+4.8%)</td><td>102.98</td><td>181.58</td></tr></table>
+
+compare the wall-clock encoding and decoding speed of the proposed TACO with the baselines in Table 2. We have measured the latency on a Linux GPU server equipped with one NVIDIA GeForce RTX 3090 and AMD EPYC 7313 16-Core CPU. From the table, we find that TACO indeed introduces some latency in compression speed. In particular, we observe that the encoding time increases by 7ms over the vanilla ELIC. We note that the overhead, however, is not very big. TACO still runs much faster than LIC-TCM and remains comparable with MS-ILLM. Most of the overhead comes from the CLIP embedding step, which can be effectively parallelized with image encoding steps. Furthermore, we have used the smallest version of CLIP, which can be run fast on conventional GPU devices. We also compare the encoding speed when using a high-resolution image  $(1788\times 2641$ , second column in the Table 2). As a result, the computational cost of attention remains relatively small and we find that the computational cost of CLIP remains constant with respect to the resolution.
+
+# Q4. How many parameters are used in TACO?
+
+A. Not too big—smaller than perception-oriented CODECs.
+
+We compare the number of models in Table 3. TACO adds 64.82M parameters to the ELIC, where 63.17M comes from the CLIP and 1.65M comes from the text adapter. In total, TACO has 101.75M parameters, which requires  $\sim 400\mathrm{MBs}$  to be loaded on memory. While this is larger than vanilla ELIC, it is  $\sim 80\%$  smaller than other perception-oriented codecs, such as MS-ILLM, and HiFiC.
+
+Table 3. Memory efficiency of TACO. We compare the model parameter sizes of the TACO and baselines. TACO shows superior memory efficiency than perception-oriented CODECs.  
+
+<table><tr><td>Modules</td><td>Parameters (M)</td></tr><tr><td>ELIC</td><td>36.93</td></tr><tr><td>LIC-TCM</td><td>45.41</td></tr><tr><td>HiFiC/MS-ILLM</td><td>181.72</td></tr><tr><td>TACO</td><td>101.75</td></tr></table>
+
+Table 4. Caption dependency. We compare TACO reconstruction results with four different captions generation methods, spanning from human to GPT-4. Human and GPT-4 achieves the best results when measured on 100 random samples of MS-COCO, but other captioning methods work very competitively (0.0148 bpp). Underlined denotes the best result.  
+
+<table><tr><td></td><td>Human (Chen et al., 2015)</td><td>OFA (Wang et al., 2022)</td><td>BLIP-2 (Li et al., 2023)</td><td>GPT-4 (Achiam et al., 2023)</td></tr><tr><td>LPIPS</td><td>0.0435</td><td>0.0435</td><td>0.0435</td><td>0.0435</td></tr><tr><td>PSNR</td><td>27.42</td><td>27.41</td><td>27.41</td><td>27.42</td></tr></table>
+
+# Q5. Can we use other captioning models?
+
+A. Yes. In Table 4, we compare the compression quality of TACO utilizing captions generated by various methods. We observe that GPT-4 achieves a human-like result, but other captioning models perform very competitively. From this observation, we conjecture that the compression quality of TACO is less dependent on its writing style, but more on the core content of the sentences.
+
+# 5. Conclusion
+
+In this paper, we have studied how one can utilize auxiliary text information when compressing images. We find that, in order to achieve high PSNR as well as high perceptual quality, utilizing the text to generate better codes is a simple yet effective strategy; using text to guide the decoding procedure may be much more challenging task.
+
+Limitations and future work. There are several limitations of the proposed framework, which we hope to address in the future work. First, the additional encoding cost for handling the text scales quadratically with respect to the sequence length. This is due to the design of our text injection mechanism which computes cross-attention between image and textual features. In the cases where we expect the text sequence to be very long, e.g., video-like data with voice transcripts as text, the encoding time can get prohibitively long. Another limitation is the scalability of training. Currently, we train TACO using the image-text dataset, which is substantially scarcer than image-only or text-only dataset. To overcome this problem, using a captioning model to generate image-text pairs for training may be a good baseline.
+
+# Acknowledgements
+
+This work was supported by Samsung Advanced Institute of Technology and Institute of Information & communications Technology Planning & Evaluation (IITP) grant funded by the Korea government(MSIT) (RS-2019-II191906, Artificial Intelligence Graduate School Program(POSTECH)). We would like to thank Sadeep Jayasumana for helpful comment on implementing the CMMD.
+
+# Impact Statement
+
+Our paper aims to advance the general field of machine learning and data compression. We thus expect many societal consequences to follow. However, we do not feel any specific consequence should be highlighted further.
+
+# References
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+
+# Appendix
+
+# A. Additional Experimental Details
+
+Implementation of TACO. As illustrated in Figure 4, the base model we use is a version of ELIC, which has an intermediate feature dimension of 192 and a final feature map dimension of 320. The CLIP encoder we use is the "CLIPTextModel" with a version of 'openai/clip-vit-base-patch32,' available through the HuggingFace. We have set the token length to be 38, therefore getting the text embedding sequences with the size of  $\mathbb{R}^{38\times 512}$  when inputting the caption to the CLIPTextModel.
+
+Baselines. As mentioned in the main text, for neural codecs, we mostly use the official model checkpoints. For BPG, we have used version 0.9.8. For VTM, we have used version 19.2.
+
+# B. Additional Metrics on Compression Results
+
+We additionally report more compression results, MS-SSIM, PieAPP, and CMMD scores, measured on the MS-COCO 30k, CLIC, and Kodak datasets.
+
+![](images/66540d5a469ab5cfeeeea54c75e3f7ace93ae40ad24d6525c36f84956c428942.jpg)
+
+![](images/9d23dcd4c82bc8a0afd08c591c6e549ea1817ae1dc956d303cd74929a46abb88.jpg)
+
+![](images/86f62eba723aed3f6a4ecabd78321d3d5a609658d9050f1681659a5edd56fac6.jpg)
+
+![](images/1c7652bd6202d332972345d8e81407e994baf4eb32adfc96035b2a0c97a27247.jpg)
+
+![](images/cf0b123ac54284ec33dca4749530b3a3a3079bc2fc698db8b4bd7ebf67c20de5.jpg)
+
+![](images/0079020536b27692e0fcbc80f5385baf1b449c93c88394e02c53c66cef4e1235.jpg)
+
+![](images/968a6c1a9354ba2f50b432824f6fa54e16c72480e210b569691fd851fb7e79d7.jpg)
+
+![](images/17041757a67618704dcfb1dd4632fa383de4c14db5e70757f352973d0f6c2743.jpg)  
+Figure A1. Additional quality metrics result on: MS-COCO 30k (top), CLIC (middle), Kodak (bottom)
+
+![](images/5fec1423be3bdd44930f9849efb2e25f966b521a8369f16d87f46221a3384638.jpg)
+
+# C. Additional Ablation Study
+
+Freezing base model. We test whether freezing the base model (ELIC) to be a fully image-trained model is beneficial for performance or not. We find that this is not true (see Figure A2).
+
+![](images/c34c788b23c1b7576baa88f6da8c406ab164e007905134b9d625605fb0dfca3c.jpg)  
+Figure A2. Ablation result: freezing base model and only training text adapter
+
+![](images/95727455c2cbeaf0e1e21624d0442a8f59fd7c9440fa6068ad547259026c925d.jpg)
+
+![](images/782773a639011ab89f6f6e378b12274a136b57adadd96ae788a6b9067dc12c86.jpg)
+
+# D. Additional Qualitative Results
+
+We provide additional visualizations of the compression results of TACO and baselines in Figure A4. As has been observed in Figure 2, we confirm that TACO suffers from fewer compression artifacts than MS-ILLM while reconstructing sharper details than LIC-TCM. We find that TACO consistently provides high-quality reconstructions, while LIC-TCM tends to overly smooth out the textures (see, e.g., the white wall under the cat) and MS-ILLM hallucinates small details (see, e.g., eyes of the cat).
+
+- In the top display, we draw attention to two features: "the reflections on glasses," and "the texture of eyebrow and hair." MS-ILLM tends to hallucinate flat reflections on the glasses, and LIC-TCM tends to generate blurry eyebrow and hair.  
+- In the middle display, we focus on the black metal staircase behind the net. MS-ILLM generates wavy textures on the metal, and LIC-TCM removes the net in front of the staircase. TACO also generates some hallucinative patterns but to a lesser degree. Also, we note that TACO is the only compression method that reconstructs (any) stud on the staircase.  
+- In the bottom display, we highlight the gold hinge. MS-ILLM smooths out the details on the hinge, and LIC-TCM generates a blurry image overall.
+
+"A young woman sitting on the grass wearing a hat and glasses"
+
+![](images/e7d7d7bee2ae930f651e4773f125e25dacae678c397cd5e0850faeb2a178e7f1.jpg)
+
+![](images/e25ede67a9a0566b6b076e1e4ce824843b9d7a0cefa684c993264708150814b8.jpg)  
+Original
+
+![](images/b54688d9721b778b2e4c8f958c160d0f49b8755c072bde30afeace7d8f05d799.jpg)  
+TACO
+
+![](images/c99408950d85a732d64b2294efdcb1308c7c6ec06682f2a00d9b1b11897d6901.jpg)  
+MS-ILLM
+
+![](images/12c0714ffad7ca955822ac06b7a06a4b77d4533f407dbb1a15ad815ff03688f6.jpg)  
+LIC-TCM
+
+"A young man riding a skateboard in a parking lot"
+
+![](images/c7c6b275f6fd203ff82e0b01c1d3c0bd086b819a1ad394a4a681b60c28f99a96.jpg)
+
+![](images/dd8c27294b4b47fba5b6ca018820d2e4fada7036fbbbb86ec0230d8b493b06d9.jpg)  
+Original
+
+![](images/afc6827d6bdcfd22ce760c3d42e3b74539919e8050d29c3e403fe3741b19cc2f.jpg)  
+TACO
+
+![](images/c38dbc1433c4465a734ada17a1db4d9e18639505434bafb5156dc2d220c6c73e.jpg)  
+MS-ILLM
+
+![](images/e66facd39d5156742667aa39efcc1a6acd704e68104cdfc4bf7eb7d22f36cf05.jpg)  
+LIC-TCM
+
+"An old piano with books and a lamp on it"
+
+![](images/b667cc7d38888b339f317d796d6dadb38239207a53abc21bbf021d1935707cf3.jpg)  
+Figure A3. Additional Qualitative Results.
+
+![](images/cc99e448f429b7c6278b38de0835ebd3b87311475392271b788af7981c67c4da.jpg)
+
+![](images/c93e76346399c42d30cd76b765104e9f5d9b0683904e9a1afdd83431a6d84179.jpg)  
+TACO
+
+![](images/051a9d7eb6cdf31a2e5220b5e38dc4e87ec05fd787567ac63dbc762ca1406333.jpg)  
+Original  
+MS-ILLM
+
+![](images/bd73a447c12b691119d57b5029465b6aacefdeac697f5f50c23311466b44b4dc.jpg)  
+LIC-TCM
+
+"A large body of water with palm trees on an island"
+
+![](images/8160fa543202cbc46432b7cd5a7bb128ffddbcd66d0b7fa1694d9e9b84c4a705.jpg)
+
+![](images/cd8789dfec8c2d8f904b07e03fedf1b6cea0f7f91d2e415aaab29c4f4841e2ae.jpg)  
+Original
+
+![](images/2c2398b528e1fc0206e206e97614ad6185d94135a6b1cb695de3403d9288528a.jpg)  
+TACO
+
+![](images/84efd74e922a3e8a92c1917f60db810dab3028e994c76e17fdfa9a3136492798.jpg)  
+MS-ILLM
+
+![](images/af715eb158ab457740d0933855e655189cb88cb4c82915872f1b3b50db32f6d0.jpg)  
+LIC-TCM
+
+"A cat sitting on top of a white sand field"
+
+![](images/c41fbf2c08f203546c16448e54957ab96ed57c362fa31c11560a95e3d22607c5.jpg)
+
+![](images/21da3d04773062cc30acde835c0dd326ba66582ca3abd88d5b81c8e81fe43398.jpg)
+
+![](images/4d2fa6df7b819911178b360f872474077b0c6735208804b24e034c7d9c205d3d.jpg)  
+TACO
+
+![](images/1aba3fdf78bd7ef898befbce503709f26d7b6304061fe2d098d8232fffa5c34e.jpg)  
+Original  
+MS-ILLM
+
+![](images/3c52dd2fe8bb4b1eca11f60e59f46969c45fabbef8bcf14eaeb7f6a75aea32f9.jpg)  
+LIC-TCM
+
+"A person walking a dog in front of a building"
+
+![](images/22136b585af70f0fa3ffd9e5434612a9101e9c7a3f1edc60d464e4c1ceab259b.jpg)  
+Figure A4. Additional Qualitative Results.
+
+![](images/6799ac94a088f7baac8b4b875a4e57a6297b21efba00a96fcd5150d0b6ee816e.jpg)
+
+![](images/10d14f8d20558548e980f21141445fc821bba3e6dcc61834e14cbe02e7477418.jpg)
+
+![](images/94379729fc651bd1280db8002ef4d9d5209377c7515db791c55994d7a401719a.jpg)  
+Original  
+MS-ILLM
+
+![](images/0d8129f53487276b88c60a23b53bc21a789c101ef31061e34f8860c552ce058d.jpg)  
+TACO  
+LIC-TCM

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2024/Neural Jump-Diffusion Temporal Point Processes/e9547c48-812c-46c0-8f9d-b917215e2546_origin.pdf

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2024/Neural Jump-Diffusion Temporal Point Processes/full.md

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+# Neural Jump-Diffusion Temporal Point Processes
+
+Shuai Zhang<sup>1</sup> Chuan Zhou<sup>12</sup> Yang Liu<sup>1</sup> Peng Zhang<sup>3</sup> Xixun Lin<sup>4</sup> Zhi-Ming Ma<sup>1</sup>
+
+# Abstract
+
+We present a novel perspective on temporal point processes (TPPs) by reformulating their intensity processes as solutions to stochastic differential equations (SDEs). In particular, we first prove the equivalent SDE formulations of several classical TPPs, including Poisson processes, Hawkes processes, and self-correcting processes. Based on these proofs, we introduce a unified TPP framework called Neural Jump-Diffusion Temporal Point Process (NJDTPP), whose intensity process is governed by a neural jump-diffusion SDE (NJDSDE) where the drift, diffusion, and jump coefficient functions are parameterized by neural networks. Compared to previous works, NJDTPP exhibits model flexibility in capturing intensity dynamics without relying on any specific functional form, and provides theoretical guarantees regarding the existence and uniqueness of the solution to the proposed NJDSDE. Experiments on both synthetic and real-world datasets demonstrate that NJDTPP is capable of capturing the dynamics of intensity processes in different scenarios and significantly outperforms the state-of-the-art TPP models in prediction tasks.
+
+# 1. Introduction
+
+Many real-world scenarios often generate a large amount of asynchronous event sequences. Each event consists of a timestamp and a type mark, indicating when and what the event occurred. Examples include user activities on social media platforms (Farajtabar et al., 2017), electronic health records in healthcare (Liu & Hauskrecht, 2021), and transaction behaviors in e-commerce systems (Xue et al., 2022).
+
+$^{1}$ Academy of Mathematics and Systems Science, Chinese Academy of Sciences  $^{2}$ School of Cyber Security, University of Chinese Academy of Sciences  $^{3}$ Cyberspace Institute of Advanced Technology, Guangzhou University  $^{4}$ Institute of Information Engineering, Chinese Academy of Sciences. Correspondence to: Chuan Zhou <zhouchuan@amss.ac.cn>.
+
+Proceedings of the  $41^{st}$  International Conference on Machine Learning, Vienna, Austria. PMLR 235, 2024. Copyright 2024 by the author(s).
+
+Modeling such data has become increasingly important for tasks such as predicting the occurrence of future events (Du et al., 2016; Mei & Eisner, 2017; Zhang et al., 2020a; Zuo et al., 2020), detecting anomalies in event sequences (Liu & Hauskrecht, 2021; Shchur et al., 2021; Zhang et al., 2023), and performing causal inference on events (Xu et al., 2016; Zhang et al., 2020b; Gao et al., 2021).
+
+Temporal point processes (TPPs) (Daley et al., 2003) serve as a useful mathematical tool for modeling sequences of discrete events in continuous time. Classical examples of TPPs include Poisson processes (Kingman, 1992), Hawkes processes (Hawkes, 1971), and self-correcting processes (Isham & Westcott, 1979). A central concept in TPPs is the intensity process<sup>1</sup> (Oakes, 1975), also known as the intensity function (Zhang et al., 2020b), which measures the expected rate of events occurrence given historical events. While these classical models exhibit favorable statistical properties, the fixed parametric form of their intensity functions prevents them from capturing complicated dynamics.
+
+To enhance the capability of TPP models, there has been a surge in modeling the intensity function as a transformation of the hidden state of neural networks. Depending on the neural network structures, these TPP models can be divided into two categories, i.e., those based on either RNNs or Transformers (Du et al., 2016; Zuo et al., 2020; Yang et al., 2022), and those based on continuous-depth neural networks (Jia & Benson, 2019; Chen et al., 2020). While being more expressive than classical TPPs, the former models usually assume a specific functional form for the intensity function. For example, RMTPP (Du et al., 2016) assumes that the intensity exponentially decreases or increases between events. However, relying on such an assumption would limit model expressiveness when the employed assumption deviates from reality (Omi et al., 2019). The latter models represent the hidden state as the solution to a neural jump stochastic differential equation (Jia & Benson, 2019). These models, however, provide no theoretical guarantee for the global existence and uniqueness of the solution.
+
+In this paper, we provide a new view for TPPs by reformulating the intensity process as the solution to a stochastic differential equation (SDE) (Ikeda & Watanabe, 2014). Specifi
+
+cally, we first derive equivalent SDE formulations of several classical TPPs mentioned above. From these SDE formulations, we observe that the coefficient functions in SDE play a key role in shaping the evolution of intensity process over time and revealing the influences between events. Based on these observations, we introduce the Neural Jump-Diffusion Temporal Point Process (NJDTPP), whose intensity process is governed by a neural jump-diffusion SDE (NJDSDE). The drift, diffusion, and jump coefficient functions in NJDSDE are parameterized by three neural networks, i.e., the drift net, diffusion net, and jump net. Concretely, the drift net captures the intrinsic evolution of the intensity process, the diffusion net models the Gaussian noise with the Brownian motion (Wang et al., 2017; 2018), and the jump net captures the influences between events, such as the excitatory and inhibitory influences. Remarkably, our NJDTPP model does not require a specific functional form for the intensity function. Instead, by using the drift, diffusion, and jump nets, the solution to NJDSDE can implicitly determine a free-form intensity process consistent with the observed event data. We summarize our contributions as follows:
+
+- Theoretical Analysis. We prove the equivalent SDE formulations of several classical TPPs. For the SDE formulation, we provide a sufficient condition for the existence of a unique positive solution. Moreover, we theoretically analyze the existence and uniqueness of the solution to the proposed neural jump-diffusion SDE.  
+- Unified Framework. By viewing the intensity process as the solution to an SDE, we propose a unified TPP framework NJDTPP which can learn a free-form intensity process consistent with the observed data. A number of classical TPPs can be interpreted as special cases of our framework with simple coefficient functions.  
+- Extensive Experiments. We conduct experiments on three synthetic and six real-world datasets to evaluate the performance of NJDTPP. Experimental results show that NJDTPP successfully captures the dynamics of intensity processes and achieves state-of-the-art results in the tasks of likelihood evaluation and event prediction.
+
+# 2. Related Work
+
+Neural Temporal Point Processes. Neural TPPs that combine TPPs with neural networks have received considerable attention (Du et al., 2016; Mei & Eisner, 2017; Zhang et al., 2020a; Zuo et al., 2020; Lin et al., 2021; Yang et al., 2022). While being more expressive than classical parametric ones, neural TPPs usually assume a specific functional form for the intensity function. For example, RMTPP (Du et al., 2016) assumes that the intensity exponentially decreases or increases between events; THP (Zuo et al., 2020) utilizes the softplus function so that the intensity
+
+between events is approximately linearly interpolated. However, relying on such an assumption can undermine model effectiveness if the employed assumption deviates from reality. In addition to the dominant paradigm of parameterizing intensity functions, alternative methods involve modeling cumulative intensity functions (Omi et al., 2019) and conditional density functions (Shchur et al., 2019). However, these methods may not fully capture the dynamics of the intensity process. In contrast to existing studies, our model formulates the intensity process as the solution to an SDE without relying on any specific functional form.
+
+Neural Differential Equations. Neural differential equations (NDEs) (Kidger et al., 2021a) are defined as differential equations in which coefficient functions are parameterized by neural networks. Many NDEs, including neural ODE and its variants (Chen et al., 2018; Rubanova et al., 2019; Kidger et al., 2020; Herrera et al., 2020), as well as neural SDEs (Li et al., 2020; Kong et al., 2020; Kidger et al., 2021a;b), have been proposed for modeling time series. However, there is a distinction between time series and event sequences (Xiao et al., 2017). In time series, time serves only as the index to order the sequence of values for the target variable. In event sequences, time is regarded as a random variable representing the timestamp of asynchronous events, with time itself being the subject of research. Therefore, many existing NDE-based models are not directly suitable for modeling event sequences. While Jia & Benson (2019) and Chen et al. (2020) utilize NDEs to model event sequences, they actually capture the dynamics of the hidden state of neural networks. Besides, they solely focus on the jump term, neglecting the diffusion term associated with randomness driven by Brownian motion. In contrast, we incorporate Brownian motion to model Gaussian noise, and more importantly our proposed NJDSDE models the dynamics of the intensity process.
+
+Equivalent SDE Formulations for TPPs. Wang et al. (2018) provided a jump-diffusion SDE framework for modeling user activities. They introduced the diffusion term to model the Gaussian noise, such as fluctuations in the dynamics caused by unobserved factors. However, their utilization of fixed linear coefficient functions in the SDE might not fully capture the actual intensity. On the contrary, we employ neural networks to parameterize coefficient functions, allowing for a more flexible modeling of the intensity that better aligns with the observed data. While De et al. (2016); Zarezade et al. (2017); Wang et al. (2018) established the equivalent SDE formulation for Hawkes processes, we provide a distinct proof method. Besides, we derive equivalent SDE formulations for several other classical TPPs, such as Poisson processes and self-correcting processes. Moreover, for the SDE formulation, we provide a sufficient condition for the existence of a unique positive solution.
+
+# 3. Background
+
+In this section, we provide a brief overview of temporal point processes and jump-diffusion stochastic differential equations.
+
+# 3.1. Temporal Point Processes
+
+A temporal point process (TPP) (Daley et al., 2003) is a stochastic process  $\{t_i\}_{i=1}^{\infty}$ , in which the non-negative random variable  $t_i$  represents the occurrence time of the  $i$ -th event and  $t_i < t_{i+1}$ . Such a process can be equivalently represented as a counting process  $\{N_t\}_{t \geq 0}$ , where  $N_t$  represents the number of events up to time  $t$ .
+
+The most common way to characterize a TPP is via its intensity process (Oakes, 1975), also known as the intensity function. Specifically, the intensity process of  $\{N_t\}_{t \geq 0}$  is a left-continuous with right-limits process  $\{\lambda(t \mid \mathcal{F}_{t-})\}_{t \geq 0}$ , denoted for simplicity as  $\{\lambda_t\}_{t \geq 0}$ , where  $\lambda_t$  measures the expected rate of events occurring in an infinitesimal window  $(t, t + \mathrm{d}t]$  given the historical events up to time  $t$ . Formally,
+
+$$
+\lambda_ {t} \mathrm {d} t = \mathbb {P} \left(\mathrm {d} N _ {t} = 1 \mid \mathcal {F} _ {t -}\right) = \mathbb {E} \left[ \mathrm {d} N _ {t} \mid \mathcal {F} _ {t -} \right], \tag {1}
+$$
+
+where  $\mathcal{F}_{t - } = \sigma (N_s:0\leq s <   t)$  and the jump size  $\mathrm{d}N_{t} = N_{t + \mathrm{d}t} - N_{t}\in \{0,1\}$
+
+In the following, we review several classical TPPs, where the intensity function has a fixed parametric form.
+
+Poisson processes (Kingman, 1992). The intensity function of the Poisson process  $\{N_t\}_{t \geq 0}$  is independent of event history. The simplest case is a homogeneous Poisson process where the intensity is a positive constant:
+
+$$
+\lambda_ {t} = \lambda > 0. \tag {2}
+$$
+
+For a more general inhomogeneous poisson process, the intensity is a function varying over time:
+
+$$
+\lambda_ {t} = g (t) > 0. \tag {3}
+$$
+
+Hawkes processes (Hawkes, 1971). The Hawkes process  $\{N_t\}_{t \geq 0}$  with the widely used exponential kernel assumes that events are self-exciting. The arrival of a new event results in a sudden increase in intensity, and this influence decays exponentially:
+
+$$
+\lambda_ {t} = \mu + \alpha \sum_ {i: t _ {i} <   t} \exp (- \beta (t - t _ {i})), \tag {4}
+$$
+
+where  $\mu > 0$ ,  $\alpha > 0$  and  $\beta > 0$ .
+
+Self-correcting processes (Isham & Westcott, 1979). In contrast to the Hawkes process, the self-correcting process  $\{N_t\}_{t \geq 0}$  assumes that a new event inhibits future events and the intensity grows exponentially over time:
+
+$$
+\lambda_ {t} = \exp (\mu t - \sum_ {i: t _ {i} <   t} \alpha), \tag {5}
+$$
+
+where  $\mu > 0$  and  $\alpha > 0$ .
+
+# 3.2. Jump-Diffusion Stochastic Differential Equations
+
+One-dimensional autonomous jump-diffusion stochastic differential equations (JDSDE) (Hanson, 2007) with initial conditions are of the form
+
+$$
+\left\{ \begin{array}{l} \mathrm {d} X _ {t} = f \left(X _ {t}\right) \mathrm {d} t + g \left(X _ {t}\right) \mathrm {d} W _ {t} + h \left(X _ {t}\right) \mathrm {d} N _ {t}, \\ X _ {0} = x _ {0}, \end{array} \right. \tag {6}
+$$
+
+where  $x_0 \in \mathbb{R}$  is the initial value,  $f: \mathbb{R} \to \mathbb{R}$  is the drift coefficient function,  $g: \mathbb{R} \to \mathbb{R}$  is the diffusion coefficient function,  $h: \mathbb{R} \to \mathbb{R}$  is the jump coefficient function,  $\{W_t\}_{t \geq 0}$  is a standard Brownian motion, and  $\{N_t\}_{t \geq 0}$  is a counting process that jumps at times  $\{t_i\}_{i=1}^{\infty}$ . Suppose that  $\{W_t\}_{t \geq 0}$  and  $\{N_t\}_{t \geq 0}$  are independent. In this paper, it is essential to highlight that the process  $\{N_t\}_{t \geq 0}$  in Eq.(6) is a general counting process introduced in Section 3.1, distinct from many previous works (Cyganowski et al., 2002; Hanson, 2007; Lamberton & Lapeyre, 2011) that focus on a Poisson process.
+
+The JDSDE Eq.(6) is interpreted as a stochastic integral equation (Cyganowski et al., 2002):
+
+$$
+X _ {t} = x _ {0} + \int_ {0} ^ {t} f (X _ {s}) \mathrm {d} s + \int_ {0} ^ {t} g (X _ {s}) \mathrm {d} W _ {s} + \int_ {0} ^ {t} h (X _ {s}) \mathrm {d} N _ {s},
+$$
+
+where the first integral is a Riemann integral, the second is an Itô integral and the third is a Riemann-Stieltjes integral. In fact, Eq.(6) behaves as a normal Itô SDE (Cyganowski et al., 2002; Hanson, 2007) between jumps of  $\{N_t\}_{t \geq 0}$ . This can be expressed as:
+
+$$
+\mathrm {d} X _ {t} = f \left(X _ {t}\right) \mathrm {d} t + g \left(X _ {t}\right) \mathrm {d} W _ {t}, \quad t \in \left(t _ {i - 1}, t _ {i} \right].
+$$
+
+On the other hand, at a jump time  $t_i$ ,  $\{N_t\}_{t \geq 0}$  has a jump size of  $\Delta N_{t_i} = 1$ , which implies that the process  $\{X_t\}_{t \geq 0}$  will have a jump of size
+
+$$
+\Delta X _ {t _ {i}} = X _ {t _ {i} +} - X _ {t _ {i}} = h \left(X _ {t _ {i}}\right) \Delta N _ {t _ {i}} = h \left(X _ {t _ {i}}\right),
+$$
+
+where  $X_{t_i + } = \lim_{s\downarrow t_i}X_s$  . Then  $X_{t_i + } = X_{t_i} + h(X_{t_i})$
+
+# 4. Equivalent SDE Formulations for TPPs
+
+In this section, we derive equivalent SDE formulations of several classical TPPs, which involves expressing their respective intensity process as a solution to the corresponding SDE. Then for the SDE formulation, we provide a sufficient condition for the existence of a unique positive solution.
+
+Theorem 1. The intensity processes of homogeneous and inhomogeneous Poisson processes can be equivalently expressed as solutions to the following ODEs, respectively. These ODEs can be viewed as degenerate forms of SDEs.
+
+$$
+\mathrm {d} \lambda_ {t} = 0, \quad \lambda_ {0} = \lambda , \tag {7}
+$$
+
+$$
+\mathrm {d} \lambda_ {t} = g ^ {\prime} (t) \mathrm {d} t, \quad \lambda_ {0} = g (0), \tag {8}
+$$
+
+where  $\lambda > 0$  and  $g(t) > 0$  is assumed to be differentiable.
+
+According to Eq.(2) and Eq.(3), Theorem 1 is evident. Subsequently, we establish equivalent SDE formulations for Hawkes processes and self-correcting processes.
+
+Theorem 2. The intensity process  $\{\lambda_t\}_{t\geq 0}$  of the Hawkes process  $\{N_t\}_{t\geq 0}$  can be equivalently expressed as the solution to the jump SDE
+
+$$
+\mathrm {d} \lambda_ {t} = \beta (\mu - \lambda_ {t}) \mathrm {d} t + \alpha \mathrm {d} N _ {t}, \quad \lambda_ {0} = \mu . \tag {9}
+$$
+
+Proof. See Appendix A.1. The proof sketch is as follows: Taking inspiration from (Björk, 2021), we now solve the above SDE. Let the jump times of  $\{N_t\}_{t \geq 0}$  be  $\{t_i\}_{i=1}^{\infty}$ , then Eq.(9) behaves as an ODE  $\mathrm{d}\lambda_t = \beta (\mu - \lambda_t)\mathrm{d}t$  between these jump points. And at a jump time  $t_i$ , the jump size is  $\alpha$ , leading to  $\lambda_{t_i+} = \lambda_{t_i} + \alpha$ . Iteratively solving this ODE between jumps with the initial value  $\lambda_{t_{i-1+}}$ , we establish that the intensity process Eq.(4) satisfies Eq.(9).
+
+Theorem 3. The intensity process  $\{\lambda_t\}_{t\geq 0}$  of the self-correcting process  $\{N_t\}_{t\geq 0}$  can be equivalently expressed as the solution to the jump SDE
+
+$$
+\mathrm {d} \lambda_ {t} = \mu \lambda_ {t} \mathrm {d} t + (e ^ {- \alpha} - 1) \lambda_ {t} \mathrm {d} N _ {t}, \quad \lambda_ {0} = 1. \tag {10}
+$$
+
+The proof of this theorem is similar to the previous one and can be found in Appendix A.2. The following result shows that under certain conditions, there exists a unique positive solution to an SDE, which means that an SDE can determine an intensity process of a TPP.
+
+Theorem 4. Assume that the ODE
+
+$$
+\mathrm {d} y _ {t} = \frac {f \left(e ^ {y _ {t}}\right)}{e ^ {y _ {t}}} \mathrm {d} t, \quad t \geq 0, \quad y _ {0} = y,
+$$
+
+has a unique global solution for every  $y \in \mathbb{R}$  and let  $h(x): \mathbb{R} \to \mathbb{R}$  be a chosen function such that  $h(x) + x > 0$  for  $x > 0$ . Then the jump SDE
+
+$$
+\mathrm {d} \lambda_ {t} = f (\lambda_ {t}) \mathrm {d} t + h (\lambda_ {t}) \mathrm {d} N _ {t}, \quad \lambda_ {0} = \lambda , \tag {11}
+$$
+
+has a unique global positive solution for every  $\lambda > 0$ .
+
+Appendix A.3 includes the detailed proof. Specially, according to this theorem, by setting  $f(x) = \mu x$  and  $h(x) = (e^{-\alpha} - 1)x$ , it follows that Eq.(10) has a unique global positive solution.
+
+From the above equivalent SDE formulations of several classical TPPs, we can clearly see that the coefficient functions within the SDE play a key role in shaping the evolution of intensity processes over time and revealing the influences between events. For example, in Hawkes processes (Eq.(9)), there exists excitatory influences between events, where each occurrence of an event leads to an instantaneous increase in intensity by  $\alpha$ . This inspires us that by defining appropriate coefficient functions, it becomes feasible to construct an intensity process consistent with the observed data. These observations motivate us to propose our model Neural Jump-Diffusion Temporal Point Processes.
+
+# 5. Neural Jump-Diffusion TPPs
+
+In this section, for symbol simplicity and reader comprehension, we first model the intensity process of the univariate TPPs. Subsequently, we extend our method to address the multivariate TPPs, proposing a more comprehensive model.
+
+# 5.1. Neural Jump-Diffusion Univariate Point Process
+
+Unlike classical TPPs with known linear drift and jump coefficient functions, we consider a general problem where the dynamics of the intensity process are completely unknown. Specifically, we assume access to a large set of event sequences, each denoted as  $S = \{t_i\}_{i=1}^n$ , representing independent realizations of a counting process  $\{N_t\}_{t \geq 0}$ . The objective is to identify the unknown dynamics governing the intensity process  $\{\lambda_t\}_{t \geq 0}$  of  $\{N_t\}_{t \geq 0}$ .
+
+To this end, we propose the Neural Jump-Diffusion Univariate Point Process whose intensity process is governed by a neural jump-diffusion SDE (NJDSDE). The drift, diffusion, and jump coefficient functions in the NJDSDE are parameterized by three neural networks which are called the drift net, diffusion net, and jump net, respectively. To ensure that the intensity  $\{\lambda_t\}_{t\geq 0}$  remains positive, we introduce the log-intensity process  $\eta_t\coloneqq \log \lambda_t$ . Then we formally present the NJDSDE for  $\{\eta_t\}_{t\geq 0}$  as follows:
+
+$$
+\left\{ \begin{array}{l} \mathrm {d} \eta_ {t} = \underbrace {f _ {\theta_ {f}} (\eta_ {t})} _ {\text {d r i f t n e t}} \mathrm {d} t + \underbrace {g _ {\theta_ {g}} (\eta_ {t})} _ {\text {d i f f u s i o n n e t}} \mathrm {d} W _ {t} + \underbrace {h _ {\theta_ {h}} (\eta_ {t})} _ {\text {j u m p n e t}} \mathrm {d} N _ {t}, \\ \eta_ {0} = \log \lambda_ {0}, \end{array} \right. \tag {12}
+$$
+
+where  $\eta_0\in \mathbb{R}$  is the initial value,  $f_{\theta_f}:\mathbb{R}\to \mathbb{R},g_{\theta_g}:\mathbb{R}\to$ $\mathbb{R}$ $h_{\theta_h}:\mathbb{R}\rightarrow \mathbb{R}$ $\{W_t\}_{t\geq 0}$  is a standard Brownian motion (Le Gall, 2016), and  $\{N_t\}_{t\geq 0}$  is the counting process mentioned above, which records the occurrence of events. Suppose that  $\{W_t\}_{t\geq 0}$  and  $\{N_t\}_{t\geq 0}$  are independent. We explain each term in Eq.(12) in detail:
+
+- The drift term  $f_{\theta_f}(\eta_t) \, \mathrm{d}t$  captures the intrinsic evolution of  $\{\eta_t\}_{t \geq 0}$ .  
+- The diffusion term  $g_{\theta_g}(\eta_t) \, \mathrm{d}W_t$  models the Gaussian noise with the Brownian motion. Inspired by (Wang et al., 2018), we add the diffusion term to model the impact of noise on the intensity process.  
+- The jump term  $h_{\theta_h}(\eta_t) \, \mathrm{d}N_t$  represents the magnitude of the jump, capturing the influence of historical events up to time  $t$ . Its sign indicates whether the influence is excitatory or inhibitory.
+
+The proposed NJDSDE Eq.(12) is a general framework. When the function  $f_{\theta_f}$  is set to  $\beta (\mu e^{-\eta_t} - 1)$ ,  $g_{\theta_g}$  is set to 0, and  $h_{\theta_h}$  is set to  $\log (1 + \alpha e^{-\eta_t})$  in Eq.(12), the NJDSDE characterizes Hawkes processes. Similarly, when  $f_{\theta_f}$  is set
+
+to  $\mu$ ,  $g_{\theta_g}$  is set to 0, and  $h_{\theta_h}$  is set to  $-\alpha$ , Eq.(12) characterizes self-correcting processes. The similar results for Poisson processes are trivial. In other words, the proposed NJDSDE encompasses the classical TPPs mentioned above. In addition, a specific class (but not all) of log-Gaussian Cox processes (Møller et al., 1998) can also be incorporated into our modeling framework Eq.(12). The proofs for these conclusions are detailed in Appendix A.4.
+
+We proceed to investigate the existence and uniqueness of the solution  $\{\eta_t\}_{t\geq 0}$  to the proposed NJDSDE. The theoretical analysis in the following theorem provides insights into designing an effective network architecture for the drift net  $f_{\theta_f}$ , diffusion net  $g_{\theta_g}$ , and jump net  $h_{\theta_h}$ .
+
+Theorem 5. Assuming that  $f_{\theta_f}(x), g_{\theta_g}(x), h_{\theta_h}(x)$  are measurable functions  $\mathbb{R} \to \mathbb{R}$ ,  $h_{\theta_h}(x)$  is continuous, and there exists a positive constant  $C$  such that for all  $x, y \in \mathbb{R}$ ,
+
+$$
+\left| f _ {\theta_ {f}} (x) - f _ {\theta_ {f}} (y) \right| + \left| g _ {\theta_ {g}} (x) - g _ {\theta_ {g}} (y) \right| \leq C | x - y |,
+$$
+
+then for every  $\lambda_0 > 0$ , there exists a unique adapted left-continuous process  $\{\eta_t\}_{t\geq 0}$  with right-limits that satisfies Eq.(12).
+
+The proof is available in Appendix A.5. According to Theorem 5, if  $f_{\theta_f}(x), g_{\theta_g}(x)$  and  $h_{\theta_h}(x)$  are uniformly Lipschitz continuous, then Eq.(12) has a unique strong solution. Thus, we utilize Lipschitz nonlinear activations, such as ReLU, sigmoid, and Tanh, within the network architectures, as highlighted in previous works (Anil et al., 2019; Kong et al., 2020; Oh et al., 2024; Lin et al., 2024). Moreover, in this paper, the drift net, diffusion net, and jump net are implemented as three multi-layer perceptrons (MLPs).
+
+Remarks. We summarize the differences of our model compared to existing TPP models:
+
+- Different from the SDE formulation of classical TPPs (e.g., Eq.(9)), the coefficient functions in our model are parameterized by neural networks rather than relying on fixed functions. This enables a more flexible modeling of the complex dynamics of the intensity process.  
+- Compared to neural TPPs (Du et al., 2016; Zuo et al., 2020), our model eliminates the need to assume a specific functional form for the intensity function. Instead, based on the NJDSDE, our model formulates the time evolution of the intensity process in a general manner.  
+- Furthermore, our model differs from previous TPP models based on neural differential equations (Jia & Benson, 2019; Chen et al., 2020; Song et al., 2024). In addition to incorporating the Brownian motion to model the Gaussian noise, a key distinction lies in our proposed NJDSDE, which models the dynamics of the intensity process rather than the hidden state.
+
+# 5.2. Model Training
+
+To learn model parameters in  $f_{\theta_f}, g_{\theta_g}, h_{\theta_h}$ , and the initial value  $\eta_0$ , we perform the Maximum Likelihood Estimation (MLE). For an event sequence  $S = \{t_i\}_{i=1}^n$  over the time interval  $[0, T]$ , given its intensity  $\lambda_t$ , the log-likelihood function (Rasmussen, 2018) is
+
+$$
+\ell (\mathcal {S}) = \sum_ {i = 1} ^ {n} \log \lambda_ {t _ {i}} - \int_ {0} ^ {T} \lambda_ {s} d s = \sum_ {i = 1} ^ {n} \eta_ {t _ {i}} - \int_ {0} ^ {T} e ^ {\eta_ {s}} d s.
+$$
+
+In general, the integral term does not have a closed-form computational method. Therefore, we apply numerical integration methods for approximate calculations, such as the trapezoidal rule (Zuo et al., 2020). This requires determining the value of  $\eta_t$  at the divided time points. Noting that this process  $\{\eta_t\}_{t \geq 0}$  is governed by our proposed NJDSDE Eq.(12). That is, on the time interval  $(t_{i-1}, t_i]$ ,  $\eta_t$  is governed by the neural SDE
+
+$$
+\mathrm {d} \eta_ {t} = f _ {\theta_ {f}} (\eta_ {t}) \mathrm {d} t + g _ {\theta_ {g}} (\eta_ {t}) \mathrm {d} W _ {t}. \tag {13}
+$$
+
+And at a jump time  $t_i$ , the jump size of  $\eta_t$  is given by
+
+$$
+\Delta \eta_ {t _ {i}} = \eta_ {t _ {i} +} - \eta_ {t _ {i}} = h _ {\theta_ {h}} \left(\eta_ {t _ {i}}\right) \Delta N _ {t _ {i}} = h _ {\theta_ {h}} \left(\eta_ {t _ {i}}\right). \tag {14}
+$$
+
+Then the right-limit of  $\eta_t$  at  $t_i$  is
+
+$$
+\eta_ {t _ {i} +} = \eta_ {t _ {i}} + h _ {\theta_ {h}} \left(\eta_ {t _ {i}}\right). \tag {15}
+$$
+
+Since the solution of neural SDEs (e.g., Eq.(13)) is generally analytically intractable, numerical approximation methods are often required (Kong et al., 2020; Kidger et al., 2021b). We adopt the Euler-Maruyama scheme (Kloeden & Platen, 1992) with fixed step size due to its computational efficiency. Under such a scheme, the time interval  $(t_{i-1}, t_i]$  is divided into  $N$  subintervals  $t_{i-1} = \tau_0^i < \dots < \tau_k^i < \dots < \tau_N^i = t_i$  with stepsize  $\Delta_k^i = \tau_{k+1}^i - \tau_k^i = (t_i - t_{i-1}) / N$ . Then we discretize Eq.(13) on  $(t_{i-1}, t_i]$  by the recursive equation
+
+$$
+\eta_ {\tau_ {k + 1} ^ {i}} = \eta_ {\tau_ {k} ^ {i}} + f _ {\theta_ {f}} \left(\eta_ {\tau_ {k} ^ {i}}\right) \Delta_ {k} ^ {i} + g _ {\theta_ {g}} \left(\eta_ {\tau_ {k} ^ {i}}\right) \Delta W _ {k} ^ {i}, \tag {16}
+$$
+
+for  $k = 0,1,\ldots ,N - 1$  with  $\eta_{\tau_0^i} = \eta_{t_{i - 1} + }$  . Here,  $\Delta W_k^i =$ $W_{\tau_{k + 1}^i} - W_{\tau_k^i}$  is sampled from  $\mathcal{N}(0,\Delta_k^i)$  for numerical computation. The advantage of introducing the log-intensity  $\eta_t = \log \lambda_t$  is evident in obtaining a numerical solution of Eq.(12) over the entire real number space, rather than being restricted to the domain of positive real numbers.
+
+Iteratively using Eq.(15) and Eq.(16), we can calculate the log-likelihood function as follows:
+
+$$
+\ell (\mathcal {S}) = \sum_ {i = 1} ^ {n} \eta_ {t _ {i}} - \sum_ {i = 1} ^ {n + 1} \sum_ {k = 1} ^ {N} \frac {\tau_ {k} ^ {i} - \tau_ {k - 1} ^ {i}}{2} \left(e ^ {\eta_ {\tau_ {k - 1} ^ {i}}} + e ^ {\eta_ {\tau_ {k} ^ {i}}}\right), \tag {17}
+$$
+
+where  $\tau_0^1 = 0$ ,  $\tau_N^{n + 1} = T$ ,  $\eta_{\tau_0^i} = \eta_{t_{i - 1}+}$  and  $\eta_{\tau_N^i} = \eta_{t_i}$ . The complete algorithm of model training is described in Algorithm 1 in Appendix B.
+
+# 5.3. Neural Jump-Diffusion Multivariate Point Process
+
+An important example of multivariate TPPs is the multivariate Hawkes process  $N_{t} = (N_{t}^{1},\dots ,N_{t}^{M})^{\mathrm{T}}$  whose intensity process  $\lambda_{t} = (\lambda_{t}^{1},\dots ,\lambda_{t}^{M})^{\mathrm{T}}$  characterizes the past event influences on future ones in an excitatory manner (Hawkes, 1971):
+
+$$
+\lambda_ {t} ^ {m} = \mu_ {0} ^ {m} + \sum_ {l = 1} ^ {M} \sum_ {t _ {i} <   t, m _ {i} = l} \alpha^ {m l} \exp \left(- \beta \left(t - t _ {i}\right)\right), m \in [ M ],
+$$
+
+where  $[M] := \{1, 2, \dots, M\}$ . For this intensity process, we derive the following equivalent SDE formulation.
+
+Theorem 6. The intensity process  $\{\lambda_t\}_{t\geq 0}$  of the multivariate Hawkes process  $\{N_{t}\}_{t\geq 0}$  can be equivalently expressed as the solution to the jump SDEs
+
+$$
+\left\{ \begin{array}{l} \mathrm {d} \lambda_ {t} ^ {m} = \beta \left(\mu_ {0} ^ {m} - \lambda_ {t} ^ {m}\right) \mathrm {d} t + \sum_ {l = 1} ^ {M} \alpha^ {m l}   \mathrm {d} N _ {t} ^ {l}, \\ \lambda_ {0} ^ {m} = \mu_ {0} ^ {m}, \quad m \in [ M ]. \end{array} \right.
+$$
+
+The proof follows a similar approach to that given earlier for Theorem 2, and is omitted here.
+
+Subsequently, we extend our approach to identify the unknown dynamics of general multivariate point processes. Let  $S = \{(t_i,m_i)\}_{i = 1}^n$  be a multi-type event sequence, where each event  $(t_i,m_i)$  indicates that the  $i$ -th event occurs at time  $t_i\in \mathbb{R}_+$  and is of type  $m_{i}\in [M]$ . We still denote  $\pmb {N}_t = (N_t^1,\dots ,N_t^M)^\mathrm{T}$  the associated multivariate counting process. Similar to Section 5.1, we introduce the  $M$ -dimensional log-intensity process  $\pmb {\eta}_t\coloneqq \log \pmb {\lambda}_t = (\log \lambda_t^1,\dots ,\log \lambda_t^M)^\mathrm{T}$ . For each  $m\in [M]$ , we propose the NJDSDE for  $\{\eta_t^m\}_{t\geq 0}$  as follows:
+
+$$
+\mathrm {d} \eta_ {t} ^ {m} = f ^ {m} (\boldsymbol {\eta} _ {t}) \mathrm {d} t + \sum_ {k = 1} ^ {K} g ^ {m k} (\boldsymbol {\eta} _ {t}) \mathrm {d} W _ {t} ^ {k} + \sum_ {l = 1} ^ {M} h ^ {m l} (\boldsymbol {\eta} _ {t}) \mathrm {d} N _ {t} ^ {l},
+$$
+
+where the superscripts are used for the indices of vectors and matrices, such as the function  $g^{mk}$  and  $h^{ml}$  are the  $(m,k)$ -th component of the  $M \times K$ -matrix  $\pmb{g}_{\theta_g}$  and the  $(m,l)$ -th component of the  $M \times M$ -matrix  $\pmb{h}_{\theta_h} = [h_{\theta_h}^1 |\dots |h_{\theta_h}^M]$  with  $h_{\theta_h}^l$  as its  $l$ -th column vector, respectively. Moreover, the components  $W_t^k$  of  $\pmb{W}_t = (W_t^1,\dots ,W_t^K)^{\mathrm{T}}$  are standard Brownian motions which are pairwise independent. Suppose that  $\pmb{W}_t$  and  $\pmb{N}_t$  are independent.
+
+We rewrite the above componentwise expression into vector form, leading to the formulation of the NJDSDE for  $\{\eta_t\}_{t\geq 0}$  as follows:
+
+$$
+\left\{ \begin{array}{l} \mathrm {d} \boldsymbol {\eta} _ {t} = \underbrace {\boldsymbol {f} _ {\theta_ {f}} (\boldsymbol {\eta} _ {t})} _ {\text {d r i f t n e t}} \mathrm {d} t + \underbrace {\boldsymbol {g} _ {\theta_ {g}} (\boldsymbol {\eta} _ {t})} _ {\text {d i f f u s i o n n e t}} \mathrm {d} \boldsymbol {W} _ {t} + \underbrace {\boldsymbol {h} _ {\theta_ {h}} (\boldsymbol {\eta} _ {t})} _ {\text {j u m p n e t}} \mathrm {d} \boldsymbol {N} _ {t}, \\ \boldsymbol {\eta} _ {0} = \log \boldsymbol {\lambda} _ {0}, \end{array} \right. \tag {18}
+$$
+
+![](images/4698573b7d82d1508efe8e3e25b39d3639c013d56efab149214c7c8da7904b94.jpg)  
+Figure 1. Illustration of the NJDSDE-governed intensity process with three event types. This example shows that the intensity process behaves as a diffusion process between event points, with a jump occurring at these specific points.
+
+where  $\pmb{\eta}_0\in \mathbb{R}^M$  is the initial value, the coefficient functions  $f_{\theta_f}:\mathbb{R}^M\to \mathbb{R}^M$ $g_{\theta_g}:\mathbb{R}^M\to \mathbb{R}^{M\times K}$  and  $h_{\theta_h}:\mathbb{R}^M\to \mathbb{R}^{M\times M}$  are the drift net, diffusion net, and jump net, respectively. In particular, when  $M = K = 1$  this NJDSDE degenerates into Eq.(12). Essentially, Eq.(18) behaves as a neural SDE between event points, with a jump occurring at these points. This can be interpreted as:
+
+$$
+\left\{ \begin{array}{l} \mathrm {d} \boldsymbol {\eta} _ {t} = \boldsymbol {f} _ {\boldsymbol {\theta} _ {f}} (\boldsymbol {\eta} _ {t}) \mathrm {d} t + \boldsymbol {g} _ {\boldsymbol {\theta} _ {g}} (\boldsymbol {\eta} _ {t}) \mathrm {d} \boldsymbol {W} _ {t}, \quad t \in (t _ {i - 1}, t _ {i} ], \\ \boldsymbol {\eta} _ {t _ {i} +} = \boldsymbol {\eta} _ {t _ {i}} + \boldsymbol {h} _ {\boldsymbol {\theta} _ {h}} ^ {m _ {i}} (\boldsymbol {\eta} _ {t _ {i}}), \quad t = t _ {i}, m = m _ {i}. \end{array} \right. \tag {19}
+$$
+
+It is evident from this expression that the numerical value of  $l$ -th element in vector  $h_{\theta_h^{m_i}}$  characterizes the influence magnitude of event type  $m_i$  on event type  $l$ , with the sign reflecting the excitatory or inhibitory influence.
+
+As discussed in Section 5.1, we employ Lipschitz continuous activations in the neural networks  $f_{\theta_f}, g_{\theta_g}$ , and  $h_{\theta_h}$ . This ensures the existence of a unique strong solution to Eq.(18). We also apply MLE to learn model parameters and provide the complete training algorithm in Appendix B. Figure 1 presents a concrete example of the intensity process governed by the NJDSDE Eq.(18).
+
+# 5.4. Event Prediction
+
+With the proposed NJDSDE-governed intensity process, we aim to predict the next event time and next event type given the historical events  $\mathcal{H}_{t_{n + 1}} = \{(t_i,m_i)\}_{i = 1}^n$ .
+
+The conditional density function of the next event time (Rasmussen, 2018) is that for  $t \geq t_n$ ,
+
+$$
+f _ {t _ {n + 1}} (t) = \lambda_ {t} ^ {\mathrm {g}} \exp \left(- \int_ {t _ {n}} ^ {t} \lambda_ {s} ^ {\mathrm {g}} \mathrm {d} s\right), \tag {20}
+$$
+
+where  $\lambda_t^{\mathrm{g}}\coloneqq \sum_{m = 1}^{M}\lambda_t^m = \sum_{m = 1}^{M}\exp (\eta_t^m)$ . Then for the next event time prediction, we employ the formula
+
+$$
+\widehat {t} _ {n + 1} = \int_ {t _ {n}} ^ {\infty} t f _ {t _ {n + 1}} (t) \mathrm {d} t. \tag {21}
+$$
+
+Note that after the jump time  $t_n$ , the process  $\pmb{\eta}_t$  governed by Eq.(18) will have the dynamics
+
+$$
+\left\{ \begin{array}{l} \mathrm {d} \boldsymbol {\eta} _ {t} = \boldsymbol {f} _ {\boldsymbol {\theta} _ {f}} (\boldsymbol {\eta} _ {t}) \mathrm {d} t + \boldsymbol {g} _ {\boldsymbol {\theta} _ {g}} (\boldsymbol {\eta} _ {t}) \mathrm {d} \boldsymbol {W} _ {t}, \quad t > t _ {n}, \\ \boldsymbol {\eta} _ {t _ {n} +} = \boldsymbol {\eta} _ {t _ {n}} + \boldsymbol {h} _ {\boldsymbol {\theta} _ {h}} ^ {m _ {n}} (\boldsymbol {\eta} _ {t _ {n}}). \end{array} \right. \tag {22}
+$$
+
+Therefore, similar to the method discussed in Section 5.2 for computing the log-likelihood function, we utilize the Euler-Maruyama scheme to discretize Eq.(22), followed by numerical integration techniques to compute the integrals mentioned above. Here, the value of  $\eta_{t_n}$  required for discretizing Eq.(22) can be obtained by discretizing Eq.(18) over the interval  $[0, t_n]$  using the Euler-Maruyama scheme and the historical events  $\mathcal{H}_{t_{n+1}}$ .
+
+Following previous works (Zuo et al., 2020; Shi et al., 2023; Xue et al., 2024), the next event type prediction is given by
+
+$$
+\widehat {m} _ {n + 1} = \operatorname {a r g m a x} _ {m} \lambda_ {t _ {n + 1}} ^ {m} / \lambda_ {t _ {n + 1}} ^ {\mathrm {g}}. \tag {23}
+$$
+
+# 6. Experiments
+
+We first test the flexibility of our NJDTPP model by recovering the ground truth dynamics of the intensity process of classical TPPs. Then, we evaluate the modeling capability for event sequences and the prediction performance of NJDTPP on six real-world datasets. Our code is available at https://github.com/Zh-Shuai/NJDTPP.
+
+# 6.1. Intensity Process Recovery for Classical TPPs
+
+Synthetic Datasets. We consider the following classical TPPs: (i) Poisson Process: the intensity is given by  $\lambda_{t} = \lambda_{0}$ , where  $\lambda_0 = 1.0$ ; (ii) Hawkes Process: the intensity is given by  $\lambda_{t} = \mu + \alpha \sum_{t_{i} < t} \exp(-\beta(t - t_{i}))$ , where  $\mu = 0.2$ ,  $\alpha = 0.8$ ,  $\beta = 1.0$ ; and (iii) Self-Correcting Process: the intensity is given by  $\lambda_{t} = \exp(\mu t - \sum_{t_{i} < t} \alpha)$ , where  $\mu = 0.5$ ,  $\alpha = 0.2$ . For each TPP, we simulate a dataset using the Ogata's thinning algorithm (Ogata, 1981). Each dataset contains 500 event sequences within the time interval [0, 100]. The train-validation-test data split is  $3:1:1$ .
+
+Experimental Setup. We fit our NJDTPP model to each dataset using the training procedure described in Section 5.2. In this experiment, the drift, diffusion, and jump nets are implemented as three MLPs, each with 2 hidden layers of 32 units. The activation function chosen for these networks is Tanh. More training details are reported in Appendix C.5. For evaluation, we visually demonstrate the similarity between the estimated intensity from the learned NJDTPP and the ground truth intensity.
+
+In addition, we compare the mean absolute percentage error (MAPE) of the estimated intensity of our model with the Poisson process (PP) model, the Hawkes process (HP) model, the self-correcting process (SC) model, an RNN-based model (Jia & Benson, 2019), and the Neural Jump SDE (NJSDE, Jia & Benson (2019)) model. The baseline results in Table 1 are extracted from (Jia & Benson, 2019). Similar to the calculation of the log-likelihood function, we first numerically solve Eq.(12) using the Euler-Maruyama scheme, and then compute MAPE through numerical integration. See Appendix C.4 for the definition of MAPE.
+
+![](images/e343a1b4f82724b6c155ca104d810fedf5edcc26ca3e9450cb941daf44395148.jpg)
+
+![](images/cb9f52ff57bfeb28af6c75f39a00661c34258ccbd7ead64de084b4447f0b1830.jpg)
+
+![](images/d2a99a191e47e76758414a09f900d1cc1dd2366b7a6d944c8fae38371a8faf92.jpg)  
+Figure 2. The estimated and ground truth intensity process of the Poisson Process, Hawkes Process, and Self-Correcting Process. Each blue cross represents an event at the corresponding time.
+
+Table 1. The MAPE comparison of the estimated intensity. Each row represents a synthetic dataset. Each column represents a model.  
+
+<table><tr><td></td><td>PP</td><td>HP</td><td>SC</td><td>RNN</td><td>NJSDE</td><td>NJDTPP</td></tr><tr><td>Poisson</td><td>0.1</td><td>0.3</td><td>98.7</td><td>3.2</td><td>1.3</td><td>0.1</td></tr><tr><td>Hawkes</td><td>188.2</td><td>3.5</td><td>101.0</td><td>22.0</td><td>5.9</td><td>0.2</td></tr><tr><td>Self-Correcting</td><td>29.1</td><td>29.1</td><td>1.6</td><td>24.3</td><td>9.3</td><td>0.1</td></tr></table>
+
+Results. Figure 2 compares the estimated intensity process (red curve) of our NJDTPP model with the ground truth (grayish purple curve). It clearly shows that NJDTPP effectively recovers the dynamics of ground truth intensities. This also indicates that our model can capture the excitatory and inhibitory influences between events. Table 1 reports the estimated MAPE for NJDTPP and the baseline models. These values are the average results for all 100 event sequences in the test set. As shown, our model fits the data well and shows a substantial performance improvement compared to baselines.
+
+Having observed that NJDTPP achieves superior experimental results, we now turn to analyze the two main factors that contributed to this success. Firstly, as discussed in Section 5.1, these classical TPPs are special cases of our modeling framework. Specifically, when the drift, diffusion, and jump functions take certain forms, our proposed NJDSDE Eq.(12) characterizes these classical TPPs. Secondly, we employ neural networks to parameterize the drift, diffusion, and jump functions in Eq.(12). Due to the powerful capability of neural networks, our model can fit the data well and effectively recover the ground truth intensity.
+
+![](images/a025431ad7da0cb52ed59d687df56dd91d2fff0f79f9a03226bcdc5333c330a7.jpg)
+
+![](images/ecfced97f1378923c6f7df100e11884b3449ac9dfaa286088e099d1a19a2abf0.jpg)  
+Figure 3. Event time prediction RMSE comparison.
+
+![](images/e9917f4424bc53858e82bc8fc8fd10a9bb518f4a15308957577aca725309d6aa.jpg)
+
+![](images/cbcba3696b66d99c8a822ee7fe0920c789a0ad7cdbfacc56b4018f26cf6f42e3.jpg)
+
+![](images/0456dbf518705e66ae5261c58c5de528c65fc81d047b9cc5eeebec690dc0bcad.jpg)
+
+![](images/6fa7fa917602f85d434e07619561a75a9061c4a91e46a15f8208fee0708ee311.jpg)
+
+Table 2. Event type prediction accuracy and  $F_{1}$  comparison.  
+
+<table><tr><td rowspan="2">Model</td><td colspan="2">Retweet</td><td colspan="2">Earthquake</td><td colspan="2">Taxi</td><td colspan="2">Taobao</td><td colspan="2">StackOverflow</td><td colspan="2">MIMIC-II</td></tr><tr><td>Acc</td><td>F1</td><td>Acc</td><td>F1</td><td>Acc</td><td>F1</td><td>Acc</td><td>F1</td><td>Acc</td><td>F1</td><td>Acc</td><td>F1</td></tr><tr><td>RMTPP</td><td>0.503</td><td>0.415</td><td>0.214</td><td>0.082</td><td>0.836</td><td>0.815</td><td>0.431</td><td>0.429</td><td>0.437</td><td>0.265</td><td>0.812</td><td>0.600</td></tr><tr><td>NHP</td><td>0.601</td><td>0.573</td><td>0.451</td><td>0.283</td><td>0.890</td><td>0.886</td><td>0.463</td><td>0.447</td><td>0.467</td><td>0.315</td><td>0.832</td><td>0.680</td></tr><tr><td>SAHP</td><td>0.522</td><td>0.497</td><td>0.417</td><td>0.246</td><td>0.847</td><td>0.802</td><td>0.442</td><td>0.320</td><td>0.452</td><td>0.301</td><td>0.827</td><td>0.639</td></tr><tr><td>THP</td><td>0.536</td><td>0.375</td><td>0.451</td><td>0.281</td><td>0.865</td><td>0.828</td><td>0.467</td><td>0.406</td><td>0.463</td><td>0.309</td><td>0.853</td><td>0.596</td></tr><tr><td>AttNHP</td><td>0.592</td><td>0.575</td><td>0.452</td><td>0.283</td><td>0.761</td><td>0.724</td><td>0.458</td><td>0.386</td><td>0.465</td><td>0.310</td><td>0.856</td><td>0.817</td></tr><tr><td>NSTPP</td><td>0.513</td><td>0.482</td><td>0.354</td><td>0.165</td><td>0.884</td><td>0.837</td><td>0.437</td><td>0.289</td><td>0.449</td><td>0.293</td><td>0.840</td><td>0.799</td></tr><tr><td>NJDTPP</td><td>0.608</td><td>0.584</td><td>0.472</td><td>0.313</td><td>0.908</td><td>0.891</td><td>0.486</td><td>0.452</td><td>0.469</td><td>0.327</td><td>0.861</td><td>0.823</td></tr></table>
+
+Table 3. Negative log-likelihood comparison.  
+
+<table><tr><td>Model</td><td>Retweet</td><td>Earthquake</td><td>Taxi</td><td>Taobao</td><td>StackOverflow</td><td>MIMIC-II</td></tr><tr><td>RMTPP</td><td>4.241</td><td>3.653</td><td>0.227</td><td>1.659</td><td>2.891</td><td>2.333</td></tr><tr><td>NHP</td><td>4.137</td><td>2.189</td><td>0.208</td><td>0.986</td><td>2.496</td><td>2.205</td></tr><tr><td>SAHP</td><td>5.009</td><td>3.941</td><td>0.478</td><td>1.640</td><td>2.952</td><td>3.394</td></tr><tr><td>THP</td><td>4.560</td><td>3.387</td><td>0.442</td><td>1.191</td><td>2.630</td><td>1.515</td></tr><tr><td>AttNHP</td><td>4.756</td><td>2.376</td><td>0.491</td><td>1.206</td><td>2.586</td><td>1.697</td></tr><tr><td>NSTPP</td><td>4.527</td><td>2.203</td><td>0.217</td><td>1.415</td><td>2.541</td><td>2.421</td></tr><tr><td>NJDTPP</td><td>4.092</td><td>1.305</td><td>-0.293</td><td>-0.440</td><td>2.347</td><td>1.398</td></tr></table>
+
+# 6.2. Likelihood Evaluation and Event Prediction
+
+We use the negative log-likelihood (NLL) as a metric to evaluate the ability of NJDTPP in modeling event sequences on real-world datasets. Moreover, we evaluate the performance of NJDTPP in the standard next-event prediction task in TPPs, predicting every next event  $(t_i, m_i)$  given the history  $\mathcal{H}_{t_i}$ . We use Eq.(21) and Eq.(23) to predict the next event time and type, respectively. We evaluate event time prediction by Root Mean Square Error (RMSE) and event type prediction by accuracy and the weighted  $F_1$  score. The training details and the configuration of our NJDTPP model are provided in Appendix C.5 and Appendix C.6.
+
+Datasets. We evaluate our model on six real-world benchmark datasets: Retweet (Zhou et al., 2013), Earthquake (Xue et al., 2024), Taxi (Whong, 2014), Taobao (Xue et al., 2022), StackOverflow (Leskovec & Krevl, 2014), and MIMIC-II (Johnson et al., 2016). The MIMIC-II dataset is available at the public Github repository $^2$ , and all other datasets are available at the public EasyTPP $^3$  library (Xue et al., 2024), an open benchmark for evaluating TPPs. See Appendix C.2 for dataset details.
+
+Table 4. Performance of the NJDTPP variant on Earthquake and Taxi. NJDTPP-BM refers to the variant without Brownian motion.  
+
+<table><tr><td rowspan="2">Model</td><td colspan="4">Earthquake</td><td colspan="4">Taxi</td></tr><tr><td>NLL</td><td>RMSE</td><td>Acc</td><td>F1</td><td>NLL</td><td>RMSE</td><td>Acc</td><td>F1</td></tr><tr><td>NJDTPP-BM</td><td>1.594</td><td>1.507</td><td>0.470</td><td>0.306</td><td>-0.290</td><td>0.302</td><td>0.905</td><td>0.887</td></tr><tr><td>NJDTPP</td><td>1.305</td><td>1.420</td><td>0.472</td><td>0.313</td><td>-0.293</td><td>0.296</td><td>0.908</td><td>0.891</td></tr></table>
+
+Baselines. We compare NJDTPP with the following models. Two RNN-based models: Recurrent Marked Temporal Point Process (RMTPP, Du et al. (2016)) and Neural Hawkes Process (NHP, Mei & Eisner (2017)). Three attention-based models: Self-Attentive Hawkes Process (SAHP, Zhang et al. (2020a)), Transformer Hawkes Process (THP, Zuo et al. (2020)), and Attentive Neural Hawkes Process (AttNHP, Yang et al. (2022)). One TPP with the hidden state governed by a neural jump SDE: Neural Spatio-Temporal Point Process (NSTPP, Chen et al. (2020)). More details in Appendix C.3. For the implementation of baselines, we use the code from EasyTPP $^{3}$  (Xue et al., 2024), and extract partial baseline results from the same paper.
+
+Results. Table 3 summarizes the per-event NLL of these models on each test set. From this table, we can observe that NJDTPP fits the data well and significantly outperforms baselines across all experiments. This demonstrates our model's capability to learn complex real-world intensity dynamics. The results for next event time and event type prediction are presented in Figure 3 and Table 2 respectively. It is evident that, in each dataset, NJDTPP outperforms all competing models, often by a substantial margin. This shows the superior performance of our model in prediction tasks. The success of NJDTPP can be attributed to its flexibility in capturing complex intensity dynamics and the influences between events.
+
+# 6.3. Ablation Study
+
+We conduct an ablation study on the Earthquake and Taxi datasets, investigating the variant of NJDTPP by removing the diffusion term. We evaluate models based on NLL and prediction performance. Table 4 reports the experimental results. As shown, the diffusion term contributes to model performance due to the fact that it models the Gaussian noise with the Brownian motion.
+
+# 7. Conclusion
+
+We have presented Neural Jump-Diffusion Temporal Point Processes, a unified TPP framework that can learn a freeform intensity process consistent with the observed event data. By modeling the intensity process as the solution to an SDE, our approach eliminates the need to pre-specified the functional form of the intensity function, thereby significantly enhancing the flexibility and capability of TPP models. Experimental results show that our model effectively captures intensity dynamics and the influences between events, as well as achieves state-of-the-art results on benchmark datasets in likelihood evaluation and event prediction tasks.
+
+# Acknowledgements
+
+We would like to thank the anonymous reviewers for their valuable comments and suggestions. This work was partially supported by the National Key Research and Development Program of China (2021YFB3100600), the Strategic Priority Research Program of the Chinese Academy of Sciences (XDB0680101), the NSFC (62376064, U2336202), and the CAS Project for Young Scientists in Basic Research (YSBR-008).
+
+# Impact Statement
+
+We develop a novel framework for modeling the intensity process of temporal point processes using neural jump-diffusion stochastic differential equations. We hope that our study will inspire new developments in the field of temporal point processes. There are many potential societal consequences of our work, none which we feel must be specifically highlighted here.
+
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+
+# Appendix
+
+# A. Proofs
+
+# A.1. Proof of Theorem 2
+
+Proof. To solve the SDE Eq.(9), we first denote the jump times of the Hawkes process  $\{N_t\}_{t \geq 0}$  as  $\{t_i\}_{i=1}^{\infty}$ . On the time interval  $(t_{i-1}, t_i]$ , Eq.(9) behaves as an ODE  $\mathrm{d}\lambda_t = \beta (\mu - \lambda_t)\mathrm{d}t$ . On the other hand, at a jump time  $t_i$ , the jump size of  $\{\lambda_t\}_{t \geq 0}$  is given by  $\Delta \lambda_{t_i} = \lambda_{t_i+} - \lambda_{t_i} = \alpha \Delta N_{t_i} = \alpha$ . The right-limit at  $t_i$  is then  $\lambda_{t_i+} = \lambda_{t_i} + \alpha$ .
+
+Up to the first jump time  $t_1$ , the solution follows the ODE  $\mathrm{d}\lambda_t = \beta (\mu -\lambda_t)\mathrm{d}t$  on  $[0,t_1]$  with the initial value  $\lambda_0 = \mu$ . Solving this ODE yields  $\lambda_{t} = \mu -C_{1}e^{-\beta t}$ , where  $C_1$  is a constant. Using the initial condition  $\lambda_0 = \mu$ , we find that  $C_1 = 0$ . Thus, we have  $\lambda_{t} = \mu$  for all  $t\in [0,t_1]$ . In particular,  $\lambda_{t_1} = \mu$  and  $\lambda_{t_1 + } = \lambda_{t_1} + \alpha = \mu +\alpha$
+
+Subsequently, we solve the ODE  $\mathrm{d}\lambda_{t} = \beta (\mu -\lambda_{t})\mathrm{d}t$  on  $(t_1,t_2]$  with  $\lambda_{t_1 + } = \mu +\alpha$  to obtain  $\lambda_{t} = \mu -C_{2}e^{-\beta t}$ , where  $C_2$  is a constant. Using  $\lambda_{t_1 + } = \mu +\alpha$  and taking the right-limit at  $t_1$  in the above equation, we obtain  $\lambda_{t_1 + } = \mu +\alpha = \mu -C_2e^{-\beta t_1}$ . Then  $C_2 = -\alpha e^{\beta t_1}$ . Thus, we get  $\lambda_{t} = \mu +\alpha e^{-\beta (t - t_{1})}$  for  $t\in (t_1,t_2]$ . This allows us to determine  $\lambda_{t_2} = \mu +\alpha e^{-\beta (t_2 - t_1)}$  and  $\lambda_{t_2 + } = \lambda_{t_2} + \alpha = \mu +\alpha e^{-\beta (t_2 - t_1)} + \alpha$ .
+
+Iterating this procedure, the solution is given by  $\lambda_{t} = \mu +\alpha \sum_{i:t_{i} < t}\exp (-\beta (t - t_{i}))$  , which is the intensity process of the Hawkes process.
+
+# A.2. Proof of Theorem 3
+
+Proof. To solve the SDE Eq.(10), we first denote the jump times of the self-correcting process  $\{N_t\}_{t \geq 0}$  as  $\{t_i\}_{i=1}^{\infty}$ . On the time interval  $(t_{i-1}, t_i]$ , Eq.(10) behaves as an ODE  $\mathrm{d}\lambda_t = \mu \lambda_t \mathrm{d}t$ . On the other hand, at a jump time  $t_i$ , the jump size of  $\{\lambda_t\}_{t \geq 0}$  is given by  $\Delta \lambda_{t_i} = \lambda_{t_i+} - \lambda_{t_i} = (e^{-\alpha} - 1)\lambda_{t_i}\Delta N_{t_i} = (e^{-\alpha} - 1)\lambda_{t_i}$ . The right-limit at  $t_i$  is then  $\lambda_{t_i+} = e^{-\alpha}\lambda_{t_i}$ .
+
+Up to the first jump time  $t_1$ , the solution follows the ODE  $\mathrm{d}\lambda_t = \mu \lambda_t \mathrm{d}t$  on  $[0, t_1]$  with the initial value  $\lambda_0 = 1$ . Solving this ODE yields  $\lambda_t = C_1 e^{\mu t}$ , where  $C_1$  is a constant. Using the initial condition  $\lambda_0 = 1$ , we can find that  $C_1 = 1$ . Thus, we have  $\lambda_t = e^{\mu t}$  for  $t \in [0, t_1]$ . In particular,  $\lambda_{t_1} = e^{\mu t_1}$  and  $\lambda_{t_1+} = e^{-\alpha} \lambda_{t_1} = e^{\mu t_1 - \alpha}$ .
+
+Subsequently, we solve the ODE  $\mathrm{d}\lambda_{t} = \mu \lambda_{t}\mathrm{d}t$  on  $(t_1,t_2]$  with  $\lambda_{t_1+} = e^{\mu t_1 - \alpha}$  to obtain  $\lambda_{t} = C_{2}e^{\mu t}$ , where  $C_2$  is a constant. Using  $\lambda_{t_1+} = e^{\mu t_1 - \alpha}$  and taking the right-limit at  $t_1$  in the above equation, we obtain  $\lambda_{t_1+} = e^{\mu t_1 - \alpha} = C_2e^{\mu t_1}$ . Then  $C_2 = e^{-\alpha}$ . Thus, we get  $\lambda_{t} = e^{\mu t - \alpha}$  for  $t\in (t_1,t_2]$ . This allows us to determine  $\lambda_{t_2} = e^{\mu t_2 - \alpha}$  and  $\lambda_{t_2+} = e^{-\alpha}\lambda_{t_2} = e^{\mu t_2 - 2\alpha}$ .
+
+Iterating this procedure, the solution is given by  $\lambda_{t} = \exp (\mu t - \sum_{i:t_{i} <   t}\alpha)$ , which is the intensity process of the self- correcting process.
+
+# A.3. Proof of Theorem 4
+
+Proof. Step 1: For every  $\lambda >0$ , we shall prove the existence and uniqueness of the solution to the SDE given by
+
+$$
+\left\{ \begin{array}{l} \mathrm {d} \eta_ {t} = \frac {f \left(e ^ {\eta_ {t}}\right)}{e ^ {\eta_ {t}}} \mathrm {d} t + \log \frac {e ^ {\eta_ {t}} + h \left(e ^ {\eta_ {t}}\right)}{e ^ {\eta_ {t}}} \mathrm {d} N _ {t}, \\ \eta_ {0} = \log \lambda , \end{array} \right. \tag {24}
+$$
+
+where the logarithmic function is well-defined due to the condition  $h(x) + x > 0$  for all  $x > 0$ .
+
+To initiate the proof, we denote the jump times of  $\{N_t\}_{t\geq 0}$  as  $\{t_i\}_{i = 1}^{\infty}$ . Since the ODE
+
+$$
+\left\{ \begin{array}{l} \mathrm {d} y _ {t} = \frac {f \left(e ^ {y _ {t}}\right)}{e ^ {y _ {t}}} \mathrm {d} t, \quad t \geq 0, \\ y _ {0} = y, \end{array} \right. \tag {25}
+$$
+
+has a unique global solution for every  $y \in \mathbb{R}$ , we can solve the ODE  $\mathrm{d}\eta_t = \frac{f(e^{\eta t})}{e^{\eta t}}\mathrm{d}t$  on the interval  $[0, t_1]$  with the initial value  $\eta_0 = \log \lambda$  for every  $\lambda > 0$ . In particular, we can obtain the value of  $\eta_{t_1}$  and then  $\eta_{t_1+} = \eta_{t_1} + \log \frac{e^{\eta t_1} + h(e^{\eta t_1})}{e^{\eta t_1}} = \log (e^{\eta_{t_1}} + h(e^{\eta_{t_1}}))$ .
+
+Since the ODE Eq.(25) is autonomous, we can similarly solve the ODE  $\mathrm{d}\eta_t = \frac{f(e^{\eta t})}{e^{\eta t}}\mathrm{d}t$  on  $(t_1,t_2]$  with the initial value  $\eta_{t_1 + }$ . This yields  $\eta_{t_2}$  and then  $\eta_{t_2 + } = \eta_{t_2} + \log \frac{e^{\eta t_2} + h(e^{\eta t_2})}{e^{\eta t_2}} = \log (e^{\eta t_2} + h(e^{\eta t_2}))$ .
+
+Iterating this procedure, we have proved that Eq.(24) has a unique global solution for every  $\lambda > 0$ .
+
+Step 2: We now aim to prove that  $\lambda_{t} = e^{\eta_{t}}$  is the unique global positive solution of the SDE Eq.(11), where  $\{\eta_t\}_{t\geq 0}$  is the solution of Eq.(24).
+
+Note that  $\lambda_{t} = e^{\eta_{t}}$  ensures that  $\lambda_{t}$  is positive and  $\lambda_0 = e^{\eta_0} = \lambda$ . According to Eq.(24), between the jumps of  $\{N_t\}_{t\geq 0}$ , the process  $\{\eta_t\}_{t\geq 0}$  follows the dynamics  $\mathrm{d}\eta_t = \frac{f(e^{\eta_t})}{e^{\eta_t}}\mathrm{d}t$ . Therefore, the dynamics of  $\lambda_{t}$  can be expressed as follows:
+
+$$
+\mathrm {d} \lambda_ {t} = \mathrm {d} e ^ {\eta_ {t}} = e ^ {\eta_ {t}} \mathrm {d} \eta_ {t} = f \left(e ^ {\eta_ {t}}\right) \mathrm {d} t = f \left(\lambda_ {t}\right) \mathrm {d} t. \tag {26}
+$$
+
+On the other hand, at a jump time  $t$ , the process  $\{N_t\}_{t \geq 0}$  has a jump size of  $\Delta N_t = N_{t+} - N_t = 1$ , implying that the process  $\{\eta_t\}_{t \geq 0}$  will have a jump of size  $\Delta \eta_t = \log \frac{e^{\eta t} + h(e^{\eta t})}{e^{\eta t}} \Delta N_t = \log \frac{e^{\eta t} + h(e^{\eta t})}{e^{\eta t}}$ . Then  $\eta_{t+} = \eta_t + \Delta \eta_t = \eta_t + \log \frac{e^{\eta t} + h(e^{\eta t})}{e^{\eta t}} = \log (e^{\eta t} + h(e^{\eta t}))$ . Since  $\lambda_t = e^{\eta t}$ , the induced jump size of  $\lambda_t$  is given by
+
+$$
+\Delta \lambda_ {t} = e ^ {\eta_ {t +}} - e ^ {\eta_ {t}} = e ^ {\log \left(e ^ {\eta_ {t}} + h \left(e ^ {\eta_ {t}}\right)\right)} - e ^ {\eta_ {t}} = h \left(e ^ {\eta_ {t}}\right) = h \left(\lambda_ {t}\right). \tag {27}
+$$
+
+Combining Eq.(26) and Eq.(27), the equation holds:  $\mathrm{d}\lambda_{t} = f(\lambda_{t})\mathrm{d}t + h(\lambda_{t})\mathrm{d}N_{t}$ . Therefore, we establish that the SDE Eq.(11) given by
+
+$$
+\left\{ \begin{array}{l} \mathrm {d} \lambda_ {t} = f \left(\lambda_ {t}\right)   \mathrm {d} t + h \left(\lambda_ {t}\right)   \mathrm {d} N _ {t}, \\ \lambda_ {0} = \lambda , \end{array} \right.
+$$
+
+has a unique global positive solution for every  $\lambda > 0$ .
+
+# A.4. Proofs of Special Cases of Our Proposed NJDSDE
+
+Proof. We first provide the proof for the case of Hawkes processes, and the proofs for self-correcting processes and Poisson processes are similar.
+
+According to Theorem 2, between the jumps of the Hawkes process  $\{N_t\}_{t \geq 0}$ , the intensity process  $\{\lambda_t\}_{t \geq 0}$  follows the dynamics  $\mathrm{d}\lambda_t = \beta (\mu - \lambda_t)\mathrm{d}t$ . Therefore, the dynamics of  $\eta_t \coloneqq \log (\lambda_t)$  can be expressed as follows:
+
+$$
+\mathrm {d} \eta_ {t} = \mathrm {d} \log (\lambda_ {t}) = \frac {1}{\lambda_ {t}} \mathrm {d} \lambda_ {t} = \frac {\beta (\mu - \lambda_ {t})}{\lambda_ {t}} \mathrm {d} t = \frac {\beta (\mu - e ^ {\eta_ {t}})}{e ^ {\eta_ {t}}} \mathrm {d} t = \beta (\mu e ^ {- \eta_ {t}} - 1) \mathrm {d} t. \tag {28}
+$$
+
+On the other hand, at a jump time  $t$ , the intensity process  $\{\lambda_t\}_{t \geq 0}$  has a jump of size  $\Delta \lambda_t = \alpha$ . Then  $\lambda_{t+} = \lambda_t + \Delta \lambda_t = \lambda_t + \alpha$ . Since  $\eta_t = \log(\lambda_t)$ , the induced jump size of  $\eta_t$  is given by
+
+$$
+\Delta \eta_ {t} = \log \left(\lambda_ {t +}\right) - \log \left(\lambda_ {t}\right) = \log \left(\frac {\lambda_ {t} + \alpha}{\lambda_ {t}}\right) = \log \left(\frac {e ^ {\eta_ {t}} + \alpha}{e ^ {\eta_ {t}}}\right) = \log \left(1 + \alpha e ^ {- \eta_ {t}}\right). \tag {29}
+$$
+
+Combining Eq.(28) and Eq.(29), the equation holds:  $\mathrm{d}\eta_{t} = \beta (\mu e^{-\eta_{t}} - 1)\mathrm{d}t + \log (1 + \alpha e^{-\eta_{t}})\mathrm{d}N_{t}$ .
+
+Therefore, when the function  $f_{\theta_f}(\eta_t)$  is set to  $\beta (\mu e^{-\eta_t} - 1)$ ,  $g_{\theta_g}(\eta_t)$  is set to 0, and  $h_{\theta_h}(\eta_t)$  is set to  $\log (1 + \alpha e^{-\eta_t})$  in Eq.(12), the proposed NJDSDE characterizes Hawkes processes.
+
+Now, we prove that a specific class (but not all) of log-Gaussian Cox processes (LGCPs) can be incorporated into our modeling framework Eq.(12).
+
+Specifically, when  $f_{\theta_f}$  takes 0,  $g_{\theta_g}$  takes 1, and  $h_{\theta_h}$  takes 0 in Eq.(12), the NJDSDE reduces to  $\mathrm{d}\eta_t = \mathrm{d}W_t$ . Given the initial value  $\eta_0 = \log \lambda_0$  and  $W_0 = 0$ , we have  $\eta_t = W_t + \log \lambda_0$ . Since  $\lambda_t = \exp(\eta_t)$ , it follows that  $\lambda_t = \lambda_0 \exp(W_t)$ . Taking  $\lambda_0$  as 1, we obtain  $\lambda_t = \exp(W_t)$ , which indeed represents a specific class of LGCPs, since the Brownian motion  $W_t$  is a Gaussian process with independent stationary increments (Bertoin, 1996). In turn, since a Gaussian process is not necessarily a Brownian motion, for example when the increments of a Gaussian process do not satisfy independence, our framework Eq.(12) cannot incorporate all LGCPs.
+
+# A.5. Proof of Theorem 5
+
+Proof. The following proof is adapted from the Theorem 9.1 in (Ikeda & Watanabe, 2014).
+
+Consider a probability space  $(\Omega, \mathcal{F}, \mathbb{P})$  on which we define a Brownian motion  $W = \{W_t\}_{t \geq 0}$  and a counting process  $N = \{N_t\}_{t \geq 0}$  that jumps at the times  $\{t_i\}_{i=1}^{\infty}$ . Suppose that  $W$  and  $N$  are independent. We define the filtration  $\{\mathcal{F}_t\}_{t \geq 0}$  as the augmented natural filtration of  $W$  and  $N$ , i.e., for all  $t \geq 0$  we set  $\mathcal{F}_t = \sigma\left(\{(W_s, N_s) : s \leq t\} \cup \mathcal{N}\right)$ , where  $\mathcal{N} = \{A \in \mathcal{F} : \mathbb{P}(A) = 0\}$ . With this,  $(\Omega, \mathcal{F}, \{\mathcal{F}_t\}_{t \geq 0}, \mathbb{P})$  is a filtered probability space that satisfies the usual conditions.
+
+It is easy to see that the jump times  $\{t_i\}_{i = 1}^{\infty}$  are stopping times of  $\{\mathcal{F}_t\}_{t\geq 0}$  since  $\{t_i\leq t\} = \{N_t\geq i\} \in \mathcal{F}_t$  , and  $\lim_{i\to \infty}t_i = \infty$  a.s. First we shall show the existence and uniqueness of solutions in the time interval  $[0,t_1]$  . For this, consider the following equation
+
+$$
+\mathrm {d} X (t) = f _ {\theta_ {f}} (X (s)) \mathrm {d} s + g _ {\theta_ {g}} (X (s)) \mathrm {d} W _ {s}, \quad X (0) = \eta_ {0}. \tag {30}
+$$
+
+The functions  $f_{\theta_f}$  and  $g_{\theta_g}$  in the above equation depend only on the variable  $x$  and are independent of the time variable  $t$ . In this case, note that the Lipschitz condition  $|f_{\theta_f}(x) - f_{\theta_f}(y)| + |g_{\theta_g}(x) - g_{\theta_g}(y)| \leq C|x - y|$  implies  $|f_{\theta_f}(x) - f_{\theta_f}(y)| \leq C|x - y|$ . Then we derive
+
+$$
+\left| f _ {\theta_ {f}} (x) \right| \leq \left| f _ {\theta_ {f}} (x) - f _ {\theta_ {f}} (0) \right| + \left| f _ {\theta_ {f}} (0) \right| \leq C | x | + \left| f _ {\theta_ {f}} (0) \right| \leq D (1 + | x |), \tag {31}
+$$
+
+where  $D = \max \{C, |f_{\theta_f}(0)|\}$ . Therefore, the linear growth condition automatically follows from the Lipschitz condition (Kuo, 2006). According to the Theorem 5.2.1 in (Oksendal, 2013), we know that the solution  $X(t)$  of Eq.(30) exists uniquely. This solution is a measurable function of  $X(0)$ ,  $W$  and  $N$  in the obvious sense. Using the continuity of  $h_{\theta_h}(x)$ , we can set
+
+$$
+\eta_ {1} (t) = X (t), \quad t \in [ 0, t _ {1} ], \tag {32}
+$$
+
+and
+
+$$
+\eta_ {1} \left(t _ {1} +\right) = X \left(t _ {1}\right) + h _ {\theta_ {h}} \left(X \left(t _ {1}\right)\right). \tag {33}
+$$
+
+The process  $\{\eta_1(t)\}_{t\in [0,t_1]}$  is clearly the unique solution of Eq.(12) in the time interval  $[0,t_1]$ . Next, set  $\widetilde{X} (0) = \eta_{1}(t_{1} + )$ $\widetilde{W} = \{\widetilde{W}_t\}_{t\geq 0}$  where  $\widetilde{W}_{t} = W_{t + t_{1}} - W_{t_{1}}$ , and  $\tilde{N} = \{\tilde{N}_t\}_{t\geq 0}$  where  $\tilde{N}_t = N_{t + t_1} - N_{t_1}$ . Since the SDE Eq.(30) is autonomous, we can determine the process  $\tilde{\eta}_2(t)$  on  $[0,\tilde{t}_1]$  with respect to  $\widetilde{X} (0)$ ,  $\widetilde{W}$  and  $\tilde{N}$  in the same way as  $\eta_1(t)$ . Clearly  $\tilde{t}_1$ , defined with respect to  $\tilde{N}$ , coincides with  $t_2 - t_1$ . Define  $\{\eta (t)\}_{t\in [0,t_2]}$  by
+
+$$
+\eta (t) = \left\{ \begin{array}{l l} \eta_ {1} (t), & t \in [ 0, t _ {1} ], \\ \tilde {\eta} _ {2} (t - t _ {1}), & t \in (t _ {1}, t _ {2} ]. \end{array} \right. \tag {34}
+$$
+
+It is easy to see that  $\{\eta(t)\}_{t \in [0, t_2]}$  is the unique solution of Eq.(12) in the time interval  $[0, t_2]$ . Continuing this process successively,  $\eta(t)$  is determined uniquely in the time interval  $[0, t_i]$  for every  $i$  and hence  $\eta(t)$  is determined globally. This completes the proof.
+
+# B. Training Algorithm
+
+The pseudo-codes of training algorithm of the Neural Jump-Diffusion Univariate Point Process (NJDUPP) and the Neural Jump-Diffusion Multivariate Point Process (NJDMPP) are presented in Algorithm 1 and Algorithm 2, respectively.
+
+Algorithm 1 Training of the NJDUPP.  
+Input: Model parameter  $\theta = \{\theta_f, \theta_g, \theta_h, \eta_0\}$ , start time  $t_0 = 0$ , event sequences  $\{\mathcal{S}^l\}_{l=1}^L$ , where  $\mathcal{S}^l = \{t_i^l\}_{i=1}^{n_l}$   
+Initialize  $f_{\theta_f}, g_{\theta_g}, h_{\theta_h}, \eta_{t_0+} = \eta_0, \mathcal{L} = 0$   
+while Not Converge do  
+for each sequence  $\mathcal{S}^l$  in batch do  
+for  $i = 1, \ldots, n_l + 1$  do  
+ $\{\eta_{\tau_k^i}\}_{k=1}^N = \text{SDESolve}(f_{\theta_f}, g_{\theta_g}, \eta_{t_{i-1}+}^l, (\tau_0^i = t_{i-1}^l, \ldots, \tau_k^i, \ldots, \tau_N^i = t_i^l))$ $\eta_{t_{i+}^l} = \eta_{t_i^l} + h_{\theta_h}(\eta_{t_i^l}')$   
+end  
+ $\mathcal{L}^l = -\ell(\{\eta_{\tau_k^i}\}_{k=0}^{N,n_l+1}; \theta)$ $\mathcal{L} + = \mathcal{L}^l$   
+end  
+back-propagate with gradient  $\nabla_\theta \mathcal{L}$   
+update model parameters by Adam optimizer  
+end
+
+Algorithm 2 Training of the NJDMPP.  
+Input: Model parameter  $\theta = \{\theta_{f},\theta_{g},\theta_{h},\eta_{0}\}$ ,  $t_0 = 0$ , multi-type event sequences  $\{S^l\}_{l=1}^L$ , where  $S^l = \{(t_i^l,m_i^l)\}_{i=1}^{n_l}$   
+Initialize  $f_{\theta_f}, g_{\theta_g}, h_{\theta_h}, \eta_{t_0+} = \eta_0, \mathcal{L} = 0$   
+while Not Converge do  
+for each sequence  $S^l$  in batch do  
+for  $i = 1,\ldots,n_l + 1$  do  
+ $\{\eta_{\tau_k^i}\}_{k=1}^N = \text{SDESolve} \left( f_{\theta_f}, g_{\theta_g}, \eta_{t_{i-1}^l}, (\tau_0^i = t_{i-1}^l, \ldots, \tau_k^i, \ldots, \tau_N^i = t_i^l) \right)$ $\eta_{t_{i}^l+} = \eta_{t_i^l} + h_{\theta_h}^{m_i^l}(\eta_{t_i^l})$   
+end  
+ $\mathcal{L}^l = -\ell \left( \left\{ \eta_{\tau_k^i} \right\}_{k=0}^{N,n_l+1}; \theta \right)$ $\mathcal{L}^+ = \mathcal{L}^l$   
+end  
+back-propagate with gradient  $\nabla_\theta \mathcal{L}$   
+update model parameters by Adam optimizer  
+end
+
+# C. Experimental Details
+
+# C.1. Experimental Environment
+
+The experiments are conducted on a Linux server with eight GPUs (NVIDIA RTX 2080 Ti * 8). We implement our model and all baselines with the deep learning library PyTorch (Paszke et al., 2017).
+
+# C.2. Dataset Descriptions
+
+- Retweet (Zhou et al., 2013). The dataset consists of sequences of time-stamped user retweet events, categorized into three types based on the users' following sizes: "small", "medium", and "large".  
+- Earthquake (Xue et al., 2024). This dataset contains timestamped earthquake events over the Conterminous U.S from 1996 to 2023. The seven event types are defined based on the magnitude of earthquakes.
+
+Table 5. Statistics of the used datasets.  
+
+<table><tr><td rowspan="2">Dataset</td><td rowspan="2"># Types</td><td colspan="3"># Sequences</td><td colspan="3">Sequence Length</td><td colspan="3"># Events</td></tr><tr><td>Train</td><td>Dev</td><td>Test</td><td>Min</td><td>Mean</td><td>Max</td><td>Train</td><td>Dev</td><td>Test</td></tr><tr><td>Retweet</td><td>3</td><td>9,000</td><td>1,535</td><td>1,520</td><td>10</td><td>40</td><td>97</td><td>369,731</td><td>62,823</td><td>61,154</td></tr><tr><td>Earthquake</td><td>7</td><td>3000</td><td>400</td><td>896</td><td>11</td><td>16</td><td>18</td><td>49,363</td><td>6,612</td><td>14,748</td></tr><tr><td>Taxi</td><td>10</td><td>1,400</td><td>200</td><td>400</td><td>36</td><td>37</td><td>38</td><td>51,854</td><td>7,404</td><td>14,820</td></tr><tr><td>Taobao</td><td>17</td><td>1,300</td><td>200</td><td>500</td><td>32</td><td>57</td><td>64</td><td>75,205</td><td>11,737</td><td>28,455</td></tr><tr><td>StackOverflow</td><td>22</td><td>1,401</td><td>401</td><td>401</td><td>41</td><td>65</td><td>101</td><td>90,497</td><td>25,762</td><td>26,518</td></tr><tr><td>MIMIC-II</td><td>75</td><td>527</td><td>58</td><td>65</td><td>2</td><td>4</td><td>33</td><td>1,930</td><td>252</td><td>237</td></tr></table>
+
+- Taxi (Whong, 2014). This dataset contains time-stamped taxi pick-up and drop-off events throughout the five boroughs of New York city. Each combination of borough, whether it's a pick-up or drop-off, defines an event type, resulting in a total of 10 event types.  
+- Taobao (Xue et al., 2022). This dataset includes the time-stamped click behavior of users in Taobao platform from November 25 to December 3, 2017. Each user has a sequence of product click events, where each event contains a timestamp and the product category.  
+- StackOverflow (Leskovec & Krevl, 2014). This dataset contains two years of user-awarded collections from the question-answering website. Each user is awarded a sequence of badges, with a total of 22 different badge types.  
+- MIMIC-II (Johnson et al., 2016). This dataset includes timestamped de-identified clinical visit events of Intensive Care Unit patients for seven years. Each patient has a sequence of hospital visit events, and each event records its timestamp and disease diagnosis.
+
+Table 5 shows statistics about each dataset mentioned above.
+
+# C.3. Baseline Descriptions
+
+We provide detailed descriptions of the used baselines as follows:
+
+- RMTPP (Du et al., 2016). RMTPP leverages RNNs to learn a hidden representation of event history, and then applies an exponential transformation on this representation for defining the intensity function.  
+- NHP (Mei & Eisner, 2017). NHP proposes a continuous-time LSTM to encode event sequences. The intensity function of NHP can decay over time and does not need to encode inter-event times as numerical inputs to the LSTM.  
+- SAHP (Zhang et al., 2020a). SAHP uses a self-attention mechanism to aggregate historical events, which enhances the expression ability of the intensity function.  
+- THP (Zuo et al., 2020). To capture the long-term dependence of events, THP proposes to model the intensity function using a Transformer architecture.  
+- AttNHP (Yang et al., 2022). AttNHP generalizes the Transformer architecture for modeling event sequences. Its architecture builds rich embeddings of actual and possible events at any given time, based on lower-level representations of these events and their context.  
+- NSTPP (Chen et al., 2020). With the goal of modeling high-fidelity distributions in continuous time and space, NSTPP uses the Neural ODE framework to parameterize the spatio-temporal TPP by combining ideas from Neural Jump SDEs (Jia & Benson, 2019) and continuous-time normalizing flows (Chen et al., 2018).
+
+# C.4. Evaluation Metrics
+
+We formulate the metrics used in this paper as follows:
+
+- The mean absolute percentage error (MAPE) of the estimated intensity is given by
+
+$$
+\mathrm{MAPE} = \frac{1}{T}\int_{0}^{T}\left|\frac{\lambda_{t}^{\mathrm{model}} - \lambda_{t}^{\mathrm{GT}}}{\lambda_{t}^{\mathrm{GT}}}\right|\mathrm{d}t\times 100\% , \tag{35}
+$$
+
+where  $T$  is the observation length,  $\lambda_t^{\mathrm{model}}$  is the trained model intensity, and  $\lambda_t^{\mathrm{GT}}$  is the ground truth intensity.
+
+- The negative log-likelihood (NLL) of a multivariate point process over a time interval  $[0, T]$  is
+
+$$
+\mathrm {N L L} = - \sum_ {i = 1} ^ {n} \log \lambda_ {t _ {i}} ^ {m _ {i}} + \sum_ {m = 1} ^ {M} \left(\int_ {0} ^ {T} \lambda_ {s} ^ {m} \mathrm {d} s\right) = - \sum_ {i = 1} ^ {n} \eta_ {t _ {i}} ^ {m _ {i}} + \sum_ {m = 1} ^ {M} \left(\int_ {0} ^ {T} \exp \left(\eta_ {s} ^ {m}\right) \mathrm {d} s\right). \tag {36}
+$$
+
+- The root mean square error (RMSE) is
+
+$$
+\operatorname {R M S E} (\boldsymbol {t}, \hat {\boldsymbol {t}}) = \sqrt {\frac {1}{N} \sum_ {i = 1} ^ {N} \left(t _ {i} - \hat {t} _ {i}\right) ^ {2}}. \tag {37}
+$$
+
+- The accuracy of multiclass classification is the fraction of correct classifications, that is
+
+$$
+\text {A c c u r a c y} = \frac {\# \text {c o r r e c t c l a s s i f i c a t i o n s}}{\# \text {a l l c l a s s i f i c a t i o n s}}. \tag {38}
+$$
+
+- The formula for the weighted  $F_{1}$  score, accounting for class imbalance, is expressed as:
+
+$$
+F _ {\text {l w e i g h t e d}} = \sum_ {i = 1} ^ {C} w _ {i} \cdot F _ {1 i}, \tag {39}
+$$
+
+where  $C$  is the number of classes,  $w_{i}$  represents the sample weight for class  $i$ , and  $F_{1i}$  is the  $F_{1}$  score for class  $i$ .
+
+# C.5. Training Details
+
+We train our NJDTPP model for all experiments by minimizing the negative log-likelihood of training sequences, as described in Appendix B. The drift net, diffusion net, and jump net of NJDTPP are implemented as three multi-layer perceptrons (MLPs) with the same network structure. The activation function chosen for these networks is Tanh. Optimizer is Adam (Kingma & Ba, 2015) with a weight decay of  $10^{-5}$ . The MLP parameters and the initial value  $\eta_0$  are initialized by the Gaussian distribution.
+
+# C.6. Hyper-parameter Setting
+
+We employ the "diagonal" noise in the diffusion term of Eq.(18), i.e.,  $g_{\theta_g}$  is a diagonal matrix. In this case, the dimension of the Brownian motion  $W_t$  is equal to the total number of event types, i.e.,  $K = M$ . Grid search is used to determine other hyper-parameters: the learning rate is selected from  $\{0.001, 0.01, 0.1\}$ , the hidden layer number is selected from  $\{1, 2, 3\}$ , and the hidden layer size is selected from  $\{16, 32, 64\}$ .

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2024/Neural NeRF Compression/full.md

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+# Neural NeRF Compression
+
+Tuan Pham<sup>1</sup> Stephan Mandt<sup>1</sup>
+
+# Abstract
+
+Neural Radiance Fields (NeRFs) have emerged as powerful tools for capturing detailed 3D scenes through continuous volumetric representations. Recent NeRFs utilize feature grids to improve rendering quality and speed; however, these representations introduce significant storage overhead. This paper presents a novel method for efficiently compressing a grid-based NeRF model, addressing the storage overhead concern. Our approach is based on the non-linear transform coding paradigm, employing neural compression for compressing the model's feature grids. Due to the lack of training data involving many i.i.d scenes, we design an encoder-free, end-to-end optimized approach for individual scenes, using lightweight decoders. To leverage the spatial inhomogeneity of the latent feature grids, we introduce an importance-weighted rate-distortion objective and a sparse entropy model employing a masking mechanism. Our experimental results validate that our proposed method surpasses existing works in terms of grid-based NeRF compression efficacy and reconstruction quality.
+
+# 1. Introduction
+
+Over the past few years, the field of 3D scene modeling and reconstruction has been revolutionized by the advent of Neural Radiance Fields (NeRF) methodologies (Mildenhall et al., 2021; Zhang et al., 2020; Barron et al., 2021). NeRFs offer a sophisticated method for 3D reconstruction, with the ability to synthesize novel viewpoints from limited 2D data. Yet, the original NeRF model requires millions of MLP queries, which causes slow training and rendering.
+
+To address these efficiency concerns, recent NeRF advancements have transitioned to the integration of an explicit grid representation (Yu et al., 2021; Sun et al., 2022; Fridovich-
+
+Keil et al., 2022; Chen et al., 2022; Fridovich-Keil et al., 2023; Chan et al., 2022). While significantly accelerating training and rendering processes, this change also poses a new challenge: the storage cost for saving the explicit grid NeRF representation increases. This problem is crucial, especially in real-world (e.g., large-scale VR and AR) applications where storage and transmission impose critical constraints.
+
+Our work seeks to significantly reduce the storage costs of NeRFs. Inspired by neural image compression methodology (Yang et al., 2023b), we apply non-linear transform coding techniques (Balle et al., 2020) to compress the explicit grid NeRF representation efficiently. However, we sidestep the conventional auto-encoder approach in favor of an iterative inference framework, in which we jointly optimize the latent code along with a lightweight decoder. We further take account of the NeRF grid importance scores while reconstructing the scene to boost the efficiency of our compression model. Lastly, we propose a novel entropy model that masks uninformative feature grid points. Utilizing a rate-distortion objective, we can choose from various compression levels. Our proposed approach departs from previous works on compressing explicit grid NeRF representations (Li et al., 2023a;b; Deng & Tartaglione, 2023) based on voxel pruning and/or vector quantization (Gray, 1984) while taking into account the varying importance levels of different voxel grid locations.
+
+To show the effectiveness of our proposed method, we perform extensive experiments on four different datasets. Our results show that our model is capable of compressing diverse NeRF scenes to a much smaller size and improves over previous works in terms of rate-distortion performance.
+
+# 2. Background
+
+# 2.1. Neural Radiance Fields
+
+Neural radiance fields (Mildenhall et al., 2021) mark a paradigm shift in 3D scene representation using deep neural networks. Unlike traditional approaches that employ discrete structures such as point clouds or meshes, NeRFs model a scene using a continuous volumetric function  $F: (\mathbf{x}, \mathbf{d}) \to (\mathbf{c}, \sigma)$ . Here, an input comprising of spatial coordinates  $\mathbf{x}$  and a viewing direction  $\mathbf{d}$  is mapped to
+
+an output representing color c and volume density  $\sigma$
+
+For each pixel, the estimated color  $\hat{C} (\mathbf{r})$  for the corresponding ray  $\mathbf{r}$  can be calculated by:
+
+$$
+\hat {C} (\mathbf {r}) = \sum_ {i = 1} ^ {N} T _ {i} \cdot \alpha_ {i} \cdot \mathbf {c} _ {i}, \tag {1}
+$$
+
+with the following definitions:
+
+-  $\alpha_{i} = 1 - \exp (-\sigma_{i}\delta_{i})$  is the probability of light being absorbed at the  $i$ -th point, dependent on the volume density  $\sigma_{i}$  and the distance  $\delta_{i}$  between adjacent sampled points on the ray.  
+-  $T_{i} = \prod_{j=1}^{i}(1 - \alpha_{j})$  represents the accumulated transmittance, or the remaining light that has not been absorbed before reaching the  $i$ -th point.
+
+NeRF is then trained to minimize total squared error loss between the rendered and true pixel colors.
+
+$$
+\mathcal {L} _ {\text {r e n d e r}} = \sum_ {\mathbf {r}} | | \hat {C} (\mathbf {r}) - C (\mathbf {r}) | | _ {2} ^ {2} \tag {2}
+$$
+
+Despite NeRF's ability to provide intricate scene details with a relatively compact neural network, the computational demand remains a significant constraint. The evaluation over the volume often requires thousands of network evaluations per pixel. To reduce training and inference time, recent research has employed explicit grid structure into NeRF. More specifically, they introduce voxel grids (Sun et al., 2022; Fridovich-Keil et al., 2022) or decomposed feature planes (Chen et al., 2022; Fridovich-Keil et al., 2023; Chan et al., 2022) into the model, and query point features via trilinear or bilinear interpolation. While this notably speeds up training and inference, it does come at the expense of greater storage needs from saving the feature grids.
+
+# 2.2. Neural Compression
+
+Neural compression utilizes neural networks to perform end-to-end learned data compression (Yang et al., 2023b). Traditional compression algorithms are handcrafted and specifically tailored to the characteristics of the data they compress, such as JPEG (Wallace, 1991) for images or MP3 for audio. In contrast, neural compression seeks to learn efficient data representations directly from the data, exemplified by the nonlinear transform coding paradigm (Balle et al., 2020).
+
+Existing lossy neural compression methods (Ballé et al., 2016; 2018; Minnen et al., 2018; Cheng et al., 2020; Matsubara et al., 2022; Yang & Mandt, 2023a;b; Yang et al., 2023a) often leverage an auto-encoder architecture (Kingma & Welling, 2014). Here, an encoder  $E$  maps data  $\mathbf{X}$  to continuous latent representations  $\mathbf{Z} = E(\mathbf{X})$ . This continuous
+
+$\mathbf{Z}$  is then quantized to integers by  $Q$ , resulting in  $\hat{\mathbf{Z}} = Q(\mathbf{Z})$ . An entropy model  $P$  is used to transmit  $\hat{\mathbf{Z}}$  losslessly. Finally, a decoder  $D$  receives the quantized latent code  $\hat{\mathbf{Z}}$  and reconstructs the original data  $\hat{\mathbf{X}} = D(\hat{\mathbf{Z}})$ . One commonly trains the encoder  $E$ , the decoder  $D$  and the entropy model  $P$  jointly using a rate-distortion objective:
+
+$$
+\begin{array}{l} \mathcal {L} (E, D, P) = \mathbb {E} _ {\mathbf {X} \sim p (\mathbf {X})} [ d (\mathbf {X}, D (Q (E (\mathbf {X}))) \tag {3} \\ \left. - \lambda \log_ {2} P (Q (E (\mathbf {X}))) \right] \\ \end{array}
+$$
+
+where  $d(\cdot, \cdot)$  is a distortion loss and the second term is the rate loss that measures the expected code length. The parameter  $\lambda$  balances between the two loss terms. At training time, the quantizer  $Q$  is typically replaced with injecting uniform noise (Balle et al., 2016). See (Yang et al., 2023b) for a detailed review of neural compression.
+
+# 3. Method
+
+In this section, we describe our method for grid-based NeRF compression. Our primary focus is on the compression of the TensoRF-VM model (Chen et al., 2022), characterized by its decomposed 2D feature plane structure (Kolda & Bader, 2009). We select TensoRF-VM because of its proficient 3D scene modeling capabilities, often outperforming alternative methods like Plenoxels (Fridovich-Keil et al., 2022) or DVGO (Sun et al., 2022). Our method has the potential to be applied to other grid-based NeRF architectures.
+
+Problem setting. We have a TensoRF-VM model that was pre-trained for a single scene, and our task is to reduce its size through compression while maintaining its reconstruction quality. We assume that we have access to the training dataset comprising view images at compressing time.
+
+Notation. The three feature planes (or matrix components) of TensoRF-VM are denoted by  $\{\mathbf{P}_i\}_{i=1}^3$ , in which subscript  $i$  signifies the index of the planes and each  $\mathbf{P}_i \in \mathbb{R}^{C_i \times H_i \times W_i}$ . In practice,  $\{\mathbf{P}_i\}_{i=1}^3$  is the channel-wise concatenation of the density planes and appearance planes of TensoRF-VM. The vector components are not considered in our compression and, hence, are not represented in our notation. For indexing a specific spatial grid location  $j$  in the feature plane  $i$ , we employ a superscript, represented as  $\mathbf{P}_i^j$ .
+
+# 3.1. Compressing the feature planes
+
+Most storage for compressing TensoRF-VM is spent on the feature grids. To illustrate this, we analyze a trained model for the Lego scene from the Synthetic-NeRF dataset (Mildenhall et al., 2021). In this model, the 2D feature planes take  $67.61\mathrm{MB}$ , while the other components, such as the rendering MLP, the rendering mask, and the vector
+
+![](images/75162d4a801854987cb4b8a24760df5c2fe605441e22aafadb47731ae40e2c18.jpg)  
+Figure 1: Overview of our model. At training time (left), we learn the three latent codes  $\{\mathbf{Z}_i\}_{i=1}^3$  to reconstruct the three frozen feature planes  $\{\mathbf{P}_i\}_{i=1}^3$ . The reconstructed feature planes  $\{\hat{\mathbf{P}}_i\}_{i=1}^3$  are used to render the scene and calculate the rendering loss. The entropy model  $P$  is used to calculate the rate loss and compress the latent codes to bitstring. At rendering time (right), we use  $P$  to decompress the bitstring to latent codes  $\{\hat{\mathbf{Z}}_i\}_{i=1}^3$  and then reconstruct the feature planes  $\{\hat{\mathbf{P}}_i\}_{i=1}^3$ .
+
+![](images/05a09feb3429c389da2dd3c39056854511f450c9965cfdcb13f20ebe2ef54b7a.jpg)
+
+components, take only 1.21 MB. Given this disparity, we focus on compressing TensoRF-VM's feature planes.
+
+In more detail, we can define an encoder  $E$  that embeds the three feature planes  $\{\mathbf{P}_i\}_{i=1}^3$  to latent codes  $\{\mathbf{Z}_i\}_{i=1}^3$ , in which the  $\mathbf{Z}_i$  may have lower resolution than  $\mathbf{P}_i$ . The latent codes are quantized to  $\{\hat{\mathbf{Z}}_i\}_{i=1}^3$  and compressed with entropy coding using an entropy model  $P$ . At rendering time, we decompress the quantized latent codes  $\{\hat{\mathbf{Z}}_i\}_{i=1}^3$  and forward them to the decoder  $D$  to reconstruct the three feature planes  $\{\hat{\mathbf{P}}_i\}_{i=1}^3$ . We then use  $\{\hat{\mathbf{P}}_i\}_{i=1}^3$  to query sampling point features and render the scene. The compressed NeRF model includes the compressed latent codes, the decoder, the entropy model, and the other components.
+
+It is crucial to highlight that we only need to reconstruct the three feature planes once, and all subsequent querying operations for the sampling points are executed on these reconstructed planes. Thus, the decompression process only adds minimal overhead to the overall rendering procedure.
+
+Per-scene optimization. The conventional approach to neural image compression (Ballé et al., 2016; 2018) involves training a compression model on a big dataset containing thousands of images. However, applying this same training method to NeRF compression presents three challenges: First, we need a dataset with numerous 3D objects. Although datasets like Objaverse (Deitke et al., 2023) or Objaverse-XL (Deitke et al., 2024) exist, they are synthetic datasets and only contain a single object for each scene. Additionally, pre-trained NeRF models are required for every 3D object in the dataset, demanding significant computational resources and storage. Finally, we cannot adapt other components of the NeRF model, such as the rendering MLP and the vector components of the TensorF-VM model. Due
+
+to these challenges, we optimize each NeRF scene individually, a process we refer to as per-scene optimization. In this approach, the compressor is overfitted to each NeRF scene, which results in improved compression performance.
+
+Transform coding without encoder. In nonlinear transform coding (Balle et al., 2020; Yang et al., 2023b), one usually employs an encoder to obtain the latent code of a new data point via amortized inference (Kingma & Welling, 2014; Gershman & Goodman, 2014). This is essential for compressing a new data point quickly in a single network pass. Nonetheless, in the case of per-scene TensoRF-VM compression, our primary objective is to compress merely the three feature planes, and our decoder is overfitted to a single scene. Moreover, using an encoder for amortized inference leads to an irreducible amortization gap in optimization (Cremer et al., 2018; Marino et al., 2018), which has been shown to degrade compression performance (Campos et al., 2019; Yang et al., 2020).
+
+For these reasons, we remove the encoder and directly learn the three latent codes  $\{\mathbf{Z}_i\}_{i=1}^3$  for each scene. More specifically, we initialize the  $\{\mathbf{Z}_i\}_{i=1}^3$  as a tensor of zeros, and jointly optimize  $\{\mathbf{Z}_i\}_{i=1}^3$  with the decoder  $D$  and the entropy model  $P$ . At decoding time, the receiver thus receives a binary code along with the entropy model and decoder to reconstruct the sample, all three of which are counted towards the bitrate.
+
+Architecture design. Since we must transmit the decoder  $D$  along with the latent code  $\{\mathbf{Z}_i\}_{i=1}^3$  to decompress the scene, it's essential for the decoder to be lightweight. Yang & Mandt (2023b) established a lightweight decoder for neural image compression. We found that a two-layer transposed convolutional neural network with SELU activation (Klambauer et al., 2017) is effective for our needs.
+
+# 3.2. Importance-weighted training loss
+
+Our model is trained end-to-end (on top of the pre-trained NeRF) with a rate-distortion loss. The rate loss is defined as the log-likelihood of the entropy model  $P$ , and it ensures that the compressed feature planes have low relative entropy to the prior  $P$ . For the distortion loss, we discover that using only the NeRF rendering loss  $\mathcal{L}_{\text{render}}$  is not sufficient; we also need to use an L2 feature plane reconstruction loss for good rendering quality.
+
+However, reconstructing the entire feature planes is not the most efficient approach for compression. Prior research (Li et al., 2023a;b; Deng & Tartaglione, 2023) has illustrated that these feature grids possess significant redundancy and could be pruned to decrease the size of the model. Consequently, if we were to reconstruct every single grid location, it would inevitably lead to additional storage costs.
+
+To address this issue, we suggest computing weight maps, defined below, that we use to re-weight the feature plane reconstruction loss. With this approach, our model is guided to reconstruct only high-density grid locations while ignoring the less populated ones, ensuring a more optimized and effectively compressed representation.
+
+For each feature plane  $\mathbf{P}_i\in \mathbb{R}^{C_i\times W_i\times H_i}$ , we define  $\mathbf{W}_i\in$ $\mathbb{R}^{W_i\times H_i}$  as the corresponding weight map, shared across all feature channels. These weight maps are constructed based on the rendering importance score  $\{\mathbf{I}_i\}_{i = 1}^3$  by Li et al. (2023a;b), defined next.
+
+As follows, we consider feature plane  $i$  and grid location  $j$ . The collection of sampling points  $\mathbf{x}_k \in \mathbb{R}^3$  in the vicinity of location  $j$  (upon projection) shall be denoted as  $\mathcal{N}_j$ . Since the coordinates of  $\mathbf{x}_k$  are continuous and the grid locations discrete, we distribute the "mass" of each  $\mathbf{x}_k$  onto the relevant grid locations using bilinear interpolation, resulting in the interpolation weights  $\omega_{kj}^i$  for sampling point  $\mathbf{x}_k \in \mathcal{N}_j$ . In addition, each sampling point  $\mathbf{x}_k$  in Eq. 1 has a corresponding transmittance coefficient  $T_k \cdot \alpha_k$  that we interpret as its importance. This leads to the following importance scores for each grid location  $j$  in plane  $i$ ,
+
+$$
+\mathbf {I} _ {i} ^ {j} = \sum_ {k \in \mathcal {N} _ {j}} \omega_ {k j} ^ {i} \cdot T _ {k} \cdot \alpha_ {k} \tag {4}
+$$
+
+In sum, each importance score  $\mathbf{I}_i^j$  is a weighted aggregate of the individual importance scores of the neighboring sampling points  $\mathbf{x}_k$  over the feature grid.
+
+Finally, we apply a log-transform to the importance maps  $\{\mathbf{I}_i\}_{i = 1}^3$  , and then normalize them to the range of [0, 1] to get the weights  $\{\mathbf{W}_i\}_{i = 1}^3$  ..
+
+$$
+\mathbf {W} _ {i} = \text {n o r m a l i z e} (\log (\mathbf {I} _ {i} + \epsilon)), \tag {5}
+$$
+
+in which  $\epsilon = 0.01$  to ensure that the log is well-defined.
+
+# 3.3. Masked entropy model
+
+Applying neural compression to TensoRF-VM enables us to use a wide range of different entropy models. In this section, we design a simple but effective entropy model that works well for TensoRF-VM compression, exploiting the spatial sparsity of the feature plane representation.
+
+Theoretically, a learned entropy model,  $P$ , should result in a close-to-optimal coding strategy, provided the model is flexible enough. In practice, we observed that a predominant portion of the learned latent code is zero, especially in the background. This observation might be attributed to our choice of initializing the latent codes as zero tensors and the fact that large parts of the feature planes are not used for rendering. Such sparsity is poorly captured using the standard entropy models used in neural image compression (Balle et al., 2016; 2018), leading to entropy coding inefficiencies.
+
+To design a better entropy model, we construct a spike-and-slab prior, oftentimes used in Bayesian statistics (Mitchell & Beauchamp, 1988). To this end, we construct binary masks  $\{\mathbf{M}_i\}_{i=1}^3$  into our entropy model  $P$ . The model  $P$  compresses grid features  $\mathbf{P}_i^j$  only when  $\mathbf{M}_i^j = 1$ , allowing selective compression of specific features and avoiding others. Those masks are learnable and can be treated as additional parameters of  $P$ .
+
+In more detail, we design  $P$  to be a fully factorized probability distribution as in Balle et al. (2016) and Balle et al. (2018). Every grid location is independent and identically distributed by binary mixture, consisting of a nonparametric distribution  $p_{\theta}(\cdot)$  with learnable  $\theta$ , and a Dirac mass  $\delta (\cdot)$  at zero. For each latent code  $\hat{\mathbf{Z}}_i$  to be compressed, we establish a corresponding binary mask  $\mathbf{M}_i$  that has the same spatial size and is shared across features channels. The conditional probability distribution  $P$  (given the mask) is then factorized across spatial locations  $j$  as:
+
+$$
+\begin{array}{l} P _ {\mathbf {M} _ {i}} (\hat {\mathbf {Z}} _ {i}) = \prod_ {j} p (\hat {\mathbf {Z}} _ {i} ^ {j} | \mathbf {M} _ {i} ^ {j}); \\ p \left(\hat {\mathbf {Z}} _ {i} ^ {j} \mid \mathbf {M} _ {i} ^ {j}\right) = \left\{ \begin{array}{l l} \delta \left(\hat {\mathbf {Z}} _ {i} ^ {j}\right) & \text {i f} \quad \mathbf {M} _ {i} ^ {j} = 0 \\ p _ {\theta} \left(\hat {\mathbf {Z}} _ {i} ^ {j}\right) & \text {i f} \quad \mathbf {M} _ {i} ^ {j} = 1. \end{array} \right. \tag {6} \\ \end{array}
+$$
+
+We stress that we could also entropy-code the masks under a hyperprior  $p(\mathbf{M})$ , but found little benefit to do so in practice.
+
+This construction implies that, if  $\mathbf{M}_i^j = 0$ , then we designate  $\hat{\mathbf{Z}}_i^j = 0$ . Thus, the input to the decoder  $D$  can be calculated as  $\hat{\mathbf{Z}}_i \odot \mathbf{M}_i$ , and the reconstructed planes are
+
+$$
+\hat {\mathbf {P}} _ {i} = D \left(\hat {\mathbf {Z}} _ {i} \odot \mathbf {M} _ {i}\right). \tag {7}
+$$
+
+However, since the masks  $\mathbf{M}_i$  are binary, they cannot be learned directly. To address this, we turn to the Gumbel-Softmax trick (Jang et al., 2016; Yang et al., 2020) to fa
+
+ciliate the learning of  $\mathbf{M}_i$ . For each  $\mathbf{M}_i$ , we define the binary probabilities denoted by  $\pi_{\mathbf{M}_i}^0$  and  $\pi_{\mathbf{M}_i}^1$  indicating  $\mathbf{M}_i = 0/1$ , respectively. At training time, we sample  $\mathbf{M}_i$  using the straight-through Gumbel-Softmax estimator (Bengio et al., 2013; Jang et al., 2016):
+
+$$
+\mathbf {M} _ {i} = \underset {j \in \{0, 1 \}} {\arg \max } \left(g _ {j} + \log \pi_ {\mathbf {M} _ {i}} ^ {j}\right) \tag {8}
+$$
+
+in which  $g_{j}$  are i.i.d samples drawn from Gumbel(0,1). The straight-through Gumbel-Softmax estimator allows us to calculate the gradients of  $\pi_{\mathbf{M}_i}^j$ . We then optimize the mask probabilities  $\pi_{\mathbf{M}_i}^j$  following the rate-distortion loss:
+
+$$
+\begin{array}{l} \mathcal {L} = \mathcal {L} _ {\text {r e n d e r}} (\{\hat {\mathbf {P}} _ {i} \} _ {i = 1} ^ {3}) \\ + \sum_ {i = 1} ^ {3} \left(\left| \left| \left(\mathbf {P} _ {i} - \hat {\mathbf {P}} _ {i}\right) \odot \mathbf {W} _ {i} \right| \right| _ {2} ^ {2} - \lambda \log_ {2} P _ {\mathbf {M} _ {i}} (\hat {\mathbf {Z}} _ {i})\right), \tag {9} \\ \end{array}
+$$
+
+where  $\mathbf{P}_i$  is calculated with Equation 7. In practice, we use an annealing softmax temperature  $\tau$  that decays from 10 to 0.1 to calculate the softmax gradients.
+
+# 4. Experiments
+
+As follows, we empirically demonstrate that our proposed approach of learning a lightweight, per-scene neural compression model without an encoder outperforms existing approaches based on vector quantization and models trained on multiple scenes in terms of rate-distortion performance.
+
+# 4.1. Experiment Setting
+
+Datasets. We perform our experiments on 4 datasets:
+
+- Synthetic-NeRF (Mildenhall et al., 2021): This dataset contains 8 scenes at resolution  $800 \times 800$  rendered by Blender. Each scene contains 100 training views and 200 testing views.  
+- Synthetic-NSVF (Liu et al., 2020): This dataset also contains 8 rendered scenes at resolution  $800 \times 800$ . However Synthetic-NSVF contains more complex geometry and lightning effects compared to Synthetic-NeRF.  
+- LLFF (Mildenhall et al., 2019): LLFF contains 8 real-world scenes made of forward-facing images with non-empty background. We use the resolution  $1008 \times 756$ .  
+- Tanks and Temples (Knapitsch et al., 2017): We use 5 real-world scenes: Barn, Caterpillar, Family, Ignatus, Truck from the Tanks and Temples dataset to experiment with. They have the resolution of  $1920 \times 1080$ .
+
+In our compression experiments, we initially train a TensoRF-VM model for every scene within the datasets
+
+listed above. We use the default TensoRF-VM 192 hyperparameters, as detailed in (Chen et al., 2022). Subsequently, we apply our proposed method to compress these trained models. All experimental procedures are executed using PyTorch (Paszke et al., 2019) on NVIDIA RTX A6000 GPUs.
+
+Baselines. We compare our compression paradigm with: The original NeRF model with MLP (Mildenhall et al., 2021), the uncompressed TensoRF-CP and TensoRF-VM from Chen et al. (2022), two prior compression methods for TensoRF-VM based on pruning and vector quantization named VQ-TensoRF from Li et al. (2023a) and Re:TensoRF from Deng & Tartaglione (2023).
+
+Hyperparameters. As discussed in Section 3.1, our decoder has two transposed convolutional layers with SELU activation (Klambauer et al., 2017). They both have a kernel size of 3, with stride 2 and padding 1. Thus, each layer has an upsampling factor of 2. Given a feature plane sized  $C_i \times W_i \times H_i$ , we initialize the corresponding latent code  $\mathbf{Z}_i$  to have the size of  $C_{Z_i} \times W_i / 4 \times H_i / 4$ .
+
+Having a decoder with more parameters will enhance the model's decoding ability while also increase its size. In light of this trade-off, we introduce two configurations: ECTensoRF-H (stands for Entropy Coded TensorRF - high compression) employs latent codes with 192 channels and a decoder with 96 hidden channels, while ECTensoRF-L (low compression) has 384 latent channels and 192 decoder hidden channels. Regarding the hyperparameter  $\lambda$ , we experiment within the set  $\{0.02, 0.01, 0.005, 0.001, 0.0005, 0.0002, 0.0001\}$ , with higher  $\lambda$  signifying a more compact model.
+
+We train our models for 30,000 iterations with Adam optimizer (Kingma & Ba, 2015). We use an initial learning rate of 0.02 for the latent codes and 0.001 for the networks, and apply an exponential learning rate decay.
+
+# 4.2. Results
+
+We first compare our results with the baselines quantitatively. We use the PSNR and SSIM (Wang et al., 2004) metrics to evaluate the reconstruction quality. The compression rate is determined by the compressed file size in MB.
+
+Quantitative Results. Table 1 showcases quantitative results in both rate and distortion in the high-quality/low-distortion regime, where the reconstruction quality of all compressed TensoRF models are close to other uncompressed performances for novel view synthesis. This regime is particularly relevant in NeRF compression applications, where high-quality renderings for compressed models are typically expected.
+
+Compared to the other two TensoRF compression baselines,
+
+Table 1: Quantitative results comparing our method versus the baselines. PSNR is measured in dB, while the sizes are in MB. We choose the  $\lambda$  to balance between the reconstruction quality and storage size.  
+
+<table><tr><td rowspan="2"></td><td rowspan="2">Methods</td><td colspan="3">Synthetic-NeRF</td><td colspan="3">Synthetic-NSVF</td><td colspan="3">LLFF</td><td colspan="3">Tanks and Temples</td></tr><tr><td>PSNR</td><td>SSIM</td><td>Size</td><td>PSNR</td><td>SSIM</td><td>Size</td><td>PSNR</td><td>SSIM</td><td>Size</td><td>PSNR</td><td>SSIM</td><td>Size</td></tr><tr><td rowspan="3">Uncom-pressed</td><td>NeRF</td><td>31.01</td><td>0.947</td><td>5.0</td><td>-</td><td>-</td><td>-</td><td>26.50</td><td>0.811</td><td>5.0</td><td>25.78</td><td>0.864</td><td>5.0</td></tr><tr><td>TensoRF-CP</td><td>31.56</td><td>0.949</td><td>3.9</td><td>34.48</td><td>0.971</td><td>3.9</td><td>-</td><td>-</td><td>-</td><td>27.59</td><td>0.897</td><td>3.9</td></tr><tr><td>TensoRF-VM</td><td>33.09</td><td>0.963</td><td>67.6</td><td>36.72</td><td>0.982</td><td>71.6</td><td>26.70</td><td>0.836</td><td>179.8</td><td>28.54</td><td>0.921</td><td>72.6</td></tr><tr><td rowspan="4">Com-pressed</td><td>VQ-TensoRF</td><td>32.86</td><td>0.960</td><td>3.6</td><td>36.16</td><td>0.980</td><td>4.1</td><td>26.46</td><td>0.824</td><td>8.8</td><td>28.20</td><td>0.913</td><td>3.3</td></tr><tr><td>Re:TensoRF</td><td>32.81</td><td>0.956</td><td>7.9</td><td>36.14</td><td>0.978</td><td>8.5</td><td>26.55</td><td>0.797</td><td>20.2</td><td>28.24</td><td>0.907</td><td>6.7</td></tr><tr><td>TC-TensoRF-L (ours)</td><td>32.93</td><td>0.961</td><td>3.4</td><td>36.34</td><td>0.980</td><td>4.0</td><td>26.44</td><td>0.826</td><td>4.9</td><td>28.42</td><td>0.915</td><td>2.9</td></tr><tr><td>TC-TensoRF-H (ours)</td><td>32.31</td><td>0.956</td><td>1.6</td><td>35.33</td><td>0.974</td><td>1.6</td><td>25.72</td><td>0.786</td><td>1.7</td><td>28.08</td><td>0.907</td><td>1.6</td></tr></table>
+
+Table 2: Relative improvement of our method versus VQ-TensoRF. BD-PSNR and BD-rate measure the average difference in PSNR and bitrate between the two methods.  
+
+<table><tr><td></td><td>Synthetic-NeRF</td><td>Synthetic-NSVF</td><td>Tanks and Temples</td></tr><tr><td>BD-PSNR</td><td>0.279 dB</td><td>0.289 dB</td><td>0.344 dB</td></tr><tr><td>BD-Rate</td><td>28.827 %</td><td>21.104 %</td><td>16.717 %</td></tr></table>
+
+VQ-TensoRF and Re:TensoRF, our variant ECTensoRF-L shows superior reconstruction performance in this regime in terms of both the PSNR and SSIM metrics while simultaneously maintaining a reduced file size across 3 datasets: Synthetic-NeRF, Synthetic-NSVF, and Tanks & Temples. In the case of the LLFF dataset, we are slightly behind VQ-TensoRF and Re:TensoRF in PSNR. Despite this, our achieved SSIM values surpass both baselines and, remarkably, the size of our compressed files is just about half of VQ-TensoRF and a mere quarter when compared to Re:TensoRF. For a smaller number of channels, our ECTensoRF-H is able to compress the model sizes to less than 2MB while maintaining a decent reconstruction quality. Notably, our ECTensoRF-H has a similar SSIM as Re:TensoRF on Synthetic-NeRF and Tanks&Temples.
+
+Qualitative Results. We compare rendered images from the Synthetic-NeRF dataset, using VQ-TensoRF and our compression method for both configurations: ECTensoRF-L and ECTensoRF-H in Figure 3. Visually, there is minimal disparity between the uncompressed and compressed TensoRF models. We further show more qualitative results for the other datasets in the Appendix.
+
+Rate-distortion performance. The rate-distortion curve is widely used in neural compression to compare the compression performance across different compression level. Here we analyze the rate-distortion curve of our ECTensoRF-L with various  $\lambda$  values versus VQ-TensoRF with various codebook size. For the VQ-TensoRF evaluations, we employed the officially released code and utilized the same pre-trained TensoRF models for consistency.
+
+![](images/4c81c3023842308852f154aa5a49529a6095e8048e3374ae7491f2f0f6653fc5.jpg)
+
+![](images/69492c9cb6cbdfc48cf459f01dc4089ed105cf6ad5f3e36db63737e309784c7a.jpg)  
+Figure 2: Comparison of rate-distortion curves between our proposed methods and the baseline VQ-TensoRF on the Synthetic-NeRF dataset. The upper figure illustrates PSNR against file size, and the lower figure showcases SSIM in relation to file size.
+
+![](images/eeea437766b3bdc7b01c725dda44813974b503a41cdef5dce6a1af70d5479b77.jpg)  
+Figure 3: Qualitative results on Chair and Mic scenes from the Synthetic-NeRF dataset. From left to right: uncompressed, VQ-TensorF (average size 3.6 MB), ECTensoRF-L (3.4 MB), ECTensoRF-H (1.6 MB). Our decompressed renderings are barely distinguishable in quality from both uncompressed and VQ-compressed versions at a significantly reduced file size.  
+Figure 2 shows that our ECTensoRF-L outpaces VQ-TensoRF across various levels of compression in Synthetic-NeRF dataset with both PSNR and SSIM metrics. Rate-distortion curves for other datasets can be found in the Appendix A.2.
+
+Moreover, Table 2 shows the relative improvement of our method over VQ-TensoRF using Bjontegaard Delta (BD) BD-PSNR and BD-rate metrics (Bjontegaard, 2001), highlighting that our model achieves better PSNR and bit-rate across various compression levels.
+
+Training and rendering time. Training an uncompressed TensoRF model for a scene from the Synthetic-NeRF dataset takes around 15 minutes on an NVIDIA A6000 GPU. Running on top of that, our compression method takes an additional 40 minutes. Our framework is slower than the baseline VQ-TensoRF, which runs in 7 minutes on the same hardware. Regarding rendering, our approach adds a negligible overhead of roughly 2 seconds for the decompression of parameters. Once decompressed, the rendering procedure is the same as TensoRF.
+
+Compression details. The average storage size breakdown of our model on the Synthetic-NeRF dataset (with the configuration from Table 1) is provided in the Table 3. For the feature planes, we compresses them with the learned entropy model. All the other components (the renderer MLP, decoder, density/appearance vectors, learned masks, entropy
+
+Table 3: Storage size breakdown.  
+
+<table><tr><td>Component</td><td>Size (MB)</td></tr><tr><td>Feature planes</td><td>1.657</td></tr><tr><td>Decoder</td><td>1.380</td></tr><tr><td>Other components</td><td>0.366</td></tr><tr><td>Total</td><td>3.403</td></tr></table>
+
+bottleneck parameters and model config) are packed into a single file and compressed with LZ77 (ziv, 1977).
+
+# 5. Ablation Studies
+
+We conduct experiments to verify our design choices. We test on the Synthetic-NeRF datasets, with our ECTensoRF-L architecture.
+
+# 5.1. Advantages of per-scene optimization
+
+As outlined in Section 3.1, our compression methodology is optimized on a per-scene basis. However, this raises the question: how is the performance of traditional nonlinear transform coding on TensoRF compression, even on a small-scale dataset? To address this, we conduct a comparative analysis. We compress the TensoRF model by first training a compression network using an encoder-decoder architecture, similar to traditional nonlinear transform cod
+
+![](images/7d960dde89e51fc4d389fb4272b21f941692059e575127a2ca574b9f99a2041d.jpg)  
+Figure 4: Rate-distortion comparison between traditional nonlinear transform coding (green), trained across 7 scenes, and our per-scene compression methods (orange, blue).
+
+ing. Specifically, we train the compression network on seven scenes from the Synthetic-NeRF dataset and tested the trained model on the remaining scene (Lego). We compare this approach with our per-scene optimized model.
+
+Figure 4 shows the results of this experiment. Compared to per-scene training, pre-trained NTC suffers from inferior reconstruction quality. More specifically, the maximum PSNR that the pre-trained NTC can achieve is only  $24.52\mathrm{dB}$ , which is  $12.03\mathrm{dB}$  lower than the uncompressed PSNR value (36.55 dB) and also much lower than the PSNR values of per-scene trained models. However, we note that the major advantage of pre-trained NTC is a much faster compression time. Using a pre-trained NTC model also avoids the need to transmit the entropy model and the decoder, as we assume that the receiver always has access to them, which is similar to the image compression setting.
+
+# 5.2. Ablation on other design choices
+
+Using the encoder We first show the sub-optimal performance of ECTensoRF-L compression with an encoder. As discussed in Section 3.1, using an encoder leads to an irreducible amortization gap in optimization, and the resulting compression performance is worse, as shown in Figure 5.
+
+Training without Importance-Weighted Loss. We examine the rate-distortion curves of ECTensoRF-L, trained both with and without importance weight, as depicted in Figure 5. At an identical PSNR of  $32.98\mathrm{dB}$ , employing importance weight in training our model helps reduce the file size from  $4.59\mathrm{MB}$  to  $3.92\mathrm{MB}$ .
+
+The Effect of the Masked Entropy Model. To demonstrate the efficacy of our masked entropy model, we undertook a comparative analysis between the compression
+
+![](images/a0beee516f613e8bde7d498a0803ed3e80f282dcc90c29be1a192a8821f897ff.jpg)
+
+![](images/577d29744ae7dd0d49fc7eedfdfb3a8ad454099e2524bece7b35411d79594e7a.jpg)  
+Figure 5: Ablation studies. Top: rate-distortion comparison of our approach against a version with encoder, trained on a single scene. Bottom: comparisons between model versions with factorized prior and without importance weight.
+
+performance of ECTensoRF-L using the conventional factorized prior (Balle et al., 2016; 2018) and our masked model. The results related to rate distortion curves can be found in the bottom plot of Figure 5.
+
+It's noteworthy that, due to the additional overhead introduced by sending the masks, our results lag slightly behind the factorized prior in a low-rate setting. Yet, in medium to high-rate regimes, our prior emerges superior compared to the traditional factorized prior. To illustrate, for a PSNR value of 32.98 dB, the compressed file with the factorized prior occupies 4.26 MB. In contrast, our method employing the proposed masked entropy model results in a reduced file size of 3.92 MB.
+
+To further understand the behavior of our masked entropy model, we visualize the masks learned for the Chair and Mic scene from Synthetic-NeRF dataset in Figure 6. We can observe that the masks resemble the rendering objects when viewed from different angles, and they inherently ignore the background. This behavior is similar to the pruning
+
+![](images/fc37bbc205d0760ef9d89758086d6f246936e0972cb7b0fd82a316df9eab5521.jpg)  
+Figure 6: Ablation studies. We show the sparsity masks of our entropy model learned on the Chair and Mic scene.
+
+strategies employed in prior grid-based NeRF compression works (Li et al., 2023a;b; Deng & Tartaglione, 2023).
+
+More experimental results. We further conduct experiments on different latents initialization and end-to-end training in the Appendix A.2.
+
+# 6. Related Works and Discussion
+
+Grid-based NeRF compression. Since storage cost is a significant challenge of grid-based NeRF, several methods were proposed to solve this problem. Li et al. (2023a) introduces a three-stage approach, integrating voxel pruning and vector quantization (Gray, 1984) through a learnable codebook. Similarly, Re:NeRF (Deng & Tartaglione, 2023) employs voxel pruning, but adopts a strategy of sequentially removing and reintegrating parameters to prevent a significant drop in performance. Meanwhile, Takikawa et al. (2022) adopts the codebook idea from Instant-NGP (Müller et al., 2022), but substitutes hash-based encoding with a learned mapping that associates grid positions to corresponding codebook indices. However this approach requires considerable training memory. Li et al. (2023b) applies downsampling to the voxels and employs a network to enhance render quality. Our method shares some similarity to Li et al. (2023b), but we learn the downsampled latent codes with a novel entropy model to effectively compress them. Additionally, while our masked factorized prior also resembles the pruning mechanism used in previous works, our method differentiates itself by adaptively learning the masks instead of relying on fixed thresholds.
+
+Neural compression for NeRF. Applying neural compression to NeRF is a relatively young field. Bird et al. (2021) learns an entropy model to compress the MLP-based NeRF (Mildenhall et al., 2021) network weights, based on
+
+the prior model compression work of Oktay et al. (2019). In contrast, our work focuses on compressing the feature grids of grid-based NeRF. We additionally improve the conventional compression procedure and propose a novel entropy model. Concurrent to our work, Li et al. (2024) also applies neural compression to TensoRF by leveraging a pretrained image compression network.
+
+Discussion. Throughout this paper, our emphasis has been on applying neural compression techniques specifically to TensoRF. Nonetheless, our method has the potential to be applied to other grid-based NeRF methods beyond just TensoRF, such as Triplanes (Chan et al., 2022; Fridovich-Keil et al., 2023), Factor Fields ( Chen et al., 2023) or DVGO ( Sun et al., 2022). Taking DVGO as an example, we can learn a 4D latent code and have an entropy model to model its probability density. Then a decoder may decode this 4D latent code to render the scene.
+
+# 7. Conclusion
+
+In this study, we present a novel approach to applying neural compression to the TensoRF model, a prominent grid-based NeRF method. Our approach adapts traditional neural compression techniques, commonly used in image and video compression, to NeRF models. We develop an efficient per-scene optimization scheme and propose various designs, such as importance-weighted feature reconstruction and a masked entropy model. Our experiments demonstrate that we can significantly reduce storage requirements of a NeRF model with only a minimal compromise in rendering quality, and outperform previous NeRF compression baselines. More importantly, our compression method only adds minimal overhead to the rendering process.
+
+Limitation and future work. One limitation of our neural compression approach is the longer training time compared to the baseline VQ-TensoRF, as mentioned in Section 4. Additionally, the final compressed model still includes the cost of transmitting the decoder. Future work could focus on reducing compression time, learning a network compression model (Oktay et al., 2019; Girish et al., 2022) to compress the decoder network, and applying our method to other NeRF architectures.
+
+# Impact Statement
+
+Neural compression is a collection of methods that advance data compression with end-to-end learning approaches. Biased training data may influence how models reconstruct data and may lead to misrepresentations, e.g., of individuals, especially at low bitrates.
+
+# Acknowledgements
+
+The authors acknowledge support from the National Science Foundation (NSF) under an NSF CAREER Award (2047418), award numbers 2003237 and 2007719, by the Department of Energy under grant DE-SC0022331, the IARPA WRIVA program, and by gifts from Qualcomm and Disney. We also thank Justus Will for his meticulous proofreading and valuable suggestions for this paper.
+
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+
+# A. Appendix
+
+# A.1. Algorithm
+
+Algorithm 1 TensoRF-VM compression  
+Input: Pretrained TensoRF-VM model  
+Output: Compressed TensoRF-VM model  
+Calculate  $\{\mathbf{W}_i\}_{i=1}^3$  using Eq 4 and 5  
+Initialize  $\{\mathbf{Z}_i\}_{i=1}^3$  as 0-tensors  
+Initialize decoder  $D$  and entropy model  $P$  with masks parameters  $\{\pi_{\mathbf{M}_i}^0\}_{i=1}^3$  and  $\{\pi_{\mathbf{M}_i}^1\}_{i=1}^3$   
+while not converged do  
+Sample  $\{\mathbf{M}_i\}_{i=1}^3$  using Gumbel-Softmax as in Eq 8  
+Reconstruct  $\{\hat{\mathbf{P}}_i\}_{i=1}^3$  by Eq 7  
+Render the scene with  $\{\hat{\mathbf{P}}_i\}_{i=1}^3$   
+Calculate the loss in Eq 9 and update the model  
+end while
+
+# A.2. More experimental results
+
+# A.2.1. RATE-DISTORTION COMPARISON ON OTHER DATASETS
+
+We further compare the rate-distortion curves of ECTensoRF and the baseline VQ-TensoRF on the Synthetic-NSVF, LLFF and Tanks&Temples datasets in Figure 7, 8 and 9.
+
+![](images/129ab238e80814536af9b0615ad8e0501d8de8c4bc21439bcd558e50ee53f83c.jpg)  
+Figure 7: Comparison on Synthetic-NSVF dataset.
+
+![](images/1a9dfeedecd3403058734561e7f65e6432c300f9756ee286e686c549834c796a.jpg)
+
+# A.2.2. ADDITIONAL EXPERIMENTS
+
+Latent initialization. We compare the performance of Gaussian initialization and Zero initialization of the latents code. The results are shown in Table 4.
+
+![](images/e522092b5cea06f9a51ce06d3f2acdd2a6174f06fe44462b5ffcce04edf40349.jpg)  
+Figure 8: Comparison on LLFF dataset.
+
+![](images/c1f3288bdefbc2b56f25e9fe7d51c87e2f3cde80cf5da88d840a3894e57c6f13.jpg)
+
+![](images/3b5382bd1c120babec3a93ad74a523805c971cf43bc7e766e9ead73830ceab91.jpg)  
+Figure 9: Comparison on Tanks and Temples dataset.
+
+![](images/876f2c8e4b0de29149094e1ca29fb9468429057d357fb1a938a7a9038d571863.jpg)
+
+Table 4: Comparison of Zero Initialization vs. Gaussian Initialization  
+
+<table><tr><td>λ Values</td><td colspan="2">Zero Initialization</td><td colspan="2">Gaussian Initialization</td></tr><tr><td></td><td>PSNR (dB)</td><td>Size (MB)</td><td>PSNR (dB)</td><td>Size (MB)</td></tr><tr><td>2e-2</td><td>31.31</td><td>1.86</td><td>31.13</td><td>1.85</td></tr><tr><td>1e-2</td><td>31.80</td><td>1.99</td><td>31.75</td><td>1.99</td></tr><tr><td>5e-3</td><td>32.25</td><td>2.20</td><td>32.26</td><td>2.20</td></tr><tr><td>1e-3</td><td>32.83</td><td>3.00</td><td>32.83</td><td>3.01</td></tr><tr><td>5e-4</td><td>32.93</td><td>3.40</td><td>32.92</td><td>3.42</td></tr><tr><td>2e-4</td><td>32.98</td><td>3.92</td><td>32.98</td><td>3.94</td></tr><tr><td>1e-4</td><td>33.00</td><td>4.24</td><td>32.99</td><td>4.26</td></tr></table>
+
+End-to-end training. We conduct experiments to compare the performance of our two-stage training (by first using a pre-trained TensoRF model, and train the compression model) and a single stage (by training the compression model from scratch). We show the results in Table 5.
+
+Table 5: Comparison of End-to-End Training vs. Two Stages Training  
+
+<table><tr><td>λ Values</td><td colspan="2">End-to-End Training</td><td colspan="2">Two Stages Training</td></tr><tr><td></td><td>PSNR (dB)</td><td>Size (MB)</td><td>PSNR (dB)</td><td>Size (MB)</td></tr><tr><td>2e-2</td><td>25.86</td><td>1.69</td><td>31.31</td><td>1.86</td></tr><tr><td>1e-2</td><td>28.30</td><td>1.71</td><td>31.80</td><td>1.99</td></tr><tr><td>5e-3</td><td>30.05</td><td>1.81</td><td>32.25</td><td>2.20</td></tr><tr><td>1e-3</td><td>31.29</td><td>2.39</td><td>32.83</td><td>3.00</td></tr><tr><td>5e-4</td><td>31.53</td><td>2.57</td><td>32.93</td><td>3.40</td></tr><tr><td>2e-4</td><td>31.86</td><td>2.69</td><td>32.98</td><td>3.92</td></tr><tr><td>1e-4</td><td>31.80</td><td>2.95</td><td>33.00</td><td>4.24</td></tr></table>
+
+Hyperprior model. We also perform experiments using a version of hyperprior model (Ballé et al., 2018) with our masking mechanism. More specifically, we apply masking on both the hyper-latents and latents. Both type of latents are directly optimized without using amortized inference. The hyper decoder has two transposed convolutional layers with SELU activation. We show the result on NeRF-Synthetic on Table 6.
+
+Table 6: Comparison of ECTensorF-L with and without Hyperprior  
+
+<table><tr><td>λ Values</td><td colspan="2">ECTensorF-L + Hyperprior</td><td colspan="2">ECTensorF-L</td></tr><tr><td></td><td>PSNR (dB)</td><td>Size (MB)</td><td>PSNR (dB)</td><td>Size (MB)</td></tr><tr><td>2e-2</td><td>31.31</td><td>1.92</td><td>31.31</td><td>1.86</td></tr><tr><td>1e-2</td><td>31.92</td><td>2.04</td><td>31.80</td><td>1.99</td></tr><tr><td>5e-3</td><td>32.35</td><td>2.25</td><td>32.25</td><td>2.20</td></tr><tr><td>1e-3</td><td>32.85</td><td>2.97</td><td>32.83</td><td>3.00</td></tr><tr><td>5e-4</td><td>32.93</td><td>3.32</td><td>32.93</td><td>3.40</td></tr><tr><td>2e-4</td><td>32.98</td><td>3.72</td><td>32.98</td><td>3.92</td></tr><tr><td>1e-4</td><td>33.00</td><td>3.95</td><td>33.00</td><td>4.24</td></tr></table>
+
+At lower bit rates, the hyperprior is slightly worse than the ECTensoRF-L baseline because of the irreducible cost to transmit the hyper decoder and hyper entropy model. At higher bit rates, the compression performance with the hyperprior method is better than using only a single entropy model, which aligns with prior observations in image compression (Ballé et al., 2018).
+
+Preliminary results for Factor Fields. We show the potential of applying our method to other grid-based NeRF architectures. We choose Factor Fields (Chen et al., 2023) to experiment with. We show the result of our method for Factor Fields in Table 7. Note that for Factor Fields, we compress the basis 4D tensors and do not compress the coefficient 4D tensors.
+
+Table 7: Factor Fields experiments  
+
+<table><tr><td>λ Values</td><td>PSNR (dB)</td><td>Rate (MB)</td></tr><tr><td>1e-3</td><td>26.19</td><td>1.12</td></tr><tr><td>1e-4</td><td>29.67</td><td>1.23</td></tr><tr><td>1e-5</td><td>31.35</td><td>1.82</td></tr><tr><td>Uncompressed</td><td>33.09</td><td>18.89</td></tr></table>
+
+# A.2.3. MORE QUALITATIVE RESULTS
+
+We show qualitative results on all scenes from Synthetic-NeRF, Synthetic-NSVF, LLFF and Tanks&Temples datasets in Figure 10, 11, 12 and 13.
+
+![](images/01f640cf29aff104335801a53d7dd11b543c417e72cab98c7a7d08afdc71e13e.jpg)  
+Figure 10: Qualitative results on Tanks and Temples dataset. From left to right: TensoRF, ECTensoRF-L, ECTensoRF-H, ECTensoRF-L difference and ECTensoRF-H difference.
+
+![](images/b4cc3e081876a60ad14daeabc896f15beb8d36e63451fa08bed15e1c2e5b01b4.jpg)  
+Figure 11: Qualitative results on Synthetic-NeRF dataset. From left to right: TensoRF, ECTensoRF-L, ECTensoRF-H, ECTensoRF-L difference and ECTensoRF-H difference.
+
+![](images/3072ecc7e61a1af72ba2298889531a937561e7577e9fc1ea19a322b2c5de0a60.jpg)  
+Figure 12: Qualitative results on Synthetic-NSVF dataset. From left to right: TensoRF, ECTensoRF-L, ECTensoRF-H, ECTensoRF-L difference and ECTensoRF-H difference.
+
+![](images/99c34d2cce2d386266e012c3216b1cba967539df94130ab7ddc7d59ebcb8d49c.jpg)
+
+![](images/c67f9e6d4f4d40c965ec9c69b6d0d47bff86171531d2fd5ae621217ae84dfb41.jpg)
+
+![](images/34e1cab0d0bf0ac446f1c8b2854813fa212087c9f4cf0695a07399df456a181d.jpg)
+
+![](images/5a1175cf87a5188a84e8fe474451997ff4580d4d3934b230e1ae9b58e260e9fd.jpg)
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+![](images/de15f2c901b02f38df11cd61c3cc9bda0bb221939c9c5f3e68567a0536f3932e.jpg)
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+![](images/b090c06a487a11f8d23fef6185ee1a192cd666625ecc11400c3a94fe0187e9ff.jpg)
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+![](images/b9dd7fa39d5857d72b399832157af1a2c815e60357242f1b2cf2756a70b6cb3d.jpg)
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+![](images/ca8e140e41d591721f0f62645a2123638e60b9b16629379a30e1c2992f6df345.jpg)
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+![](images/879dad31b8c36f9702f03ab1bfa78ae6ac05b1e2743e511663b12cffbcd0f7d4.jpg)
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+![](images/38fbb0390263c26ff6bf32dca2ec6e57d31d1bc4b350122183e777cf7503f042.jpg)  
+Figure 13: Qualitative results on LLFF dataset. From left to right: TensoRF, ECTensoRF-L, ECTensoRF-H, ECTensoRF-L difference and ECTensoRF-H difference.
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2024/Neural Networks Learn Statistics of Increasing Complexity/b3f9046f-6da2-4702-aa86-d3792a81ed04_origin.pdf

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2024/Neural Networks Learn Statistics of Increasing Complexity/full.md

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+# Neural Networks Learn Statistics of Increasing Complexity
+
+Nora Belrose<sup>1</sup> Quintin Pope<sup>2</sup> Lucia Quirke<sup>1</sup> Alex Mallen<sup>1</sup> Xiaoli Fern<sup>2</sup>
+
+# Abstract
+
+The distributional simplicity bias (DSB) posits that neural networks learn low-order moments of the data distribution first, before moving on to higher-order correlations. In this work, we present compelling new evidence for the DSB by showing that networks automatically learn to perform well on maximum-entropy distributions whose low-order statistics match those of the training set early in training, then lose this ability later. We also extend the DSB to discrete domains by proving an equivalence between token  $n$ -gram frequencies and the moments of embedding vectors, and by finding empirical evidence for the bias in LLMs. Finally we use optimal transport methods to surgically edit the low-order statistics of one class of images to match those of another, and show early-training networks treat the edited images as if they were drawn from the target class.
+
+Code is available at https://github.com/EleutherAI/features-across-time.
+
+# 1. Introduction
+
+Neural networks exhibit a remarkable ability to fit complex datasets while generalizing to unseen data points and distributions. This is especially surprising given that deep networks can perfectly fit random labels (Zhang et al., 2021), and it is possible to intentionally "poison" networks so that they achieve zero training loss while behaving randomly on a held out test set (Huang et al., 2020).
+
+A recently proposed explanation for this phenomenon is the distributional simplicity bias (DSB): neural networks learn to exploit the lower-order statistics of the input data first–e.g. mean and (co)variance– before learning to use its higher-order statistics, such as (co)skewness or (co)kurtosis. Refinetti et al. (2023) provide evidence for the DSB by training networks on a sequence of synthetic datasets that
+
+act as increasingly precise approximations to the real data, showing that early checkpoints perform about as well on real data as checkpoints trained directly on the real data.
+
+We build on Refinetti et al. (2023) by inverting their experimental setup. We train our models on real datasets, then test them throughout training on synthetic data that probe the model's reliance on statistics of different orders. We believe this experimental design provides more direct evidence about the generalization behavior of commonly used models and training practices.
+
+Our primary theoretical contributions are to (1) motivate the DSB through a Taylor expansion of the expected loss, (2) propose criteria quantifying whether a model "uses" statistics up to order  $k$  by checking that the model is sensitive to interventions on the first  $k$  statistics, while being robust to interventions on higher-order statistics, (3) describe efficient methods of producing synthetic data that let us investigate whether models satisfy the above criteria, and (4) extend the DSB to discrete domains by proving an equivalence between token  $n$ -gram frequencies and the moments of sequences of embedding vectors.
+
+We use a Taylor series expansion to express a model's expected loss as a sum over the central moments of an evaluation dataset. This connection provides some motivation for the DSB. Specifically, if during training, a network's loss is well approximated by the first  $k$  terms of its Taylor expansion, then the model should only be sensitive to statistics up to order  $k$ , and we argue that earlier terms of the expansion will generally become relevant before later terms.
+
+We describe two intuitive criteria that a model sensitive to statistics up to order  $k$  should satisfy: (1) changing the first  $k$  statistics of data from class A to match class B should cause the model to classify the modified data as class B, and (2) models should be unaffected by "deleting" higher-order data statistics. We evaluate whether image classification networks satisfy the above criteria during training through extensive empirical experiments across a variety of network architectures and image datasets.
+
+We evaluate whether the network satisfies criterion (1) by generating synthetic datasets where we "graft" the means and covariances of one class onto images of another class, and evaluating whether the network's classifiers the result
+
+![](images/49f246ca8a05b1546cd8335652b199c29620374ca60ce4d6dc25c9f61707b532.jpg)  
+Figure 1. (left) Pekinese dog image from the ImageNet training set. (center) Image after quantile normalizing its pixels to match the marginal distribution of the goldfish class on ImageNet. The grass is now a slightly darker shade of green and the dog's fur has a reddish hue. (right) Synthetic "goldfish" generated by sampling each pixel independently from its marginal distribution.
+
+![](images/cffcdde6543af5ccb10e98c967546aeac9d066aec6220d8f78729648b7f57749.jpg)
+
+![](images/86db1f81f6d65fff4b2235f2604cdd35826429374eddf81185ea37596c0ea7f9.jpg)  
+Figure 2. Non-cherrypicked "fake" images produced by maximum entropy sampling using only the first two moments of the class-conditional distributions, and a hypercube constraint. Fake MNIST digits are clearly recognizable, SVHN digits less so, whereas fake CIFAR-10 images look nothing like their respective classes: airplane, automobile, bird, cat, deer, dog, frog, horse, ship, truck.
+
+ing data as belonging to the target class. We formalize this notion of "grafting" statistics with optimal transport (OT) theory, using an analytic formula to map samples from one class-conditional distribution to another, while minimizing the expected squared Euclidean distance the samples are moved. We also describe coordinatewise quantile normalization, an OT method that changes the marginal distribution of each coordinate of the input to match a target class.
+
+We evaluate the degree to which networks satisfy criterion (2) by generating synthetic data that match the class-conditional means and covariances, but are otherwise maximum entropy. We generate two datasets for this purpose. One dataset comes from sampling from a Gaussian distribution with matching mean and covariances. The other dataset comes from incorporating the constraints on image pixel values. We propose a novel gradient-based optimization method to produce samples from a hypercube-constrained maximum entropy distribution. We additionally describe independent coordinate sampling, a first order method of generating hypercube-constrained maximum entropy samples using only means.
+
+Across models and datasets, we find a common pattern where criteria (1) and (2) hold early in training, with networks largely classifying images according to the means and covariances of the distributions from which they're drawn. But as training progresses, networks become sensitive to higher-order statistics, resulting in a U-shaped loss curve.
+
+We also evaluate EleutherAI's Pythia autoregressive language models (Biderman et al., 2023) on synthetic data sampled from unigram and bigram models trained on the Pile (Gao et al., 2020). We find a fascinating "double descent" (Vallet et al., 1989; Belkin et al., 2019) phenomenon where models initially mirror the same U-shaped scaling observed in image classifiers, then use in-context learning to achieve even lower loss later in training.
+
+![](images/b3582feb0b2287e9430060ca6a889c5412f9e4c0433bfb95c4353c449dffca44.jpg)
+
+For a thorough review of related work on simplicity biases in machine learning, see Appendix A.
+
+# 2. Theory and Methods
+
+Let  $\mathcal{L}(\mathbf{x})$  denote the loss of a neural network evaluated on input  $\mathbf{x}$ . If  $\mathcal{L}_{\theta}$  is analytic with an adequate radius of convergence, we can Taylor expand the loss for any given  $\mathbf{x}$  around the mean input  $\pmb{\mu}$  as:
+
+$$
+\mathcal {L} (\mathbf {x}) = \sum_ {\alpha \in \mathbb {N} ^ {d}} \frac {(\mathbf {x} - \boldsymbol {\mu}) ^ {\alpha}}{\alpha !} (\partial^ {\alpha} \mathcal {L}) (\boldsymbol {\mu}), \tag {1}
+$$
+
+where  $\alpha$  is a multi-index, or a  $d$ -tuple assigning an integer to each coordinate of  $\mathbf{x}$ . Recall that taking a vector to the power of a multi-index denotes a product of the components of the vector, where each component of the index indicates the multiplicity: e.g. if  $\alpha = (1,4,6)$ , the expression  $(\mathbf{x} - \boldsymbol{\mu})^{\alpha}$  denotes the product  $(x_{1} - \mu_{1})(x_{2} - \mu_{2})^{4}(x_{3} - \mu_{3})^{6}$ . Similarly,  $(\partial^{\alpha}\mathcal{L})$  is shorthand for the mixed partial derivative  $\frac{\partial^{11}}{\partial x_1\partial x_2^4\partial x_3^6}\mathcal{L}$ . The factorial  $\alpha!$  denotes the product of the factorials of the components:  $1! \times 4! \times 6! = 17280$ .
+
+If  $\mathbf{x}$  is drawn from a distribution with compact support, which is true for images and text, we can take the expectation of both sides of Eq. 1. This leads to an expression summing over all the central moments of  $\mathbf{x}$  multiplied by the corresponding partial derivatives of  $\mathcal{L}$  evaluated at  $\mu$ :
+
+$$
+\mathbb {E} [ \mathcal {L} (\mathbf {x}) ] = \sum_ {\alpha \in \mathbb {N} ^ {d}} \frac {(\partial^ {\alpha} \mathcal {L}) (\boldsymbol {\mu})}{\alpha !} \underbrace {\mathbb {E} [ (\mathbf {x} - \boldsymbol {\mu}) ^ {\alpha} ]} _ {\text {c e n t r a l m o m e n t}} \tag {2}
+$$
+
+Equation 2 suggests a close connection between the moments of the data distribution and the expected loss of a neural network evaluated on that distribution.4
+
+![](images/abb5d471a6a53b33408c5faf56731022b40690f300f840c008f21f7496d59bde.jpg)  
+Figure 3. Accuracy of computer vision models when evaluated on images edited with optimal transport maps as described in Sec. 2.3, using the target class, not the source class, as the label. Between roughly  $2^{4}$  and  $2^{12}$  training steps, all models classify the CQN-edited images coming from target class, with a peak in accuracy at  $2^{9}$ .
+
+![](images/a693a40a4eea2d9b2b3cd1b38cd6d62ece8da33ed20c5520c703f3cac74b0181.jpg)
+
+![](images/ad227c9cf67693f97941fa35874d579cf3d7c54122ef16307b66fc2b671e9ebd.jpg)
+
+# 2.1. Intuition
+
+Without loss of generality, assume the input is constrained to the unit hypercube. Since every coordinate of  $\mathbf{x}$  is no greater than 1, the moments will have magnitudes that monotonically decrease with increasing order; for example,  $\mathbb{E}[x_i x_j] \leq \mathbb{E}[x_i]$  for any  $i, j \in 1 \ldots d$ .
+
+Indeed, we would expect the moment magnitude to decay exponentially with order when the coordinates are independent, roughly counterbalancing the exponential increase in the number of distinct moments at higher orders. Assuming the higher derivatives of  $\mathcal{L}$  are reasonably well-behaved at initialization, the  $\frac{1}{n!}$  Taylor coefficients will then cause the contribution of higher-order moments to Eq. 2 to decay monotonically and factorially fast with order.
+
+As training progresses however, the derivatives of  $\mathcal{L}$  become correlated with the corresponding moments, potentially inflating the magnitude of higher-order terms in Eq. 2. It then seems natural to suppose that the magnitude of higher-order terms will grow in roughly monotonic order- that is, the second order term will become important first, followed by the third order term, and so on<sup>6</sup>.
+
+when the support is compact.
+
+4Expanding around an an arbitrary point a would yield an expression containing moments about a, and our analysis would otherwise be unchanged.  
+This is true of images with standard PyTorch preprocessing and one-hot encoded token sequences; other inputs can be rescaled to match this criterion, given our assumption of compact support.  
+Another argument for monotonicity is that earlier terms account for factorially more of the loss at initialization, and are thus plausibly higher-priority targets for gradient descent, until the optimizer is no longer able to easily reduce the loss further by better matching the associated statistics and moves on to higher-order terms.
+
+# 2.2. Criteria
+
+Intuitively, if a model only "uses" low-order statistics of the input distribution, this means its behavior should be strongly affected by interventions on the lower-order statistics of the input, but largely unaffected by interventions on the higher-order statistics. More specifically:
+
+1. "Grafting" the low-order statistics of class  $B$  onto class  $A$  should cause the model to treat examples from  $A$  as if they were from  $B$ .  
+2. "Deleting" the information contributed by higher-order statistics should not harm the model's performance.
+
+We operationalize both criteria more precisely below and explain how we produce synthetic data that lets us evaluate the degree to which a given model satisfies each criterion.
+
+# 2.3. Optimal Transport
+
+We operationalize Criterion 1 using optimal transport (OT) theory, which provides tools for transforming samples from one probability distribution into samples from another while minimizing the average distance that samples are moved. We use three OT methods in our experiments: bounded shift and coordinatewise quantile normalization, which primarily affect the first order moments of the distribution, and Gaussian OT, which affects all first and second-order moments.
+
+Bounded Shift In Appendix E, we derive an algorithm for shifting the mean of an empirical distribution to a desired value, while keeping it constrained to the interval [0, 1], and minimizing the transport cost. We use this algorithm to graft the mean of one class onto another, while ensuring the pixel intensities of the edited images are valid.
+
+![](images/cfdc3c5248af636c0f578a59dfdcf806b97520b3a8facee987144c4eab170a92.jpg)  
+Figure 4. Rows 1-3 show how Gaussian optimal transport affects the example CIFAR-10 airplane, bird and truck images. Each row starts with the original unedited image on the left, with each subsequent column showing the effects of editing that image's first two moments to match the class-conditional distributions of a particular target class (top). The bottom left image is the average of all test set images. Black borders indicate either an original image, or a "self-to-self" edited image.
+
+Coordinatewise Quantile Normalization (CQN) Quantile normalization is a technique for making two scalar random variables identical in their statistical properties. When applied coordinatewise to the input of a neural network, such as an image, it ensures that the coordinatewise margins match those of a target distribution, while keeping the correlations between coordinates largely intact, as illustrated by how the edited Pekinese dog image in Fig 1 (center) remains a recognizable dog image.
+
+CQN works as follows. If a random variable  $X$  has cumulative distribution function  $F_{X}(x)$ , the transformed variable  $F_{X}(X)$  will have the standard uniform distribution  $\mathrm{Unif}(0,1)$ . Conversely, given variables  $U \sim \mathrm{Unif}(0,1)$  and  $Y$ , the transformed variable  $F_{Y}^{-1}(U)$  will be equal in distribution to  $Y$ . Composing these transformations together yields quantile normalization. It can be shown that  $F_{Y}^{-1} \circ F_{X}$  is the optimal transport map from  $X$  to  $Y$  for a large class of cost functions (Santambrogio, 2015, Ch. 2.2), and is thus ideal for editing the first order statistics of a distribution while minimally perturbing higher-order statistics.
+
+Gaussian Optimal Transport Given two Gaussians  $P = \mathcal{N}(\pmb{\mu}_P, \pmb{\Sigma}_P)$  and  $Q = \mathcal{N}(\pmb{\mu}_Q, \pmb{\Sigma}_Q)$  supported on  $\mathbb{R}^d$ , the map  $T(\mathbf{x}) = \mathbf{A}(\mathbf{x} - \mathbf{m}_P) + \mathbf{m}_Q$  is the optimal transport map from  $P$  to  $Q$  under the squared Euclidean cost function, where
+
+$$
+\mathbf {A} = \boldsymbol {\Sigma} _ {P} ^ {- 1 / 2} \left(\boldsymbol {\Sigma} _ {P} ^ {1 / 2} \boldsymbol {\Sigma} _ {Q} \boldsymbol {\Sigma} _ {P} ^ {1 / 2}\right) ^ {1 / 2} \boldsymbol {\Sigma} _ {P} ^ {- 1 / 2}. \tag {3}
+$$
+
+More generally, if  $P$  is an arbitrary distribution with finite second moments,  $T(\mathbf{x})$  will transport it to a distribution with mean  $\pmb{\mu}_{Q}$  and covariance  $\pmb{\Sigma}_{Q}$ , and this map will minimize the cost  $\mathbb{E}_P[\| \mathbf{x} - T(\mathbf{x})\|_2^2]$  (Dowson & Landau, 1982).
+
+Given  $k$  image classes, each containing tensors of shape  $C \times H \times W$ , we unroll the tensors into vectors of size
+
+$CHW$ , then compute their sample means and covariance matrices, and plug these statistics into Eq. 3 to get the  $k(k - 1)$  optimal transport maps from each class to every other class.
+
+# 2.4. Maximum Entropy Sampling
+
+We can operationalize Criterion 2 using the principle of maximum entropy, which provides a principled method for constructing probability distributions based on "partial knowledge" (Jaynes, 1957). Here the partial knowledge consists of low-order statistics derived from a training dataset, but we otherwise want to minimize the information content of the higher-order statistics. We therefore want to construct the maximum entropy distribution  $P$  consistent with these low-order statistics, $^{8}$  then evaluate a neural network on samples drawn from  $P$ .
+
+Famously, the maximum entropy distribution supported on  $\mathbb{R}^d$  with known mean  $\pmb{\mu}$  and covariance matrix  $\pmb{\Sigma}$  is the Gaussian distribution  $\mathcal{N}(\pmb{\mu},\pmb{\Sigma})$ . We therefore use Gaussians in many of our experiments in Sec 3. In addition to Gaussians, we use hypercube constrained sampling to generate a set of synthetic samples using  $2^{\mathrm{nd}}$  order statistics and another set using both  $2^{\mathrm{nd}}$  and  $3^{\mathrm{rd}}$  order statistics, though that last set suffered from significant convergence issues when trying to match the  $3^{\mathrm{rd}}$  order statistics (see Appendix C for  $3^{\mathrm{rd}}$  order results and Appendix D for discussion of convergence issues). We also use two first-order methods
+
+![](images/5d66ad5f8a86a4c5deb8653db6e662b3017bf31794ef0ab4e74480c6436f3363.jpg)  
+Figure 5. Accuracy of computer vision models being trained on the standard CIFAR-10 training set, and being evaluated on maximum-entropy synthetic data with matching statistics of  $1^{\mathrm{st}}$  or  $2^{\mathrm{nd}}$  order.  $3^{\mathrm{rd}}$  order results are in Appendix C, with Appendix D discussing issues with those results.
+
+![](images/8461dc79946376581b32e7dfd29411c4d1566b5bc97ec4f90f8caa455c208c54.jpg)  
+Training steps
+
+![](images/db65c791091ca6b809067d82796163bde5a99583f80ce052a3a3b58df20889b9.jpg)
+
+(Conrad sampling and independent coordinate sampling). We describe these methods below.
+
+Hypercube Constraints One problem with using Gaussians to generate synthetic images is that natural images are constrained to a hypercube: RGB pixel intensities are in the range [0, 255], but nonsingular Gaussian distributions assign positive probability density to all of  $\mathbb{R}^d$ , so that a typical sample will often lie outside the hypercube of natural images. We might expect neural networks to quickly adapt to such a simple box constraint on the support, so we would ideally like to subject our synthetic images to this constraint.
+
+Conrad Distribution We prove in Theorem F.1 that the maximum entropy distribution supported on  $[0, 1]$  with a fixed mean  $\mu \neq \frac{1}{2}$  and unconstrained variance takes the form  $p(x) = \frac{b\exp(-bx + b)}{\exp(b) - 1}$ , where the parameter  $b$  can be found using Newton's method. This formula is not well-known, although an alternative derivation can be found in Conrad (2004). To isolate the effect of first-order statistics, we first fit a Conrad distribution to the mean of each coordinate of the images. We then generate synthetic images using inverse transform sampling to produce a value for each coordinate independently.
+
+Approximate Sampling For many sets of constraints, there is no known closed-form solution for the density, precluding standard sampling techniques like Markov chain Monte Carlo. For example, in the 1D case the maxi
+
+mum entropy probability density with known mean and variance supported on a finite interval  $[a,b]$  has the form  $p(x) = \exp (-\lambda_0 - \lambda_1x - \lambda_2x^2)$  (Dowson & Wragg, 1973),<sup>10</sup> but we are unaware of any such analytical formula for the multidimensional case.<sup>11</sup>
+
+For these cases, we propose a novel technique for approximate sampling: use gradient-based optimization to directly produce a finite set of samples whose statistics match the desired ones, while maximizing the Kozachenko-Leonenko estimate for the entropy of the implicit population distribution (Kozachenko & Leonenko, 1987; Sablayrolles et al., 2018). See Appendix H for implementation details and Appendix I for a discussion of computational requirements.
+
+Independent Coordinate Sampling (ICS) In the preceding sections, we decomposed the input distribution into its moments. Another possible decomposition is given by Sklar's theorem, which states that the distribution of any random vector  $(X_{1},\ldots ,X_{d})$  is uniquely determined by its coordinatewise marginal CDFs  $F_{X_i}(x) = \mathbb{P}(X_i\leq x)$  and a copula function  $C:[0,1]^d\to [0,1]$  that combines the marginal CDFs into a multivariate CDF  $F_{X}(\mathbf{x}) = \mathbb{P}(X_{1}\leq x_{1},\dots ,X_{d}\leq x_{d})$  (Sklar, 1959). The maximum entropy copula simply takes the product of the marginal CDFs, and corresponds to a random vector with independent coordinates. We can efficiently sample from this distribution by estimating an empirical CDF for each coordinate, then sampling from each CDF independently.
+
+By constraining the coordinatewise marginals, we ensure that all of the homogeneous moments, or moments of the form  $\mathbb{E}[(x_i)^n]$ , match those of the true data distribution, while the mixed moments, e.g.  $\mathbb{E}[x_i x_j]$  for  $i \neq j$ , will generally not match. In high dimension, almost all moments of order greater than one are mixed rather than homogeneous, so ICS matches the first order moments and almost none of the higher order ones. For this reason we classify it as a "first order" method.
+
+# 2.5. Discrete Domains
+
+Neural networks use embeddings to convert discrete inputs into vectors of real numbers. The embedding operation can be viewed as a matrix multiplication, wherein the discrete inputs are converted into one-hot vectors we then multiply by the embedding matrix. If the input is a sequence, the result is a sequence of one-hot vectors, or a one-hot matrix.
+
+Just as we unroll images into vectors to compute their moments, we can similarly unroll one-hot matrices to compute their moments. Strikingly, we find that these moments correspond to token  $n$ -gram frequencies: $^{12}$
+
+Theorem 2.1. [n-gram statistics are moments] Let  $\mathcal{V}^N$  be the set of token sequences of length  $N$  drawn from a finite vocabulary  $\mathcal{V}$ , let  $P$  be a distribution on  $\mathcal{V}^N$ , and let  $f:\mathcal{V}^N\to \{0,1\}^{N\cdot |\mathcal{V}|}$  be the function that encodes a length- $N$  sequence of tokens as a flattened concatenation of  $N$  one-hot vectors of dimension  $|\mathcal{V}|$ . Let  $f_{\sharp}P$  be the pushforward of  $P$  through this one-hot encoding, i.e., its analogue in  $\{0,1\}^{N\cdot |\mathcal{V}|}$ .
+
+Then every moment of  $f_{\sharp}P$  is equal to an  $n$ -gram statistic of  $P$  and vice versa.
+
+Furthermore, for a fixed embedding matrix  $\mathbf{E}$ , two distributions over token sequences that have the same  $n$ -gram frequencies up to order  $k$  will induce distributions over embedding space with the same moments up to  $k$ :
+
+Theorem 2.2. [Equal embedding moments] Let  $\mathbf{E} \in \mathbb{R}^{|\mathcal{V}| \times d}$  be an embedding matrix, and let  $P$  and  $Q$  be two distributions over  $\mathcal{V}^N$ . Then if  $P$  and  $Q$  have the same n-gram statistics up to order  $k \geq 1$ , their embeddings under  $\mathbf{E}$  have the same moments up to order  $k$ .
+
+For proofs, see Appendix G.
+
+Given this equivalence, we can test the DSB in language
+
+12Our formal definition of the term "  $n$ -gram statistic" is non-standard in two respects: first, we include skip-grams (e.g. the dog, where the underscore is a wildcard token), and second, it is tied to an absolute position in the sequence. However, the Pythia language models we will consider in this paper were trained on chunks of text of uniform length sampled from larger documents (Biderman et al., 2023), so the absolute position should not significantly affect the  $n$ -gram probabilities. We therefore assume in what follows that  $n$ -gram statistics exhibit translation invariance.
+
+models with maximum entropy sampling, just like the computer vision case. Given known  $n$ -gram frequencies up to order  $k$ , we produce maximum entropy samples using a  $k$ -gram autoregressive language model. For example, if only bigram frequencies are known, this corresponds to a Markov chain where the distribution of each token depends only on the token immediately preceding it.
+
+# 3. Image Classification
+
+# 3.1. Datasets.
+
+Because Gaussian optimal transport (Sec. 2.3) requires  $O(d^{3})$  compute and  $O(d^{2})$  memory, $^{13}$  we focus on datasets with  $32 \times 32$  or  $64 \times 64$  resolution images for our primary experiments. Specifically, we examine the popular image classification datasets CIFAR-10 (Krizhevsky et al., 2009), Fashion MNIST (Xiao et al., 2017), MNIST (LeCun et al., 1998), and SVHN (Netzer et al., 2011).
+
+We also build a new image classification dataset, CIFAR-Net, consisting of 200K images at  $64 \times 64$  resolution sampled from ImageNet-21K, using ten coarse-grained classes that roughly match those of CIFAR-10. The larger number of images per class (20K) allows us to get a good estimate of the class-conditional covariance matrices needed for Gaussian optimal transport, which at this resolution contain  $(3 \times 64 \times 64)^{2} \approx 1.5 \times 10^{8}$  entries each. See Appendix B for more details on CIFARNet.
+
+# 3.2. Architectures.
+
+We focus on state-of-the-art computer vision architectures in our experiments. Specifically, we use ConvNeXt V2 (Woo et al., 2023) and Swin Transformer V2 (Liu et al., 2022), which Goldblum et al. (2023) recently found to have the best performance on a variety of tasks. We train for  $2^{16}$  steps with batch size 128, using the AdamW optimizer (Loshchilov & Hutter, 2018) with  $\beta_{1} = 0.9$ ,  $\beta_{2} = 0.95$ , and a linear learning rate decay schedule starting at  $10^{-3}$  with a warmup of 2000 steps (Ma & Yarats, 2021). For data augmentation, we employ RandAugment (Cubuk et al., 2020) followed by random horizontal flips and random crops.
+
+To examine the effect of model scale on our results, we sweep over the Atto, Femto, Pico, Nano, and Tiny sizes for ConvNeXt V2, and we also construct Swin Transformers of
+
+roughly analogous sizes. $^{15}$
+
+To ensure our results are insensitive to the choice of optimizer and learning rate schedule, we also perform experiments with RegNet-Y (Radosavovic et al., 2020) using SGD with momentum and no LR warmup.
+
+# 3.3. Results
+
+We display our results on CIFAR10 in Figures 3 and 5, see Appendix C for other datasets.
+
+Optimal transport We measure the effect of optimal transport interventions by computing the accuracy or loss of the model with respect to the target class, rather than the source class. If the intervention is ineffective, we would expect the accuracy to be much lower than the random baseline of  $10\%$ , because the model should confidently classify the images as belonging to the source class. Strikingly, all models we tested get substantially higher than  $10\%$  accuracy w.r.t. the target labels, with ConvNeXt peaking at over  $40\%$  accuracy on  $2^{\text{nd}}$  order-edited images after  $2^{10}$  training steps.
+
+Maximum entropy sampling We include five different conditions in our maximum entropy sampling experiments:  $1^{\text{st}}$  order (Conrad and ICS),  $2^{\text{nd}}$  order (Gaussian sampling), and both  $2^{\text{nd}}$  and  $3^{\text{rd}}$  order approximate sampling plus a hypercube constraint. However, we place the  $3^{\text{rd}}$  order approximate sampling results in Appendix C because we were unable to match the coskewness and covariance statistics simultaneously when generating the synthetic datasets, making their interpretation difficult. See Appendix D for discussion.
+
+Overall, we find that accuracy on first order samples peaks earlier in training and has a lower maximum than accuracy on second order samples, followed by the  $2^{\mathrm{nd}}$  order hypercube-constrained samples. Remarkably, for some datasets early in training, we find some models achieve higher accuracy on the independent pixel samples than they do on images sampled from the real validation set!
+
+Non-monotonicity Across all datasets, we observe some degree of non-monotonicity in the accuracy curves: while models are quite sensitive to low-order moments early in training, they become less sensitive by the end, with accuracy often dipping below the random baseline. The degree of non-monotonicity varies by dataset, however. Very simple datasets like MNIST and Fashion MNIST show very little non-monotonicity, likely because the first and second moments of the data distribution are sufficient to produce
+
+very realistic-looking samples (Fig. 2).
+
+Overall, we found that model scale has a remarkably small effect on the learning curves, so we display curves averaged over scales in bold, with individual model scales shown as translucent lines. Each curve is itself averaged over three random seeds.
+
+# 4. Language Modeling
+
+To test the distributional simplicity bias in a discrete domain, we study EleutherAI's Pythia language model suite (Biderman et al., 2023), for which checkpoints are publicly available at log-spaced intervals throughout training. Model parameter counts range from 14 million to 12 billion.
+
+There are ten independent training runs for Pythia 14M, 70M, and 160M publicly available on the HuggingFace Hub, each using a different random seed. There are also five available seeds for Pythia 410M. We take advantage of these additional runs to examine the effect of random seed on our results. We also trained custom variants of Pythia 14M and 70M with an extended learning rate warmup period of 14,300 steps to isolate the effect of LR warmup.
+
+While we include skip-grams (e.g. the __ dog) in our formal definition of  $n$ -gram frequency (Def. G.1), we do not include them in these experiments for tractability reasons: they would greatly increase the memory and storage requirements of maximum entropy sampling. We hope to explore the effect of skip-gram statistics in future work.
+
+$n$ -gram language models We compute token unigram and bigram frequencies across Pythia's training corpus, the Pile (Gao et al., 2020), and use these statistics to construct maximum entropy  $n$ -gram language models. We autoregressively sample sequences of length 2049 from the  $n$ -gram LMs, and evaluate Pythia's cross entropy loss on these maximum entropy samples at each checkpoint.
+
+Additionally, we evaluate Pythia 12B's cross entropy loss over each token position in the maximum entropy  $n$ -gram sequences for four different checkpoints in order to detect the development of in-context learning. If in-context learning is involved in making predictions for these sequences at a training step, cross entropy loss should decrease over successive token positions in the sequence for that step (Olsson et al., 2022).
+
+# 4.1. Results
+
+We display our results on the Pythia suite in Fig. 6. See Appendix C for alternate model seeds and learning rate warmup. Overall we find that the random seed has very little effect on the learning curves, and lengthening the LR warmup period did not consistently affect their overall shape.
+
+![](images/3295599d6e187eff53e6841496dc09a2961fc9588ef605012fc7162aae358341.jpg)  
+Figure 6. Cross entropy loss of Pythia suite evaluated on unigram and bigram sequences (N=1024). For comparison, the Shannon entropy of the unigram distribution is 2.89 bits per byte (bpb), and is 2.04 bpb for the bigram distribution. Both plots exhibit "double descent" scaling: loss reaches a trough between  $2^{6}$  and  $2^{8}$  steps, increases until  $2^{10}$  steps, then decreases again as the model learns to match the data generating process in-context (Fig. 7).
+
+$n$ -gram sequence loss Consistent with the image classification tasks, unigram sequence loss consistently reaches its lowest point before bigram sequence loss and has a higher minimum value.
+
+Across all models, we observe non-monotonicity in the  $n$ -gram sequence loss curves, where loss steeply reduces and then increases to a lesser extent. However, unlike in the image classification tasks, the loss reverts to a monotonic regime later in training. We hypothesize that this is caused by the development of in-context learning sufficient to improve  $n$ -gram sequence predictions. We observe correlational evidence in the  $n$ -gram sequence loss over increasing token indices and training steps in Pythia 12B (Fig. 7), where in-context learning seems to emerge in the same training step where the non-monotonic regime ends. Fascinatingly, smaller models seem to resume the standard 'U'-shaped loss pattern in the later portions of training.[16]
+
+We speculate that this behavior may arise from a form of "catastrophic forgetting", in which all models initially learn low-order  $n$ -gram statistics, which are eventually eclipsed by more sophisticated features. Larger models have greater representational capacity, and so are better able to retain these early  $n$ -gram features.
+
+In-context learning We follow Kaplan et al. (2020) in defining in-context learning as decreasing loss at increasing token indices. We find that loss is uniform across token positions in early training steps, but slowly decreases at increasing token indices in later steps, consistent with the
+
+presence of in-context learning (Fig. 7).
+
+We observe an initial increase in loss early in each sequence. This is likely due to the fact unigram sequences are indistinguishable from real sequences at the first position, and bigram model predictions are indistinguishable from real sequences at the first and second positions.
+
+# 5. Conclusion
+
+We propose two criteria that operationalize what it means for models to exploit moments of a given order, then describe methods of generating synthetic data that test whether a network satisfies both criteria, using theoretically grounded approaches relying on optimal transport theory and the principle of maximum entropy. We extend our analysis to discrete sequences by proving a novel equivalence between  $n$ -gram statistics and statistical moments.
+
+We find new compelling empirical evidence that neural networks learn to exploit the moments of their input distributions in increasing order, and further find "double descent" in the degree to which language model predictions match low-order data statistics, driven by in-context learning on longer sequences. Our contributions strengthen the case for the distributional simplicity bias (DSB), refine our understanding of how DSB influences early learning dynamics, and provide a foundation for further investigations into DSB.
+
+# Acknowledgements
+
+We are thankful to Open Philanthropy for funding this work. We also thank New Science and Stability AI for providing computing resources. Finally, we thank Brennan Dury for
+
+![](images/b524615a451a83395d8f32e64b5aff8919de242724ca34a3b1c0ff4659b55cbe.jpg)  
+Token index
+
+![](images/8e6d19858c4a56a175684734f2aa46baf06f14afc379d5b03f0378e3d188d72a.jpg)  
+Figure 7. In-context learning of the maximum entropy bigram sequence occurs after step 1,000 in Pythia 12B. Some in-context learning of the maximum entropy unigram sequence occurs by step 1000, with more at step 143,000.
+
+independently deriving the Conrad distribution.
+
+# Impact statement
+
+The goal of this work was to advance our understanding of the generalization behavior of neural networks throughout training, in the hope that this will enable the development of more robust and predictable machine learning models.
+
+# References
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+
+# A. Additional Related Work
+
+Extensive prior work has investigated neural network simplicity bias and learning dynamics. We highlight several prior research directions that usefully contrast our own approach.
+
+# A.1. Simplicity bias
+
+One common approach studies simplicity biases in the parameter-function maps of neural network architectures. Such explanations posit that neural networks implement favorable priors, meaning that most network parameterizations, under commonly used initialization distributions, that reach good performance on the training data will also generalize to the test data, regardless of specific details about the optimization process used to find such parameterizations.
+
+Valle-Perez et al. (2018) investigated such architectural simplicity biases by using Gaussian process-based approximations to neural networks (Lee et al., 2018) to estimate the Bayesian posterior produced by randomly sampling neural network parameterizations, conditional on those networks achieving perfect training loss, and showed the resulting posterior correlated well with the odds of SGD-based training finding a given function. Chiang et al. (2023) validate this perspective by showing that a variety of non-gradient based optimizers, including unbiased sampling of random initializations, are still able to generalize from training to testing data.
+
+Another approach is to construct a simplified, theoretically tractable model of neural network learning dynamics, then analyzing the resulting model to find which types of functions it predicts networks will be most inclined to learn. The neural tangent kernel (Jacot et al., 2018), scales network widths to infinity, whereupon networks are limited to performing kernel regression with their initialization kernel. Thus, model inductive biases are determined by the spectrum of the initialization kernel's eigenfunctions, which have strong simplicity biases for commonly used architectures (Canatar et al., 2021; Baratin et al., 2021; Bietti & Mairal, 2019).
+
+# A.2. Learning order
+
+Xu et al. (2019b) proposed the Frequency Principle, the tendency of neural networks to first fit low-frequency Fourier components of a given target function, before moving on to fit higher frequency components, and empirically demonstrated this tendency on real image classification problems and synthetic datasets. Subsequent works further explored how neural network learning dynamics relate to the representation of training data in the frequency domain (Rahaman et al., 2019; Xu et al., 2019a; Basri et al., 2020; Xu & Zhou, 2021). Our work is similar in that we also aim to connect neural network learning order to simple mathematical properties of the training data, though we use distributional statistics, rather than frequency.
+
+Choshen et al. (2022) empirically studied learning dynamics of neural language models by tracking which grammatical patterns different networks learn to model across their training trajectories, and comparing network behavior across training to alternative language modeling approaches, such as  $n$ -gram models. They found that neural language models initially match the behaviors of unigram and bigram models early in training, then diverge as training progresses. These results are inline with our own findings on learning order in neural language models, and are consistent with a DSB-driven perspective on neural network learning dynamics.
+
+# B. CIFARNet dataset
+
+CIFARNet is based on the Winter 2019 version of ImageNet-21K. We selected the ten synsets from the ImageNet hierarchy which most closely matched the ten CIFAR-10 classes, with a bias toward broader synsets to maximize the dataset size:
+
+Airplane: n02691156  
+Automobile: n02958343  
+Bird: n01503061  
+Cat: n02121620  
+- Deer: n02430045  
+Dog: n02083346  
+Frog: n01639765  
+Horse: n02374451  
+- Ship: n04194289  
+Truck: n04490091
+
+We ensured class balance by randomly sampling  $20\mathrm{K}$  images from each synset. Images were directly resized to  $64\times 64$  resolution without center cropping.
+
+# C. Detailed experimental results
+
+![](images/bf94b715c503bf5fad8058c4041faca7cf15eb17b0eb3186392adad76f6791cb.jpg)  
+C.1. CIFAR-10
+
+![](images/cfaf2b58d0d1908ea32016b2f867bedf60e64bad23bcc4581714b177fbfa3b75.jpg)
+
+# C.2. CIFARNet
+
+![](images/435118596da502254caf82639f7a0ca18f0a1b30454334dfd6745cc87bfac8a1.jpg)
+
+![](images/bcd604d7515effe91236c79a9749dc95b5d740203d0b9c35c499fecbb5294a96.jpg)  
+Figure 8. These results qualitatively mirror those of the lower resolution CIFAR-10 dataset (see above), except that the maximum accuracies attained on  $2^{\mathrm{nd}}$  order samples are somewhat lower. This may suggest that networks more quickly learn to use higher-order statistics when the input has higher dimensionality.
+
+# C.3. Fashion MNIST
+
+![](images/5bccf7403f6c52130a60ae58a4c99543e8293c6a6846c36176e06a4ff5a9834c.jpg)
+
+![](images/449caa1c7a1e2ead10e3fff052e6851412fc90c1e2a7f0816e388250a4034dcd.jpg)  
+Figure 9. Fashion MNIST learning curves exhibit only a modest degree of non-monotonicity, likely because the first and second moments of the data distribution are sufficient to produce very realistic-looking samples (Fig. 2)
+
+# C.4.MNIST
+
+![](images/64b7e930a8904b95127d76f6c90badbe315af2832e547a51735152ed305e6126.jpg)
+
+![](images/84123831be997adfd5f03f630fb459d7eced5769c56e28765cdcbb8f0bf7687e.jpg)  
+Figure 10. MNIST learning curves exhibit only a modest degree of non-monotonicity, likely because the first and second moments of the data distribution are sufficient to produce very realistic-looking samples (Fig. 2)
+
+# D. Note on third-order results
+
+We tried to use a variant of the algorithm described in Appendix H for  $3^{\mathrm{rd}}$  order max entropy sampling. However, the samples invariably failed to match the desired  $3^{\mathrm{rd}}$  order statistics sufficiently to trigger the gradient  $10^{-7}$  gradient tolerance stopping condition. We believe this is due to the very large number of entries in the coskewness tensor leading to more complex constraints on the acceptable values of the synthetic dataset. In comparison, we have a far smaller number of free parameters in the synthetic dataset. For example, for CIFAR-10 we optimized one thousand samples, each with  $3 \times 32 \times 32 = 3072$  coordinates, for a total of  $3 \times 10^{6}$  parameters. By comparison, the coskewness tensor has  $3072^{3} \approx 2.9 \times 10^{10}$  entries, each one placing an additional constraint on the dataset. It may be possible to generate higher quality samples by increasing the number of synthetic datapoints, but this would further increase the computational burden, which is already quite high in the  $3^{\mathrm{rd}}$  order case (Appendix I).
+
+Nevertheless, in the interest of completeness and transparency, we report our  $3^{\mathrm{rd}}$  order results in Appendix C. These results are difficult to interpret because they are generated using synthetic data whose  $3^{\mathrm{rd}}$  order statistics only approximately match the desired ones. We thus place little weight on them. However, we find it interesting that even partially matching the coskewness tensor will improve model performance on the synthetic data in most settings, with the notable exceptions of the SVHN dataset and RegNet-Y model, where the  $3^{\mathrm{rd}}$  order datasets either consistently underperform their  $2^{\mathrm{nd}}$  order counterparts on all models (in the case of the SVHN dataset) or selectively underperform on the CIFAR10 and CIFARNet datasets (in the case of the RegNet-Y model). We thus see that a partial match of  $3^{\mathrm{rd}}$  order statistics improves performance in 10 out of the 15 model and dataset pairs we investigate.
+
+# D.1. Street View Housing Numbers
+
+![](images/ac49e9fd31a7636b9890993a8aaafd25865da84a1fc9b71fd791a2d6509623bc.jpg)
+
+![](images/d33c047ae0f9816b2c6625b3925a01fcc4a25e15c22298528fef44049fb41df2.jpg)  
+Figure 11. The Street View Housing Numbers dataset is somewhat of an outlier in that none of the models ever exceed random baseline accuracy on  $1^{\text{st}}$  order synthetic images. We hypothesize this is because of the extreme diversity of colors, fonts, and background textures in SVHN, which make "simple" first order features less discriminative for classifying digits. We also found it necessary to use a smaller learning rate to achieve convergence on this dataset (Footnote 14).
+
+# D.2. Pythia Language Models
+
+![](images/a7de0961ee2f0d44abe8fcedcbb1f5f7c2e614bbb7112917ebfc4d033b9fa2e6.jpg)
+
+![](images/b29f9fab8681712e77667fa9b7dd2f3ba327be9b852a601845c773effd9363f3.jpg)  
+Figure 12. (top) Average cross entropy loss of Pythia models evaluated on unigram and bigram sequences, (bottom) KL divergence between the predictions of our  $n$ -gram language models and the predictions of Pythia checkpoints when evaluated on chunks of tokens sampled from the Pile ( $N = 1024$ ).
+
+![](images/10365a3a7c4841938a9a05a5a6a3f26d400c68310af7da749d2123753e46fa4d.jpg)
+
+![](images/fd36e02dc824c58bdb9dea857ddf8724786c08e663e4e3f342731f8e909a3375.jpg)
+
+![](images/a0913cddd5e92577d00947e177b6c55ba7641828137581e124abc44f6f60849e.jpg)  
+Training step
+
+![](images/c2d2b2e5168283e6e9f6b1ed4e22ff0e0827909dfa3c206e084707f1148c05e6.jpg)
+
+![](images/ab4ac7ac07eb1fb09ddb1cf67f999b2ee2065ad903341a3e02950fe33f2084f8.jpg)
+
+![](images/1a616801eda8ab1da2fc27a94e464cda5a36b03d6ad766e5fcb05fcbaca5296c.jpg)
+
+![](images/839368bead54714b79469c88ddb53aec2c820ca5f6057e369a85d8c08cbfc498.jpg)  
+Training step
+
+![](images/bf7eef409f59ef55ad1f93258b24b342c3324ca805fb7095e5ea8d1373eba31c.jpg)  
+Figure 13. Effects of fast and slow learning rate warmup on n-gram sequence loss and KL divergence, Pythia 14M and 70M.
+
+![](images/192a7e4722e8ef84292cd62d85d2903d11dc97e8fab634e2749d91f866f8fec8.jpg)  
+Unigram sequence loss across time
+
+![](images/81778e353ff321d5b4b3d668d22f1274d8ff221abce909c7389c6ab1d25260b1.jpg)  
+Bigram sequence loss across time
+
+![](images/4ab68a3c5038ca34140ed137e4833b52fb7a9d431dc372c27ee73ce5c374aa5d.jpg)  
+$D_{\mathrm{KL}}$  (unigram model || Pythia) across time
+
+![](images/c9e411ef34d73400e707105d0b4a040e249dfe597743b2357fb3b6795431b3aa.jpg)  
+$\mathrm{D}_{\mathrm{KL}}$  (bigram model || Pythia) across time  
+Figure 14. N-gram sequence loss and KL divergence over Pythia sizes and random seeds (1024 sequences sampled at each step, 9 seeds for Pythia 14M, 70M and 160M, 4 seeds for Pythia 410M.)
+
+# E. First order optimal transport under a boundary constraint
+
+We would like to surgically change the mean of a set of images while keeping their pixel intensities constrained to the range  $[0,1]$ . The least-squares optimal algorithm for this task is described in Alg. 1, and we prove its correctness in the following theorem.
+
+Theorem E.1. Let  $\mathbf{x}$  be a vector in  $[0,1]^n$  and let  $m\in [0,1]$  be a desired mean. Then the optimization problem
+
+$$
+\min _ {\mathbf {y} \in [ 0, 1 ] ^ {n}} \| \mathbf {x} - \mathbf {y} \| _ {2} ^ {2} \quad \mathrm {s . t .} \frac {1}{n} \sum_ {i = 1} ^ {n} y _ {i} = m
+$$
+
+has a unique solution given by Algorithm 1.
+
+Proof. Let  $\bar{x} = \frac{1}{n}\sum_{i=1}^{n}x_{i}$ . If  $\bar{x} = m$ , we immediately have the optimal solution  $\mathbf{y}^{*} = \mathbf{x}$ , because our constraints are already satisfied and we achieve zero loss by leaving  $\mathbf{x}$  unchanged.
+
+# Algorithm 1 Optimal constrained mean shift
+
+Require: Input vector  $\mathbf{x}\in [0,1]^n$
+
+Ensure: Desired mean  $m \in [0,1]$
+
+1: Sort the coordinates of  $\mathbf{x}$  
+2:  $\bar{x} \gets \sum_{i=1}^{n} x_i$  
+3:  $\mathbf{y} \gets \mathbf{0}_n$  
+4: for  $i \in 1 \dots n$  do  
+5:  $y_{i} \gets x_{i} + m - \bar{x}$  
+6: if  $y_{i} > 1$  then  
+7:  $\bar{x}\gets \sum_{j = i}^{n}x_{j}$  
+8:  $y_{i}\gets 1$  
+9:  $m\gets \frac{nm - i}{n - i}$  
+10: Put coordinates of  $\mathbf{y}$  in their original order.  
+11: return y
+
+We can reduce the case where  $m < \bar{x}$  to the case where  $\bar{x} < m$  (or vice versa) by exploiting the reflection-symmetry of the problem. Specifically, if the solution to the analogous problem in  $\mathbf{x}'$  and  $m'$ , where  $\mathbf{x}' = \frac{1}{2} - \mathbf{x}$  and  $m' = \frac{1}{2} - m$ , is  $\mathbf{y}'$ , then the solution to the original problem is  $\mathbf{y}^* = \frac{1}{2} - \mathbf{y}'$ . This is due to the reflection-invariance of the Euclidean distance, the linearity of our mean constraint, and the fact that reflecting about  $\frac{1}{2}$  cannot move points in  $[0, 1]^n$  outside  $[0, 1]^n$ . Because of this symmetry, in what follows we will assume without loss of generality that  $\bar{x} < m$ .
+
+Note also the optimal solution must have the property that  $\forall i: x_i \leq y_i$ . Assume for the sake of contradiction that  $x_i > y_i$  for some  $i$ . Then we can improve upon this solution by increasing  $y_i$  by some  $\epsilon > 0$ , and compensating for this by moving another entry  $y_j$  for which  $x_j < y_j$  closer to its original value by  $\epsilon$ .
+
+Setting up the Lagrangian. Using the Karush-Kuhn-Tucker conditions, we encode the problem with the Lagrangian
+
+$$
+\mathcal {L} (\mathbf {y}, \lambda , \boldsymbol {\mu}, \boldsymbol {\nu}) = \sum_ {i = 1} ^ {n} (y _ {i} - x _ {i}) ^ {2} + \underbrace {\lambda \left(\frac {1}{n} \sum_ {i = 1} ^ {n} y _ {i} - m\right)} _ {\text {m e a n c o n s t r a i n t}} - \underbrace {\sum_ {i = 1} ^ {n} \mu_ {i} y _ {i} + \sum_ {i = 1} ^ {n} \nu_ {i} (y _ {i} - 1)} _ {\text {i n e q u a l i t y c o n s t r a i n t s}}. \tag {4}
+$$
+
+Differentiating  $\mathcal{L}$  with respect to  $y_{i}$  yields the stationarity condition
+
+$$
+\frac {\partial \mathcal {L}}{\partial y _ {i}} = 2 \left(y _ {i} - x _ {i}\right) + \frac {\lambda}{n} - \mu_ {i} + \nu_ {i} = 0. \tag {5}
+$$
+
+The KKT complementary slackness condition requires that  $\mu_i y_i = 0$  and  $\nu_i (y_i - 1) = 0$  for each  $i$ . This implies that  $\mu_i$  must be zero if  $y_i > 0$ , and  $\nu_i$  must be zero if  $y_i < 1$ . For each  $i$  where  $y_i < 1$ , we can use Eq. 5 and complementary slackness to write  $y_i$  as  $x_i - \frac{\lambda}{2n}$ .
+
+Putting it all together. Assume  $\mathbf{x}$  and  $\mathbf{y}$  are written in a basis that ensures the coordinates of  $\mathbf{x}$  are sorted in descending order, so that  $x_{1} \geq x_{2} \geq \ldots \geq x_{n}$ . Our problem is invariant to permutation of indices, so this does not affect the solution.
+
+We can now solve for  $y_{1}$ , the final position of the largest coordinate, in the following way. Suppose that  $y_{1} < 1$ . Then we have  $\forall i: y_{i} < 1$ , and the mean constraint can be written as  $\frac{1}{n}\left(\sum_{i=1}^{n} x_{i}\right) - \frac{\lambda}{2n} = m$ . This allows us to solve for all  $y_{i}$ :
+
+$$
+y _ {i} = x _ {i} - \frac {\lambda}{2 n} = x _ {i} + m - \frac {1}{n} \left(\sum_ {i = 1} ^ {n} x _ {i}\right). \tag {6}
+$$
+
+Note Eq. 6 may "overshoot" and violate the inequality constraint  $y_{i} \leq 1$ . If it does, then we know our supposition is false and  $y_{1} = 1$ . If it does not violate the constraint, then it must be optimal because it is also the solution to the relaxed version of this problem without the [0, 1] constraint. In the latter case, we are done.
+
+Given that  $y_{1} = 1$ , the subproblem of solving for  $y_{2},\ldots ,y_{n}$  is a smaller instance of the original problem: the target mean for these coordinates is  $m^{\prime} = \frac{nm - 1}{n - 1}$ . We can recursively apply this reasoning to solve for all other  $y_{i}$ . This procedure coincides with Algorithm 1.
+
+# F. Derivation of the Dury distribution
+
+Theorem F.1. Among all distributions supported on  $[0, 1]$  with desired mean  $m \neq \frac{1}{2}$ , the Dury distribution with density  $p(x) = \frac{b\exp(-bx + b)}{\exp(b) - 1}$  has maximum entropy, where the parameter  $b \neq 0$  is chosen to satisfy the equation  $m = -\frac{b - \exp(b) + 1}{b(\exp(b) - 1)}$ . In the special case of  $m = \frac{1}{2}$ , the maximum entropy distribution is  $\mathrm{Unif}(0, 1)$ .
+
+Proof. Consider the density function  $p(x) = \exp(-a - bx)$ , where  $a$  and  $b$  are selected to satisfy normalization  $\int_{[0,1]} p(x) dx = 1$  and mean  $\int_{[0,1]} p(x) x dx = m$  constraints, and another arbitrary density  $q(x)$  which satisfies the same constraints. We will show that the entropy of  $q$  can be no greater than the entropy of  $p$ .
+
+$$
+\begin{array}{l} H (q) \leq H (q, p) \quad (\text {i n e q u a l i t y o f e n t r o p y a n d c r o s s - e n t r o p y}) \\ = \mathbb {E} _ {q} [ \log p (x) ] \quad (\text {d e f i n i t i o n o f c r o s s - e n t r o p y}) \\ = - a - b m \quad (\text {d e f i n i t i o n} p (x) \text {a n d l i n e a r i t y}) \\ = H (p) \tag {QED} \\ \end{array}
+$$
+
+We can now analytically solve for  $a$  in terms of  $b$ . Integrating  $p(x)$  from 0 to 1 yields  $-\frac{\exp(-a - b)}{b} + \frac{\exp(-a)}{b} = 1$ . Solving for  $a$  we get  $a = -b + \log \left( \frac{\exp(b) - 1}{b} \right)$ , which when plugged back into the original formula gives us  $p(x) = \frac{b \exp(-bx + b)}{\exp(b) - 1}$ .
+
+Integration by parts yields the following formula for the mean:  $\int_{[0,1]}p(x)xdx = -\frac{b - \exp(b) + 1}{b(\exp(b) - 1)}$ . We can use a root-finding algorithm such as Newton's method to solve this expression for  $b$  given a desired mean  $m$ . Note, however, that there is a singularity in the mean formula where  $b = 0$ . Applying l'Hôpital's rule twice yields the limit:
+
+$$
+\lim  _ {b \rightarrow 0} - \frac {b - \exp (b) + 1}{b (\exp (b) - 1)} = \lim  _ {b \rightarrow 0} \frac {\exp (b) - 1}{b \exp (b) + \exp (b) - 1} = \lim  _ {b \rightarrow 0} \frac {\exp (b)}{b \exp (b) + 2 \exp (b)} = \frac {1}{2}. \tag {7}
+$$
+
+The maximum entropy distribution supported on  $[0,1]$  with no mean constraint is known to be  $\mathrm{Unif}(0,1)$ . Since it happens to have the mean  $\frac{1}{2}$ , we may conclude that the Dury distribution approaches  $\mathrm{Unif}(0,1)$  as  $b$  approaches 0.
+
+# G. Discrete domain proofs
+
+Please refer to Section 2.5 for context pertinent to this section.
+
+Definition G.1 ( $n$ -gram statistic). Let  $\mathcal{V}^N$  be the set of token sequences of length  $N$  drawn from a finite vocabulary  $\mathcal{V}$ . Given some distribution over  $\mathcal{V}^N$ , an  $n$ -gram statistic is the probability that an  $n$ -tuple of tokens  $(v_1, \ldots, v_n) \in \mathcal{V}^n$  will co-occur at a set of unique indices  $(t_1, \ldots, t_n) \in \mathbb{N}^n$ .
+
+Theorem 2.1. [n-gram statistics are moments] Let  $\mathcal{V}^N$  be the set of token sequences of length  $N$  drawn from a finite vocabulary  $\mathcal{V}$ , let  $P$  be a distribution on  $\mathcal{V}^N$ , and let  $f:\mathcal{V}^N\to \{0,1\}^{N\cdot |\mathcal{V}|}$  be the function that encodes a length- $N$  sequence of tokens as a flattened concatenation of  $N$  one-hot vectors of dimension  $|\mathcal{V}|$ . Let  $f_{\sharp}P$  be the pushforward of  $P$  through this one-hot encoding, i.e. its analogue in  $\{0,1\}^{N\cdot |\mathcal{V}|}$ .
+
+Then every moment of  $f_{\sharp}P$  is equal to an  $n$ -gram statistic of  $P$  and vice versa.
+
+Proof. While it is natural to view one-hot sequences as Boolean matrices of shape  $N \times |\mathcal{V}|$ , where each row corresponds to a sequence position, we instead consider flattened one-hot encodings in order to make use of the standard mathematical machinery for moments of random vectors.
+
+In this flattened representation, the component at index  $i$  indicates whether the token at  $t(i)$  is equal to the token  $v(i)$ , where  $(t(i), v(i)) := \mathrm{divmod}(i, |\mathcal{V}|)$ . For example, if  $\mathcal{V} = \{\text{"apple", "pear"}\}$  and  $N = 3$ , the sequence "apple apple pear" will be encoded as the vector  $(1, 0 \mid 1, 0 \mid 0, 1)$ :
+
+$$
+\underbrace{\text{apple}}_{(1,0)}\underbrace{\text{apple}}_{(1,0)}\underbrace{\text{pear}}_{(0,1)}\xrightarrow{f}_{\setminus}(1,0,1,0,0,1).
+$$
+
+Now consider the moment corresponding to some arbitrary multi-index  $\alpha \in \mathbb{N}^{N\cdot |\mathcal{V}|}$ . For illustration, let  $\alpha = (0,2|0,0|1,0)$ . Then the corresponding moment is
+
+$$
+\mathbb {E} [ f (x) ^ {\alpha} ] = \mathbb {E} [ \underbrace {f (x) _ {1} ^ {0} f (x) _ {2} ^ {2}} _ {(0, 2)} \underbrace {f (x) _ {3} ^ {0} f (x) _ {4} ^ {0}} _ {(0, 0)} \underbrace {f (x) _ {5} ^ {1} f (x) _ {6} ^ {0}} _ {(1, 0)} ], \tag {8}
+$$
+
+where  $f(x)_{1}^{0}$  denotes the first component of  $f(x)$  raised to the power 0. Since each component of  $f(x)$  is a Boolean indicator for the presence or absence of a vocabulary item at a given position, we can rewrite it with Iverson brackets:
+
+$$
+\begin{array}{l} = \mathbb {E} \left[ [ x (1) = “ \text {p e a r}" ] ^ {2} \cdot [ x (3) = “ \text {a p p l e}" ] \right] (9) \\ = \mathbb {P} [ x (1) = “ \text {p e a r}" \wedge x (3) = " \text {a p p l e}" ], (10) \\ \end{array}
+$$
+
+or the probability that the first and third tokens will be "pear" and "apple" respectively. Note that the exponent on the  $[x(1) =$  "pear"] makes no difference here as long as it is nonzero. We can always binarize  $\alpha$ , replacing all nonzero values with 1, and the moment will be unchanged since any nonzero power of  $\{0,1\}$  is still  $\{0,1\}$ .
+
+In general, since the coordinates are all Booleans in  $\{0,1\}$ , multiplication corresponds to logical conjunction and expectation corresponds to probability:
+
+$$
+\mathbb {E} [ f (x) ^ {\alpha} ] = \mathbb {E} \left[ \prod_ {i = 1} ^ {N} f (x) _ {i} ^ {\alpha_ {i}} \right] = \mathbb {P} \left[ \bigwedge_ {i \in A} x \left(t _ {i}\right) = v _ {i} \right], \tag {11}
+$$
+
+where  $A$  is the set of indices in  $1 \dots |\mathcal{V}| \times N$  where  $\alpha$  is nonzero. By Def. G.1, this probability is an  $n$ -gram statistic of order  $k = |A|$ .
+
+Conversely, we can convert any  $n$ -gram statistic with tokens in  $\mathcal{V}^n$  and sequence positions in  $\mathbb{N}^n$  into a moment of  $f_{\sharp}P$  by first flattening the indices, then plugging them into Eq. 11. The sequence positions correspond to rows, and the tokens correspond to columns, of a one-hot matrix representation of a sequence. Here we need to multiply the row and column indices together to yield indices into the flattened vector.
+
+There will be infinitely many moments which correspond to any given  $n$ -gram, because multi-indices with components larger than one are redundant.
+
+Theorem 2.2. [Equal embedding moments] Let  $\mathbf{E} \in \mathbb{R}^{|\mathcal{V}| \times d}$  be an embedding matrix, and let  $P$  and  $Q$  be two distributions over  $\mathcal{V}^N$ . Then if  $P$  and  $Q$  have the same  $n$ -gram statistics up to order  $k \geq 1$ , their embeddings under  $\mathbf{E}$  have the same moments up to order  $k$ .
+
+Proof. By Thm. 2.1, we know that the one-hot analogues of  $P$  and  $Q$ , i.e.  $f_{\sharp}P$  and  $f_{\sharp}Q$ , have equal moments up to order  $k$ . That is, for every multi-index  $\alpha \in \mathbb{N}^{N\cdot |\mathcal{V}|}$  where  $|\alpha| \leq k$ ,
+
+$$
+\mathbb {E} _ {\mathbf {x} \sim f _ {\sharp} P} \left[ \mathbf {x} ^ {\alpha} \right] = \mathbb {E} _ {\mathbf {x} \sim f _ {\sharp} Q} \left[ \mathbf {x} ^ {\alpha} \right]. \tag {12}
+$$
+
+Now let  $g: \{0,1\}^{|\mathcal{V}| \times N} \to \mathbb{R}^{d \times N}$  be the function that multiplies each one-hot vector in a sequence by  $\mathbf{E}$ , returning a sequence of embedding vectors. Each side of Eq. 12 is the expectation of a polynomial in the components of  $\mathbf{x}$ , and since  $g$  is a linear map,  $g(\mathbf{x})^{\alpha}$  is also a polynomial with the same degree.
+
+Now consider  $(g\circ f)_{\sharp}P$  and  $(g\circ f)_{\sharp}Q$ , the analogues of  $P$  and  $Q$  in embedding space. Its moments take the form
+
+$$
+\mathbb {E} _ {\mathbf {x} \sim (g \circ f) _ {\sharp} P} [ \mathbf {x} ^ {\alpha} ] = \mathbb {E} _ {\mathbf {x} \sim f _ {\sharp} P} [ g (\mathbf {x}) ^ {\alpha} ]. \tag {13}
+$$
+
+Because  $g(\mathbf{x})^{\alpha}$  is a polynomial with degree  $|\alpha| \leq k$ , the expectation  $\mathbb{E}_{\mathbf{x} \sim f_{\sharp}P}[g(\mathbf{x})^{\alpha}]$  must be a linear combination of moments of  $f_{\sharp}P$  with order no greater than  $k$ . But by Eq. 12, all of these moments are equal between  $P$  and  $Q$ , and hence all the moments of  $(g \circ f)_{\sharp}P$  and  $(g \circ f)_{\sharp}Q$  up to order  $k$  must be equal.
+
+# H. Sampling from maximum entropy distributions with complex constraints
+
+We used the following code to generate maximum entropy samples subject to a mean, covariance, and hypercube constraint.  $3^{\mathrm{rd}}$  order sampling code is similar but more complex, as it includes a form of "minibatching" for the entries of the coskewness tensor in order to reduce memory usage.
+
+```python
+from torch import nn, optim, Tensor
+import torch
+```
+
+```python
+def koleo(x: Tensor) -> Tensor:
+    ""Kozachenko-Leonenko estimator of entropy."
+    return torch.cdist(x, x).kthvalue(2).values.log().mean()
+```
+
+```python
+def psd_sqrt(A: Tensor) -> Tensor:
+    '''Compute the unique p.s.d. square root of a positive semidefinite matrix.'''
+    L, U = torch.linalg.eigh(A)
+    L = L[., None, :].clamp_min(0.0)
+    return U * L_sqrt() @ U.mH
+```
+
+```python
+def hypercube_sample( n: int, mean: Tensor, cov: Tensor, *, koleo_weight: float = 1e-3, max_iter: int = 100, seed: int = 0, ): """Generate `n` samples from max-ent distribution on [0, 1]^d with given moments.""" d = mean.shape[-1] eps = torch+finfo(mean.dtype).epsrng = torch.Generator(device=mean_device).manual_seed(seed) # Initialize with max-ent samples matching `mean` and `cov` but without hypercube # constraint. We do so in a way that is robust to singular `cov` z = mean.new_empty([n, d]).normal_(generator=rng) x = torch.clamp(z @ psd_sqrt(cov) + mean, eps, 1 - eps) # Project into hypercube # Reparametrize to enforce hypercube constraint z = nn_PARAMETER(x.logit()) opt = optim.LBFGS([z], line_search_fn="strong_wolfe", max_iter=max_iter) def closure(): opt.zero_grad() x = z.sigmoid() loss = torch(norm(x.mean(0) - mean) + torch(norm(x.T.cov() - cov) loss -= koleo_weight * koleo(x) loss.backup() return float(loss) opt step(closure) return z.sigmoid().detach()
+```
+
+# I. Computational requirements
+
+At the scales of the datasets we use in this study, both maximum entropy second order hypercube-constrained sampling and Gaussian optimal transport are extremely cheap to run. In the most expensive configuration (generating around 200K  $64 \times 64$  CIFARNet images), the optimization loop takes roughly 65 seconds on a single NVIDIA L40 GPU, while requiring approximately 29 gigabytes of GPU memory. Based on hourly pricing of $1.10 per hour from vast.ai, this will cost around
+
+$0.02 to generate a full set of synthetic CIFARNet images, with all other first and second order methods described in this paper requiring fewer computational resources than that.
+
+However, the memory required for both hypercube-constrained sampling and Gaussian optimal transport rise with the square of the number of image features (meaning the fourth power of the image size). The compute requirements of the hypercube sampling also rise with the fourth power of the image size, while the requirements for Gaussian optimal transport rise with the sixth power. This means our methods can quickly become computationally infeasible with larger image sizes, which is why we limit ourselves to at most  $64 \times 64$  images in this study.
+
+Additionally, the maximum entropy third order hypercube-constrained sampling is much more expensive than the second order methods, since the size of the statistic tensor grows as  $O(d^{order})$ . This means the coskewness tensor for CIFARNet images has dimensions  $12288 \times 12288 \times 12288$ . This would require nearly eight terabytes to store in full precision, which exceeds the memory capacity of our computing hardware by a significant degree.
+
+We therefore want to generate fake data that matches CIFARNet's coskewness statistics, without ever computing those statistics in full. Each step of our optimization process for generating third order fake thus only computes the coskewness statistics along matching length  $l$  slices of coskewness tensors of the fake and real data, meaning we only need to store two  $12288 \times 12288 \times l$  tensors at each step of optimization.
+
+CIFARNet is the most expensive dataset to imitate with maximum entropy third order hypercube-constrained sampling, as matching its first, second, and third order statistics at the same time is challenging. We used 10,000 optimization steps per class, taking a total of 36 hours on a single NVIDIA A40 GPU. Using an hourly price of $0.403 from vast.ai, this would cost roughly $14.5.

2024/Neural Networks Learn Statistics of Increasing Complexity/images.zip

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2024/Neural Operators with Localized Integral and Differential Kernels/3932d016-37c4-4572-a91b-464a12932a32_origin.pdf

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2024/Neural Operators with Localized Integral and Differential Kernels/full.md

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+# Neural Operators with Localized Integral and Differential Kernels
+
+Miguel Liu-Schiaffini *1 Julius Berner *1 Boris Bonev *2 Thorsten Kurth 2 Kamyar Azizzadenesheli 2 Anima Anandkumar 1
+
+# Abstract
+
+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  $L^2$ -error by  $34 - 72\%$  in our experiments, which include a turbulent 2D Navier-Stokes and the spherical shallow water equations.
+
+# 1. Introduction
+
+Deep learning holds the promise to greatly accelerate advances in computational science and engineering, which often require numerical solutions of partial differential equa
+
+*Equal contribution  $^{1}$ Department of Computing and Mathematical Sciences, California Institute of Technology, Pasadena CA 91125  $^{2}$ NVIDIA, Santa Clara, CA 95051. Correspondence to: Miguel Liu-Schiaffini <mliuschi@caltech.edu>, Julius Berner <jberner@caltech.edu>, Boris Bonev <bbonev@nvidia.com>.
+
+Proceedings of the  $41^{st}$  International Conference on Machine Learning, Vienna, Austria. PMLR 235, 2024. Copyright 2024 by the author(s).
+
+![](images/0f665e7044391822dfe8525d0b3e52334c610a2f44bd01522b2516546c67f5b1.jpg)
+
+![](images/974f96541446fead47dda4b51a73b2b694e4b59e7d9e9e62477f2d801ed1291a.jpg)  
+Regular kernel
+
+![](images/e7cd4a447926080795b196798f73d5a3cc969d13d4de8c7bb630c4f924f55e3d.jpg)
+
+![](images/4cb99b952c4363e19214ff20e8889bcfac7d7b80e388ba619b2fab35ce491980.jpg)  
+Local integral kernel
+
+![](images/0061bf604058e8921e292b89326ff14ceaaabfd1a81bd273298120b84e9ad500.jpg)
+
+![](images/d4fcbeaeb6f50eb661eca168cac8e46104590e3e12f1b17a3885ea25b45a09cb.jpg)
+
+![](images/d04089d8b4c02954caa2fb079ee5640f723fc601ef27c068cb9de1cbbd659b43.jpg)
+
+![](images/d1dd80460d4d1c8e39a27347b7d9e255b936d4e11dc760cd10ab2fe4d9d9fe04.jpg)  
+Differential kernel
+
+![](images/26bcccd3e3a6b0587c109fae7f98e6f648ccc8de2bcb9da7da750231182fcb2a.jpg)
+
+![](images/c7cd97d05fcbed1b90a60d35a6636313ed10336bacdf5ddd94dd58b275effacc.jpg)  
+Figure 1: Visualization of different limits of a convolution with a discretized function  $v$  as the grid width  $h$  is refined, i.e.,  $h \rightarrow 0$ . (Top) A regular convolution is collapsing to a pointwise linear operator. (Middle) Instead, we can use a kernel that can be evaluated at arbitrary resolutions and keep the receptive field unchanged, to converge to a local integral operator, see Section 3.4. (Bottom) Alternatively, we can let it collapse while constraining the kernel appropriately, converging to a differential operator, see Section 3.2.
+
+tions (PDEs) (Azizzadenesheli et al., 2024; Zhang et al., 2023; Cuomo et al., 2022). Recent advances in deep learning have enabled applications such as weather forecasting (Pathak et al., 2022; Bonev et al., 2023; Lam et al., 2023), seismology (Sun et al., 2023; Shi et al., 2023), reservoir engineering for carbon capture (Wen et al., 2022; 2023), and many other applications with orders of magnitude speedup over traditional methods.
+
+Many of the above results are achieved by neural operators, which learn mappings between function spaces, enabling
+
+operator learning for function-valued data (Li et al., 2021; Azizzadenesheli et al., 2024; Raonic et al., 2023). In particular, they are agnostic to the discretization of the input and output functions—a vital property in the context of PDEs where data is often provided at varying resolutions and high-resolution data is costly to generate (Kovachki et al., 2021). In contrast, standard neural networks such as convolutional neural networks (CNN) (Ronneberger et al., 2015; Gupta & Brandstetter, 2022) require the functions to be discretized at a fixed resolution on a regular grid, which is limiting.
+
+In the past few years, various architectures of neural operators have been developed. Among them, the Fourier neural operator (FNO) (Li et al., 2020a), which performs global convolutions in Fourier space, has gained popularity and shown good performance in a number of applications. However, such global operations are often prone to oversmoothing and may fail to capture local details. Several other architectures instead learn global operators in physical space (e.g., (Li et al., 2022b)), but they likewise lack the inductive bias of local operations.
+
+There are many applications that require a local neural operator. For instance, solution operators of several relevant PDEs are of local nature. Examples include hyperbolic PDEs, which have real-valued characteristic curves (LeVque, 1992). As a result, a solution at a given point will only depend on the initial condition within a neighborhood of that point. As such, their solution operators only have a local receptive field and can, therefore, be efficiently learned by locally supported kernels.
+
+Further examples of local operators are differential operators, which can be expressed in terms of pointwise multiplication with the frequency in the spectral domain. Consequently, they introduce large errors when approximated by a finite number of parameters in Fourier space. In this context, we note that the emulation of classical numerical methods to solve partial differential equations, such as finite-difference methods, relies on the usage of local stencils for differentiation (Thomas, 2013). This calls for the presence of local and differential operators in neural operator architectures.
+
+Naturally, every local operation can also be represented by a global operation. However, this is typically vastly parameter-inefficient and does not provide a good inductive bias for learning local operations. In the context of neural operators, spectral variants of neural operators, such as the FNO (Li et al., 2020a) and Spherical FNO (SFNO) (Bonev et al., 2023), are theoretically able to approximate local convolutions. However, representing local kernels requires the approximation of a global signal in the spectral domain, in turn demanding a large number of parameters (due to the uncertainty principle (Cohen-Tannoudji et al., 1977)).
+
+A few prior works have explored the usage of local oper
+
+Table 1: Comparison of different architectures for the solution of PDEs. The top half enumerates architectures for planar domains and the bottom half for spherical domains. Our proposed architectures are highlighted in bold and the differential and integral kernels are detailed in Section 3.  
+
+<table><tr><td>Architecture</td><td>Efficient</td><td>Receptive field</td><td>no input downsampling</td></tr><tr><td>GNO</td><td>✗</td><td>local/global</td><td>✓</td></tr><tr><td>FNO</td><td>✓</td><td>global</td><td>✓</td></tr><tr><td>CNO / U-Net</td><td>✓</td><td>local</td><td>✗</td></tr><tr><td>FNO + integral</td><td>✓</td><td>local/global</td><td>✓</td></tr><tr><td>FNO + differential</td><td>✓</td><td>local/global</td><td>✓</td></tr><tr><td>SFNO</td><td>✓</td><td>global</td><td>✓</td></tr><tr><td>SFNO + integral</td><td>✓</td><td>local/global</td><td>✓</td></tr></table>
+
+ations in the context of neural operators. Li et al. (2020b) introduce graph neural operators (GNOs), which parameterize local integral kernels with a neural network. However, evaluating this network on all combinations of points to compute the integrals can make GNOs computationally expensive and slower than hardware-optimized convolutional kernels. Alternatively, Ye et al. (2022; 2023) propose local neural operators by combining FNOs with convolutional layers and, similarly, Wen et al. (2022) integrate U-Nets and FNOs. Moreover, Raonic et al. (2023) propose convolutional neural operators (CNOs) by leveraging U-Nets that (approximately) respect the bandlimits.
+
+However, since all of the above approaches rely on standard convolutional layers on equidistant grids, they have the following shortcomings. First, such approaches do not allow for a natural extension to unstructured grids or other geometries, which are ubiquitous in PDE problems (Li et al., 2023). Moreover, they can only be applied to higher resolutions by downsampling of the (intermediate) inputs to the training resolution. Important high-frequency content can be lost through downsampling, which is particularly problematic for multi-scale data in the context of PDEs. In contrast, we develop convolutional layers that can be applied at any resolution without downsampling. We achieve this by appropriately scaling the receptive field or the values of the kernel (see Figure 1).
+
+To summarize, current neural operator architectures suffer from at least one of the following limitations (see Table 1): (1) they cannot succinctly represent operations with a local receptive field, e.g., FNO, or (2) they cannot be applied to different resolutions without relying on explicit up-/downsampling which may degrade performance, e.g., CNO, or (3) they cannot be scaled to obtain sufficient expressivity, since they incur prohibitively high computationally costs, such as, for instance, GNOs.
+
+![](images/0ce05ca54c85be440bb022caa8b73191a8d821a22211eb15f3b0d64f22a4158e.jpg)  
+Figure 2: A single layer of our local neural operator. We add (up to) two local operations using the convolutions with differential kernel (Section 3.2) and local integral kernel (Section 3.4).
+
+Our approach: In this work, we develop computationally efficient and principled approaches to include operations in neural operators that capture local receptive fields while retaining the ability to approximate operators and, hence, extend to multiple resolutions. We consider two kinds of localized operators: differential operators and integral kernel operators with a locally supported kernel (see Figure 1).
+
+For the first, we draw inspiration from stencils of finite-difference methods. We derive conditions to modify convolutional layers such that they converge to a unique differential operator when the discretization is refined. For the second case of local integral operators, we adapt discrete-continuous (DISCO) convolutions (Ocampo et al., 2022) to provide an efficient, discretization-agnostic framework that can be applied to general meshes on both planar and spherical geometries. To our knowledge, we are the first to connect the DISCO framework to operator learning.
+
+Finally, we devise efficient implementations of both layers and show that the inductive bias of local operations (see Figure 2) can significantly improve the performance of FNOs and SFNOs on three different benchmarks. In particular, we learn a differential operator in a Darcy flow setting for motivating the need for neural operators with local inductive biases, and we improve over FNO by  $87\%$ . Additionally, we improve over the baseline (S)FNO by  $34\%$  on turbulent 2D Navier-Stokes equations, by  $72\%$  on the shallow water equations on the sphere, and by  $63\%$  on the 2D diffusion-reaction equation. We also apply our method on modeling a flow past a cylinder, discretized on an irregular grid. We improve upon the best baseline by  $42\%$ .
+
+Outline: The remainder of the paper is organized as follows: Section 2 outlines connections to other architectures and ideas within the neural operator literature. Section 3
+
+introduces the main ideas of the paper as well as the macroarchitecture of our proposed neural operator. Section 4 discusses numerical experiments and results, and Section 5 summarizes the findings of our work.
+
+# 2. Related work and connections to other frameworks
+
+We strive to formulate local operations that are consistent with the neural operator paradigm, which stipulates that operations within the model are independent of the discretization of input functions (Kovachki et al., 2021). In principle, this allows the evaluation of the neural operator on arbitrary meshes<sup>1</sup>. Several discretization-independent local operations have been introduced in the deep learning literature. We outline the connections between these works in the following.
+
+GNOs and Hypernetworks: In general, local operations can be provided by graph neural operators (GNOs) (Li et al., 2020b). However, in their general form, GNOs are typically slow, hard-to-train, and not equivariant w.r.t. the symmetry group of the underlying domain (Li et al., 2020c). Taking a convolutional kernel, GNOs subsume a series of works on hypernetwork approaches for convolutional layers (Wang et al., 2018; Shocher et al., 2020).
+
+DISCO Convolutions: We show that DISCO convolutions (Ocampo et al., 2022) represent special cases of GNOs with a convolutional kernel, which enables generalization to different geometries and an efficient implementation, since the kernel can be pre-computed (see Section 3). In particular, DISCO convolutions are typically implemented as learnable linear combinations of fixed basis functions instead of neural networks. Also, they define maps that are equivariant up to the error resulting from the quadrature rule.
+
+Scale-Equivariant CNNs: In the area of computer vision, there has been a series of adaptations of CNNs to be (locally) scale-equivariant by using filter dilation, filter rescalings in the discrete domain (Rahman & Yeh, 2023; Sosnovik et al., 2021; Worrall & Welling, 2019) or the continuous domain (Xu et al., 2014; Sosnovik et al., 2019; Ghosh & Gupta, 2019), or input rescalings (Marcos et al., 2018). Moreover, filter rescaling has also been explored for more general group-convolutions (Bekkers, 2019).
+
+Neural operators and convolutional layers: While neural operators have been developed independently of the above approaches, one can leverage similar ideas. In partic-
+
+ular, one can rescale (i.e., up- and downsample) input and output functions of a convolutional layer or the kernel itself to obtain neural operators. We note that these approaches are, in principle, also applicable to a trained network (without the need for retraining).
+
+In combination with FNOs, interpolation of the input and output functions has been explored by Wen et al. (2022). Also, Ye et al. (2023; 2022) emphasized the need for local convolutions in FNOs, however, they do not discuss the application to different discretizations. Combined with correct treatment of the bandlimit of the functions (based on Karras et al. (2021)), the convolutional neural operator (Raonic et al., 2023) also uses up- and downsampling of the input and output functions to apply a U-Net architecture.
+
+We note that for bandlimited functions sampled above the Nyquist frequency, the discrete convolutions have a unique correspondence to convolutions in function space, representing a special case of the DISCO framework. However, downsampling the input function introduces errors in cases where the function is not bandlimited.
+
+# 3. Local Layers
+
+In this section, we present two conceptually different types of local layers for neural operators. First, we present efficient integral transforms with local kernels. Then, we focus on the approximation of differential operators. We consider an input function
+
+$$
+v \colon \mathbb {R} ^ {d} \supset D \to \mathbb {R} ^ {n}
+$$
+
+that is discretized on meshes  $D^h \subset D$  with width  $h$  on a domain  $D \subset \mathbb{R}^d$ . For notational convenience, we present most ideas for the case  $d = 1$ , but it is straightforward to extend them to higher-dimensional domains.
+
+# 3.1. Motivation: Convolutional Layer
+
+We take inspiration from convolutional layers since they represent the prototypical version of an efficient, local operation in neural networks. However, we will see that they are not consistent in function spaces; in particular, they converge to a pointwise linear operator when we refine the discretization of the input function  $v$ .
+
+Let us start by recalling the definition of a convolutional layer, specifically a stride-1 convolution with  $n$  input channels, a single $^3$  output channel, and kernel  $K = (K_{i})_{i = 1}^{S}\subset \mathbb{R}^{n}$  of (odd) size  $S$ . Assuming a regular grid, i.e.,  $D^{h} = \{x_{j}\}_{j = 1}^{m}\subset \mathbb{R}$  with  $x_{j + 1} - x_{j} = h$ , we can define the output
+
+of the convolutional layer at  $y \in D^{h}$  as
+
+$$
+\begin{array}{l} \operatorname {C o n v} _ {K} [ v ] (y) = \left(K \star \{v \left(x _ {j}\right) \} _ {j = 1} ^ {m}\right) (y) \\ = \sum_ {i = 1} ^ {S} K _ {i} \cdot v \left(z _ {i} + y\right), \tag {1} \\ \end{array}
+$$
+
+with
+
+$$
+z _ {i} = h \left(i - 1 - \frac {S - 1}{2}\right), \tag {2}
+$$
+
+where we use zero-padding, i.e.,  $v(x) = 0$  for  $x \notin D$ .
+
+If we now take the same kernel  $K$  for finer discretizations, i.e.,  $h\to 0$ , we see from (2) that  $z_{i}\rightarrow 0$  and therefore,
+
+$$
+\lim  _ {h \to 0} \operatorname {C o n v} _ {K} [ v ] (y) = \bar {K} \cdot v (y) \quad \text {w i t h} \quad \bar {K} = \sum_ {i = 1} ^ {S} K _ {i},
+$$
+
+given that the function  $v$  is continuous at  $y$ . In other words, the receptive field with respect to the underlying domain  $D$  is shrinking to a point, and the convolutional layer is converging to a pointwise linear operator. One way to circumvent this issue would be to downsample the function  $v$  appropriately to a pre-defined grid. This is done for previous approaches mentioned in Section 2, at the cost of losing high-frequency information in the input.
+
+In the following, we will present two ways of working on different input resolutions, while not collapsing to a pointwise operator, see also Figure 1. First, we show that rescaling (1) by the reciprocal resolution  $\frac{1}{h}$  and constraining the kernel  $K$  leads to differential operators. Then, we define the kernel  $K$  as the evaluation of a function over a fixed input domain, leading to integral operators.
+
+# 3.2. Differential Layer
+
+In this section, we construct a layer that converges to a differential operator when the width  $h$  of the discretization  $D^{h}$  tends to zero. To prevent the operator from collapsing to a pointwise operation, we constrain and rescale the values of the kernel (according to the discretization width). Note that without rescaling, we would again recover a pointwise operator; however, due to the rescaling by a factor of  $\frac{1}{h}$ , the limit might not exist without constraining the values of the kernel. The next proposition shows that it is sufficient to subtract the average kernel value. See Appendix A.1 for a more general statement and a corresponding proof.
+
+Proposition 3.1 (First-order differential layer). Let  $D_h \subset \mathbb{R}^d$  be a regular grid of width  $h$  and let  $v \in C^1(D, \mathbb{R}^n)$ . Then, for every kernel  $(K_i)_{i=1}^S \subset \mathbb{R}^n$ , there exists  $(b_j)_{j=1}^n \subset \mathbb{R}^d$  such that
+
+$$
+\lim  _ {h \rightarrow 0} \frac {1}{h} \operatorname {C o n v} _ {K - \bar {K}} [ v ] (y) = \sum_ {j = 1} ^ {n} \nabla v _ {j} (y) \cdot b _ {j}
+$$
+
+for every  $y \in D_h$ , where  $\bar{K} = \sum_{i=1}^{S} K_i$ .
+
+Proposition 3.1 shows that we can learn different directional derivatives using an appropriate adaptation of standard convolutional layers as in Section 3.1. Specifically, we center the kernel  $K$  by subtracting its mean  $\bar{K}$  and scale the result by the reciprocal resolution  $\frac{1}{h}$ . In Appendix A.2, we empirically evaluate the convergence in resolution to a unique differential operator on a simple example.
+
+Remark 3.2 (Higher-order differential operator). Similarly, we could approximate  $k$ -th order differential operator with further constraints on the elements of  $K$  and a scaling factor of  $\frac{1}{h^k}$ . However, we do not implement them in practice since we can also approximate higher-order derivatives by composing first-order differential layers.
+
+# 3.3. Integral Kernel Layers
+
+Instead of rescaling the kernel, we can also adapt the size of the kernel such that the receptive field stays the same, i.e., is independent of the resolution  $h$ . In this section, we begin at the general graph neural operator and demonstrate how (considering the desirable properties of translation equivariance and efficiency) we arrive at local convolutional layers with adaptive kernel sizes using a fixed number of parameters.
+
+Graph neural operator (GNO): One of the most general instantiations of a neural operator is arguably the graph neural operator (GNO)
+
+$$
+\begin{array}{l} \mathrm {G N O} _ {k} [ v ] (y) = \int_ {U (y)} k (x, y) \cdot v (x) \mathrm {d} x (3) \\ \approx \sum_ {x \in D ^ {h} \cap U (y)} k (x, y) \cdot v (x) q _ {x}, (4) \\ \end{array}
+$$
+
+where  $k$  is a kernel (typically parametrized by a neural network) and  $q_{x} \in \mathbb{R}$  are suitable quadrature weights. While the GNO can represent local integral operators by picking a suitably small neighborhood  $U(y) \subset D$ , the evaluation of the kernel and aggregation in each neighborhood  $U(y)$  is slow and memory-intensive for general kernels  $k \colon D \times D \to \mathbb{R}^n$ . Moreover, for an arbitrary kernel, GNO is not equivariant w.r.t. the symmetry group of the underlying domain. In particular, we lose the translation equivariance of convolutional layers for planar domains.
+
+Fourier Neural Operator (FNO): To retain translation equivariance, we can consider kernels of the form
+
+$$
+k (x, y) = \kappa (x - y)
+$$
+
+and  $U(y) = y + \operatorname{supp}(\kappa)$ . Then, we can rewrite the GNO as a convolution, i.e.,
+
+$$
+\mathrm {G N O} _ {\kappa} [ v ] = \kappa \star v. \tag {5}
+$$
+
+If we are dealing with periodic functions on the torus  $D$ , we can leverage the convolution theorem to compute (5), i.e.,
+
+$$
+\mathcal {F} [ \kappa \star v ] = \overline {{\mathcal {F} [ \kappa ]}} \cdot \mathcal {F} [ v ], \tag {6}
+$$
+
+where  $\mathcal{F}$  maps functions to their Fourier series coefficients. The Fourier Neural Operator (FNO) now directly parametrizes  $\overline{\mathcal{F}[\kappa]}$  and approximates  $\mathcal{F}$  using the fast Fourier transform given that  $D^{h}$  is an equidistant grid. While this leads to an efficient version of (4), it assumes that  $\overline{\mathcal{F}[\kappa]}$  has only finitely many nonzero Fourier modes—or, equivalently, that the kernel  $\kappa$  has full support, making (5) a global convolution.
+
+Localized convolutions: To construct a locally supported kernel  $\kappa$ , we can directly discretize the convolution in (5), i.e.,
+
+$$
+\mathrm {G N O} _ {\kappa} [ v ] (y) \approx \sum_ {x \in D ^ {h}} \kappa (x - y) \cdot v (x) q _ {x}. \tag {7}
+$$
+
+Note that the sum can be taken only over the  $x \in D^h$  with
+
+$$
+x - y \in \operatorname {s u p p} (\kappa).
+$$
+
+We remark that for an equidistant grid and constant<sup>4</sup> quadrature weights,  $q = q_{x}$ , evaluating  $\mathrm{GNO}_{\kappa}[v]$  at  $y \in D^{h}$  corresponds to a standard convolution as in (1) with kernel
+
+$$
+K _ {i} = q \kappa (z _ {i}), \quad i = 1, \dots , S,
+$$
+
+where  $z_{i}$  is defined as in (2) and  $S$  is sufficiently large such that  $\operatorname{supp}(\kappa) \subset [z_1, z_S]$ . See Appendix B.2 for details. However, the advantage of the formulation in (7) is the fact that we can reuse the same kernel  $\kappa$  across different resolutions (with the same receptive field  $\operatorname{supp}(\kappa)$ ).
+
+The remaining question is centered around the parametrization of the kernel. One could draw inspiration from works on hypernetworks in computer vision and parameterize it using a neural network (Shocher et al., 2020). However, this is significantly more costly than a standard convolution. Another idea is to interpolate a fixed-sized kernel, e.g., using sinc or bi-linear interpolation. A more general version of the latter, based on a learnable linear combination of hat functions, is also known as discrete-continuous (DISCO) convolution and has shown to be effective in different applications (Ocampo et al., 2022). We note that this formulation is closest to the original convolutional layer typically used in computer vision. However, in the next section, we see that it also allows us to use unstructured meshes and formulate the operation on more general domains  $D \subset \mathbb{R}^d$ .
+
+# 3.4. General discrete-continuous convolutions
+
+The local support of the kernel in equation (7) allows us to efficiently evaluate local convolutions in subdomains
+
+of  $\mathbb{R}^d$  using sparse matrix-vector products. However, the operation  $x - y$ , which shifts the convolution kernel, is not well-defined on manifolds such as the sphere. To generalize the previous discussion to a Lie group  $G$ , we first replace the shift operator with the group action  $g$  and obtain the so-called group convolution
+
+$$
+\begin{array}{l} \operatorname {G r o u p C o n v} _ {\kappa} [ v ] (g) = (\kappa \star v) (g) \\ = \int_ {G} \kappa (g ^ {- 1} x) \cdot v (x) d \mu (x), \tag {8} \\ \end{array}
+$$
+
+where  $g, x \in G$  and  $\mathrm{d}\mu(x)$  is the invariant Haar measure on  $G$ . This formulation presents us with the challenge of being non-trivial to discretize. The framework of DISCO convolutions (Ocampo et al., 2022) achieves this by approximating the integral with a quadrature rule while evaluating the group action continuously:
+
+$$
+\begin{array}{l} (k \star v) (g _ {i}) \approx \sum_ {j = 1} ^ {m} \kappa \left(g _ {i} ^ {- 1} x _ {j}\right) \cdot v \left(x _ {j}\right) q _ {j} \\ = \sum_ {j = 1} ^ {m} K _ {i j} \cdot v \left(x _ {j}\right) q _ {j}. \tag {9} \\ \end{array}
+$$
+
+Here  $(x_{j})_{j = 1}^{m}\subset G$  represent quadrature points with corresponding quadrature weights  $q_{j}$ . For a given set of output positions  $g_{i}$ , we obtain the fully discrete formulation as matrix-vector product with the matrix  $K_{ij} = \kappa (g_i^{-1}x_j)$  which is sparse due to the local support of the kernel. To parameterize the kernel, we choose a linear combination of basis functions  $\kappa^{(\ell)}$ , such that  $\kappa = \sum_{\ell = 1}^{L}\theta^{(\ell)}\kappa^{(\ell)}$  with trainable parameters  $\theta^{(\ell)}\in \mathbb{R}$ .
+
+Although we have presented the general idea only for Lie groups  $G$ , it is possible to construct the DISCO convolution also on manifolds with a group action acting on them, such as the 2-sphere  $\mathbb{S}^2$ . For a detailed discussion of DISCO convolutions and a construction in one dimension, we point the reader to Appendix B and Ocampo et al. (2022).
+
+Remark 3.3 (Exact integration and equivariance). We note that DISCO convolutions satisfy equivariance properties for function classes that can be exactly integrated. For instance, on planar domains, these could be polynomials when using Legendre points. For equidistant grids on the torus, exact integration holds for bandlimited functions that are sampled above the Nyquist frequency.
+
+# 3.5. Local neural operator architecture
+
+To design our neural operator, we want to combine pointwise, local, and global operations. To this end, we take an FNO (or SFNO for spherical problems) as a starting point, which already features global operations in Fourier space and pointwise operations using its residual connections. Then, we augment the operators by incorporating our
+
+![](images/54d453b927088201ddd761b2c04e8fca4bf2edbf7ea0997ec5bc315237e670b4.jpg)  
+initial condition
+
+![](images/bdb6b9ec2bee76d37087c89d35d107a8a36ea585a43cf728b2a21046e32bf704.jpg)  
+initial condition
+
+![](images/e0ad9d90318fa4c47f386446edd3f659e5b948d54ddf47c755613ed60c6eed6a.jpg)  
+ground truth
+
+![](images/56253caf99d3161817c4c6ba6968c899c18b2b53d061e6cd721977d5688edc73.jpg)  
+ground truth
+
+![](images/d82493be12a4ec7e9645b4ab2715ad0c0628d6dcd2c7e05e0732c73fb189ef3f.jpg)  
+prediction (5 steps)  
+Figure 3: Initial condition, ground truth, and corresponding autoregressive predictions of our proposed models for the Navier-Stokes problem and the shallow water equations.
+
+![](images/6b0de6feb81466d42c836f3ad16947bb1625ce43b3f8a5d4ecd00eb0b9dd3e47.jpg)  
+prediction (5 steps)  
+(a) Navier-Stokes Eqns.  
+(b) Shallow Water Eqns.
+
+proposed local convolutions from the previous section as additional branches in the respective layers (see Figure 2). These four branches are summed pointwise within each local neural operator layer. The resulting architecture can be trained end-to-end with any standard operator learning training procedure and loss.
+
+# 4. Experiments
+
+To validate the effectiveness of local operators, we evaluate the architecture on five PDE problems. Figure 3 and Figure 4 show samples on three of these problems. In all five cases, we incorporate our proposed local operators into existing FNO and SFNO architectures to demonstrate that a significant improvement in performance can be achieved by introducing the inductive bias of local convolutions. This
+
+![](images/83d9741840c6571c5094b17c6e73d427c8f3480a1a710265bc5e118a0a8acc8f.jpg)  
+ground truth
+
+![](images/6c529fa01d1edcc512c16406eb4a655e5cdb0c4d5bcdeb0091a4a35cde4c6fae.jpg)  
+prediction  
+Figure 4: Ground truth and prediction of the horizontal velocity for the flow past a cylinder. The data is represented on an unstructured mesh, which is visualized in gray color. Our proposed architecture can be readily applied to data on unstructured meshes such as this one. For this figure, we learn the residual to the previous time step.
+
+section describes the experimental setting for our problems. We provide further experiments, specific implementation details<sup>5</sup>, and hyperparameters in Appendix C.
+
+# 4.1. Darcy flow
+
+We first consider the steady-state, two-dimensional Darcy flow equation
+
+$$
+- \nabla \cdot (a \nabla u) = f, \quad u | _ {\partial D} = 0, \tag {10}
+$$
+
+on the domain  $D = (0,1)^2$ . In this problem, we motivate the need to incorporate local operators as inductive biases in neural operator architectures. To this end, we explicitly construct a problem that requires approximating the forcing function  $f$  for a given diffusion coefficient  $a$  and pressure  $u$  in (10). In other words, the task consists of learning the differential operator
+
+$$
+u \mapsto - \nabla \cdot (a \nabla u). \tag {11}
+$$
+
+Following the setup of Hasani & Ward (2024), we take
+
+$$
+a (x) = \left[ \begin{array}{c c} x _ {1} ^ {2} & \sin (x _ {1} x _ {2}) \\ x _ {1} + x _ {2} & x _ {2} \end{array} \right]
+$$
+
+for  $x \in D$ . We then compare the performance of our proposed models: FNO in parallel with differential kernels, FNO with integral kernels, and FNO with both kernels. We also compare with a baseline FNO and the U-Net architecture from Gupta & Brandstetter (2022). Moreover, we perform zero-shot super-resolution on this problem; these results can be found in Appendix C.6.
+
+# 4.2. Navier-Stokes equations
+
+Next, we consider the two-dimensional Navier-Stokes equation on the torus. In particular, we consider Kolmogorov flows, which can be described by
+
+$$
+\begin{array}{l} \partial_ {t} u + u \cdot \nabla u - \frac {1}{\operatorname {R e}} \Delta u = - \nabla p + \sin (m y) \hat {x} \tag {12} \\ \nabla \cdot u = 0 \\ \end{array}
+$$
+
+on the spatial domain  $D = (0,2\pi)^2$  equipped with periodic boundary conditions. In the above,  $u$  and  $p$  denote the velocity and pressure, and Re denotes the Reynolds number of the flow. We want to learn the solution operator mapping initial conditions  $u(\cdot ,0) = u_0$  to the time-evolved solution in vorticity form  $w(\cdot ,\tau)$  at time  $\tau \in (0,\infty)$ , where the vorticity is given by
+
+$$
+w = (\nabla \times u) \hat {z} \tag {13}
+$$
+
+with  $\hat{z}$  being a unit vector normal to the plane.
+
+We consider  $m = 4$  and  $\mathrm{Re} = 5000$ , and we emphasize that learning the solution operator for such a high Reynolds number is very challenging due to the turbulent nature and small-scale features of the flow. We conjecture that the baseline FNO is prone to over-smoothing over the finer scales, consequently leading to a degradation of performance.
+
+To validate this conjecture, we compare the performance of the local neural operator with a baseline FNO and U-Net and evaluate the effectiveness of our proposed differential and integral kernels, respectively.
+
+# 4.3. Diffusion-Reaction equation
+
+Moreover, we consider the 2D Diffusion-Reaction equation from Takamoto et al. (2022) which can, for instance, be used for modeling biological pattern formation. In particular, we want to predict the time-evolution of two non-linearly coupled variables, i.e., the activator  $u$  and the inhibitor  $v$ , solving the equation
+
+$$
+\partial_ {t} u = c _ {u} \Delta u + R _ {u} (u, v), \tag {14}
+$$
+
+$$
+\partial_ {t} v = c _ {v} \Delta v + R _ {v} (u, v), \tag {15}
+$$
+
+for the spatial domain  $D = (-1, 1)^2$  with no-flow Neumann boundary condition. The reaction functions  $R_u$  and  $R_v$  are given by the Fitzhugh-Nagumo equation
+
+$$
+R _ {u} (u, v) = u - u ^ {3} - k - v, \quad R _ {v} (u, v) = u - v \tag {16}
+$$
+
+and we consider  $c_{u} = 10^{-3}, c_{v} = 5 \cdot 10^{-3}$ , and  $k = 5 \cdot 10^{-3}$ . The task is to learn the operator mapping the state  $(u, v)$  at consecutive time-steps to the time-evolved state and we can compare against the baselines from Takamoto et al. (2022).
+
+Table 2: Results for the 2D diffusion-reaction equation from PDEBench (Takamoto et al., 2022). We present their U-Net baseline results and metrics. In particular, we report the RMSE errors for the low, medium, and high frequency ranges (fRMSE low/med./high), and the (absolute) RMSE, the maximum error, RMSE errors at the boundary (bRMSE), as well as RMSE for the conserved values (cRMSE).  
+
+<table><tr><td>Model</td><td># Params</td><td>Rel. L2-error</td><td>fRMSE low</td><td>fRMSE med.</td><td>fRMSE high</td><td>RMSE</td><td>max. error</td><td>bRMSE</td><td>cRMSE</td></tr><tr><td>U-Net</td><td>7.8·105</td><td>8.4·10-1</td><td>1.7·10-2</td><td>8.2·10-4</td><td>5.7·10-2</td><td>6.1·10-2</td><td>1.9·10-1</td><td>7.8·10-2</td><td>3.9·10-2</td></tr><tr><td>FNO</td><td>9.3·105</td><td>8.3·10-2</td><td>6.2·10-4</td><td>5.6·10-4</td><td>2.4·10-4</td><td>5.2·10-3</td><td>7.3·10-2</td><td>1.5·10-2</td><td>1.2·10-3</td></tr><tr><td>FNO + loc. int (ours)</td><td>8.8·105</td><td>6.3·10-2</td><td>4.0·10-4</td><td>4.6·10-4</td><td>1.5·10-4</td><td>3.6·10-3</td><td>5.0·10-2</td><td>1.0·10-2</td><td>4.8·10-4</td></tr><tr><td>FNO + diff. (ours)</td><td>8.8·105</td><td>3.4·10-2</td><td>4.4·10-4</td><td>1.9·10-4</td><td>6.1·10-5</td><td>1.9·10-3</td><td>3.5·10-2</td><td>4.4·10-3</td><td>1.1·10-3</td></tr><tr><td>FNO + loc. int. + diff. (ours)</td><td>8.9·105</td><td>3.1·10-2</td><td>3.2·10-4</td><td>1.8·10-4</td><td>6.2·10-5</td><td>1.7·10-3</td><td>3.3·10-2</td><td>4.2·10-3</td><td>7.4·10-4</td></tr></table>
+
+# 4.4. Shallow water equations
+
+To test the approach on the sphere, we consider the shallow water equations on the rotating sphere. They represent a system of hyperbolic partial differential equations used to model a variety of geophysical flow phenomena, such as atmospheric flows, tsunamis, and storm surges. They can be formulated in terms of the evolution of two state variables  $\varphi$  and  $u$  (geopotential height and the tangential velocity of the fluid column), governed by the equations
+
+$$
+\partial_ {t} \varphi + \nabla \cdot (\varphi u) = 0, \tag {17}
+$$
+
+$$
+\partial_ {t} (\varphi u) + \nabla \cdot F = f,
+$$
+
+on the sphere  $D = \mathbb{S}^2$  with suitable initial conditions  $\varphi (\cdot ,0) = \varphi_0$  and  $u(\cdot ,0) = u_{0}$ , a momentum flux tensor
+
+$$
+F _ {i j} = \varphi u _ {i} u _ {j} + \frac {1}{2} \varphi^ {2}, \tag {18}
+$$
+
+and a source term
+
+$$
+f = - 2 \Omega x \times (\varphi u), \tag {19}
+$$
+
+which models the Coriolis force due to the rotation of the sphere with angular velocity  $\Omega$ .
+
+As baselines, we use a planar U-Net architecture, a spherical U-Net, where convolutions are performed using local integral kernels on the sphere, and an SFNO architecture. On the sphere, we only consider the impact of the integral kernel on the SFNO architecture, as it allows for a natural extension to the sphere (see Section 3.4).
+
+# 4.5. Flow past a cylinder
+
+Due to the formulation of the discrete-continuous convolutions, our approach can be readily generalized to unstructured meshes. To demonstrate this, we consider the Navier-Stokes equations in a two-dimensional channel with a cylindrical structure and a membrane attached to it (see Rahman et al., 2024). Figure 4 depicts a sample from the validation dataset alongside predictions from our model.
+
+The dataset considers the flow past the cylinder with Reynolds number  $\mathrm{Re} = 2000$ . Rahman et al. (2024) provide several baselines for a low-data regime (250 samples),
+
+Table 3: Results for the cylinder flow problem in a low-data regime with 250 training samples. Baseline results are taken from Rahman et al. (2024) and all architecture including ours perform a direct prediction (no residual prediction) of the velocity and pressure at the next time step.  
+
+<table><tr><td>Model</td><td># Parameters</td><td>MSE error</td></tr><tr><td>GINO</td><td>6·107</td><td>2.09·10-2</td></tr><tr><td>DeepONet</td><td>6·106</td><td>1.39·10-1</td></tr><tr><td>GNN</td><td>6·105</td><td>5.00·10-3</td></tr><tr><td>ViT</td><td>3·107</td><td>1.19·10-2</td></tr><tr><td>U-net</td><td>3·107</td><td>9.34·10-2</td></tr><tr><td>FNO + local integral kernel (ours)</td><td>10·106</td><td>2.88·10-3</td></tr></table>
+
+where the task is to predict the velocity and pressure after a given time step. To deal with the unstructured grid, we adapt the architecture to utilize local integral operators as encoders and decoders to transform the data into a latent representation on an equidistant grid. Then, we can use an FNO with our local integration layers in the latent space.
+
+# 4.6. Results and discussion
+
+The results of our numerical experiments are reported $^{6}$  in Tables 2 to 4. We observe significant performance gains over the baselines in all five problem settings. In particular, we notice that the best performance in the reported relative  $L^2$  errors is achieved with the inclusion of both global and local operations, despite an overall reduction in parameter count with respect to their corresponding FNO/SFNO baselines. This is particularly pronounced for the Navier-Stokes and shallow water equations, where errors can accumulate in autoregressive roll-outs. We attribute this to our models' inductive bias, allowing it to better capture the fine-grained scales and thus achieve better performance. In Table 2, we also show that our local layers achieve lower errors at high frequencies for the diffusion-reaction equation, indicating that local, high-frequency features are correctly captured.
+
+The Navier-Stokes problem demonstrates the efficacy of
+
+Table 4: Experimental results for Darcy flow, Navier-Stokes, and the spherical shallow water problems. For all three problems, the test error is reported in terms of the relative  $L^2$ -error after a single step. For the time-dependent Navier-Stokes and shallow water equations, we also predict the error after 5 autoregressive steps.  
+
+<table><tr><td rowspan="2">Model</td><td colspan="4">Parameters</td><td colspan="2">Relative L2-Error</td></tr><tr><td># Layers</td><td># Modes</td><td>Embedding</td><td># Parameters</td><td>1 step</td><td>5 steps</td></tr><tr><td colspan="7">Darcy Flow</td></tr><tr><td>U-Net</td><td>17</td><td>-</td><td>18</td><td>2.850 · 106</td><td>1.380 · 10-2</td><td>-</td></tr><tr><td>FNO</td><td>4</td><td>20</td><td>41</td><td>2.715 · 106</td><td>5.867 · 10-2</td><td>-</td></tr><tr><td>FNO + diff. kernel (ours)</td><td>4</td><td>12</td><td>65</td><td>2.638 · 106</td><td>7.357 · 10-3</td><td>-</td></tr><tr><td>FNO + local integral kernel (ours)</td><td>4</td><td>20</td><td>40</td><td>2.617 · 106</td><td>6.034 · 10-2</td><td>-</td></tr><tr><td>FNO + local integral + diff. kernel (ours)</td><td>4</td><td>12</td><td>64</td><td>2.639 · 106</td><td>9.032 · 10-3</td><td>-</td></tr><tr><td colspan="7">Navier-Stokes Equations</td></tr><tr><td>U-Net</td><td>17</td><td>-</td><td>56</td><td>2.758 · 107</td><td>1.674 · 10-1</td><td>5.115 · 10-1</td></tr><tr><td>FNO</td><td>4</td><td>40</td><td>65</td><td>2.711 · 107</td><td>1.381 · 10-1</td><td>2.360 · 10-1</td></tr><tr><td>FNO + diff. kernel (ours)</td><td>4</td><td>40</td><td>65</td><td>2.726 · 107</td><td>1.073 · 10-1</td><td>2.129 · 10-1</td></tr><tr><td>FNO + local integral kernel (ours)</td><td>4</td><td>20</td><td>129</td><td>2.716 · 107</td><td>1.110 · 10-1</td><td>2.183 · 10-1</td></tr><tr><td>FNO + local integral + diff. kernel (ours)</td><td>4</td><td>20</td><td>127</td><td>2.691 · 107</td><td>9.022 · 10-2</td><td>1.956 · 10-1</td></tr><tr><td colspan="7">Spherical Shallow Water Equations</td></tr><tr><td>U-Net</td><td>17</td><td>-</td><td>32</td><td>2.898 · 106</td><td>1.341 · 10-3</td><td>1.226 · 10-2</td></tr><tr><td>Spherical U-Net (with local integral kernel)</td><td>17</td><td>-</td><td>32</td><td>1.639 · 106</td><td>6.160 · 10-4</td><td>3.265 · 10-3</td></tr><tr><td>SFNO</td><td>4</td><td>128</td><td>32</td><td>1.066 · 106</td><td>9.220 · 10-4</td><td>3.185 · 10-3</td></tr><tr><td>SFNO + local integral kernel (ours)</td><td>4</td><td>128</td><td>31</td><td>1.019 · 106</td><td>2.624 · 10-4</td><td>5.392 · 10-4</td></tr></table>
+
+combining both differential and local integral kernels with the global convolution of the FNO. Our proposed model with these three components together outperforms models with only two of these three operations. While hyperbolic PDEs likely require only local receptive fields (due to finite information propagation (LeVeque, 1992)) and elliptic problems instead require global information, we observe that many practical problems involving a mixture of operators benefit from a hybrid approach such as ours.
+
+We note that the Darcy flow problem is meant to motivate the need for our proposed differential kernels (and indeed, our best-performing model achieves  $87\%$  lower relative  $L^2$ -error than FNO). Since the ground truth operator in (11) is a differential operator, we expect (and observe) that for high accuracies, very local (i.e., differential) kernels are needed. In particular, we see that the baseline FNO performs poorly, and the local integral kernels do not provide additional benefits. However, we include them in Table 4 for completeness.
+
+Finally, we perform super-resolution experiments, in which we train the model at a given resolution and then evaluate it at another resolution without fine-tuning. Figure 8 depicts super-resolution results at twice the training resolution for both the Darcy and shallow water equations. The corresponding numerical results are listed in Table 5. We observe that our approach generalizes well across resolutions and outperforms the baselines across all tested resolutions. For a detailed discussion of the super-resolution experiments,
+
+we refer the reader to Appendix C.6.
+
+# 5. Conclusion
+
+In this paper, we have demonstrated a novel framework for local neural operators. We have shown how convolutional layers can be constrained to realize neural operators that approximate differential operators in the continuous limit. Moreover, we have derived convolutions with local integral kernels from the general notion of an integral transform and the related graph neural operator. Finally, we have constructed localized neural operators on the sphere by using discrete-continuous convolutions (Ocampo et al., 2022).
+
+The resulting neural operators introduce a strong inductive bias for learning operators with local receptive fields. In particular, their formulation ensures the same local operation everywhere in the domain. This equivariance (w.r.t. the underlying symmetry group) reduces the required number of learnable parameters and improves generalization.
+
+Our numerical experiments demonstrate consistent improvements when existing neural operators with global receptive fields are augmented with the proposed localized convolutions, resulting in reductions in relative  $L^2$ -error of up to  $72\%$  over the corresponding baselines. We thus expect local neural operators to play an important role in solving real-world scientific computing problems with machine learning.
+
+# Acknowledgements
+
+The authors thank Md Ashiqur Rahman for useful discussions about benchmarks and baselines. M. Liu-Schiaffini is grateful for support from the Mellon Mays Undergraduate Fellowship. J. Berner acknowledges support from the Wally Baer and Jeri Weiss Postdoctoral Fellowship. A. Anandkumar is supported in part by Bren endowed chair and by the AI2050 senior fellow program at Schmidt Sciences.
+
+# Impact Statement
+
+The aim of this work is to advance the field of machine learning and scientific computing. While there are many potential societal consequences, none of them are immediate to require specifically being highlighted here.
+
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+
+# A. General differential kernels
+
+# A.1. Theoretical construction
+
+In the following, we present the general idea for irregular grids, from which Proposition 3.1 follows as a special case. Let us first define the assumptions on our grids.
+
+Regular discrete refinement: Let  $\| \cdot \|$  be a norm on  $\mathbb{R}^d$  and denote by  $B_{h}(x)\subset \mathbb{R}^{d}$  the ball of radius  $h\in (0,\infty)$  around  $x\in \mathbb{R}^d$  w.r.t.  $\| \cdot \|$ . Further, let  $D\subset \mathbb{R}^d$  be a domain and let  $(h_\ell)_{\ell \in \mathbb{Z}}$  be a sequence that converges to zero. We call  $(D_{\ell})_{\ell \in \mathbb{Z}}\subset D$  a regular discrete refinement with widths  $(h_\ell)_{\ell \in \mathbb{Z}}$  if there exists  $N\in \mathbb{N}$  such that for all  $\ell \in \mathbb{Z}$  and  $x\in \mathbb{R}^d$  we have that
+
+$$
+\left| B _ {h _ {\ell}} (x) \cap D _ {\ell} \right| \leq N \quad \text {a n d} \quad \text {s p a n} \left(\left\{\left[ \begin{array}{c} 1 \\ y \end{array} \right] - \left[ \begin{array}{c} 1 \\ x \end{array} \right]: y \in B _ {h _ {\ell}} (x) \cap D _ {\ell} \right\}\right) = \mathbb {R} ^ {d + 1}.
+$$
+
+The second assumption states that we can find an affinely independent subset in each ball. Note that, for instance, equidistant grids satisfy these assumptions.
+
+First-order differential operator: We want to find bounded kernels  $k(x,y) \colon \mathbb{R}^d \times \mathbb{R}^d \to \mathbb{R}$  with the following property: There exist  $c \in \mathbb{R}$  and  $b \in \mathbb{R}^d$  such that for all  $v \in C^1(D, \mathbb{R})$ , all  $y \in \mathbb{R}^d$ , and any regular discrete refinement  $(D_\ell)_{\ell \in \mathbb{Z}} \subset \mathbb{R}^d$  with widths  $(h_\ell)_{\ell \in \mathbb{Z}}$  it holds that
+
+$$
+\lim  _ {\ell \rightarrow \infty} \sum_ {x \in B _ {h _ {\ell}} (y) \cap D _ {\ell}} k (x, y) v (x) = c v (y) + \nabla v (y) \cdot b. \tag {20}
+$$
+
+We will now investigate which additional assumptions are needed. Let us fix  $y \in \mathbb{R}^d$  and enumerate
+
+$$
+(x _ {j}) _ {j = 1} ^ {m} := B _ {h _ {\ell}} (y) \cap D _ {\ell}.
+$$
+
+Then, we can use Taylor's theorem to show that
+
+$$
+\begin{array}{l} \sum_ {x \in B _ {h _ {\ell}} (y) \cap D _ {\ell}} k (x, y) v (x) = \sum_ {j = 1} ^ {m} k \left(x _ {j}, y\right) \left(v (y) + \nabla v (y) \cdot \left(x _ {j} - y\right) + \mathcal {O} \left(h _ {\ell}\right)\right) (21) \\ = v (y) \sum_ {j = 1} ^ {m} k \left(x _ {j}, y\right) + \nabla v (y) \cdot \left(\sum_ {j = 1} ^ {m} k \left(x _ {j}, y\right) \left(x _ {j} - y\right)\right) + \mathcal {O} \left(m \| k \| _ {L ^ {\infty}} h _ {\ell}\right). (22) \\ \end{array}
+$$
+
+Since the kernel and  $m$  are bounded (uniformly over  $\ell \in \mathbb{N}$ ), we have that  $\mathcal{O}(m\| k\|_{L^{\infty}}h_{\ell}) = \mathcal{O}(h_{\ell})$ . As we want to guarantee convergence to  $cv(y) + \nabla v(y)\cdot b$  for suitable  $c\in \mathbb{R}$  and  $b\in \mathbb{R}^d$  (independent of  $y$ ), we need to satisfy that
+
+$$
+c = \sum_ {j = 1} ^ {m} k \left(x _ {j}, y\right) \tag {23}
+$$
+
+and that
+
+$$
+b = \sum_ {j = 1} ^ {m} k \left(x _ {j}, y\right) \left(x _ {j} - y\right) \tag {24}
+$$
+
+This yields the linear system
+
+$$
+\underbrace {\left[ \begin{array}{c c c} 1 & \dots & 1 \\ x _ {1} - y & \dots & x _ {m} - y \end{array} \right]} _ {\in \mathbb {R} ^ {(d + 1) \times n}} \left[ \begin{array}{c} k (x _ {1}, y) \\ \vdots \\ k (x _ {m}, y) \end{array} \right] = \left[ \begin{array}{c} c \\ b \end{array} \right]. \tag {25}
+$$
+
+Our assumptions on the refinement guarantee that we can find linearly independent columns such that we can solve the system. However, generally and by abuse of notation, the value of  $k(x_{j},y)$  depends on all the points  $(x_{j})_{j = 1}^{m}$ . However, for an equidistant grid, one can directly see that the convolutional kernel given in Proposition 3.1 satisfies (25) with  $c = 0$ .
+
+![](images/9d5efda2b9c994b5cd5f7099ad22b2808191a3116e7e11b5b6626accc91e9162.jpg)  
+(a) Convergence  
+Figure 5: Empirical evaluation of the proposed differential kernel. (a)  $L^2$  errors at various resolutions and quadratic coefficient scales  $c$ . (b) True differential operator for  $c = 1$ . (c) Output of the differential kernel at a resolution of  $32 \times 32$ . (d) Output of the differential kernel at a resolution of  $64 \times 64$ . (e) Output of the differential kernel at a resolution of  $4096 \times 4096$ .
+
+![](images/4ea8a65eb9e2d36d559b17f6e2ae97e13312ed378346ad29ca6ecffd346819d8.jpg)  
+(b) Ground truth
+
+![](images/bb4e79fb6f155bfeab99049aadf4bc76802e3a240d4eb626038b61d67d8fcf91.jpg)  
+(c) Low-res prediction
+
+![](images/f95872889963a603ed2fc978938bb52fee54b113010e31c974d06a61a3c9fca1.jpg)  
+(d) Medium-res prediction
+
+![](images/8448f21a3770183553e4688a0a55933389d6df2e257e10087ceb53208747a084.jpg)  
+(e) High-res prediction
+
+# A.2. Empirical evaluation
+
+In this section, we empirically evaluate the convergence of our proposed differential kernels to a differential operator as the resolution increases. In particular, we consider a randomly-initialized differential kernel satisfying the constraints discussed in Section 3.2 (mean zero and scaled by the resolution  $\frac{1}{h}$ , where  $h$  is the width of each grid cell in an equidistant grid).
+
+To be able to compute the theoretical differential operator in closed form, we consider a simple parabola  $v \colon [0,1]^2 \to \mathbb{R}^n$ , given by
+
+$$
+x \mapsto \| x \| ^ {2} \left[ c _ {1}, \dots , c _ {n} \right] ^ {\top}, \tag {26}
+$$
+
+with coefficients  $c_{1},\ldots ,c_{n}$
+
+We define a randomly-initialized first-order differential kernel  $K = (K_{ij})_{i,j=1}^{3,3} \subset \mathbb{R}^n$  (i.e., one differential kernel for each of the  $n$  input channels) subject to the constraints described in Section 3.2. Let  $b_j \in \mathbb{R}^n$  denote the direction corresponding to the directional derivative corresponding to the differential kernel for channel  $j$ , see Appendix A.1. With the parabola as defined in (26), the output function of the true differential operator corresponding to  $K$  is given by
+
+$$
+x \mapsto 2 \sum_ {i = j} ^ {n} x \cdot b _ {j}. \tag {27}
+$$
+
+In this simple experiment, we set  $n = 10$  and generate  $c_{1},\ldots ,c_{n}$  uniformly on the unit interval, multiplied by some scaling factor  $c\in \{1,2,4,16\}$ . We discretize the parabola in (26) over different resolutions, convolve the kernel  $K$  with each of these discretizations, and compute the  $L^2$  error with respect to the true differential operator. Figure 5 shows the  $L^2$  error for these values of  $c$  and resolutions, as well as visualizations of the true differential operator and the one outputted by the differential kernel at three different resolutions.
+
+From Figure 5, we observe a clear convergence to the differential operator as the resolution increases. For a fixed resolution, convergence is slower for greater magnitudes of the second derivative (i.e., larger  $c$ ). This can be theoretically shown by estimating the remainder of the Taylor expansion in Appendix A.1.
+
+# B. Discrete-continuous convolutions
+
+# B.1. General Ideas
+
+The following section outlines the basic ideas behind discrete-continuous convolutions as introduced by (Ocampo et al., 2022). To generalize the (continuous) convolution (5) to Lie groups and quotient spaces of Lie Groups, we consider the group convolution (see e.g. Cohen & Welling (2016)).
+
+Definition B.1 (Group Convolution). Let  $\kappa, v: G \to \mathbb{R}$  be functions defined on the group  $G$ . The group convolution is given by
+
+$$
+(\kappa \star v) (g) = \int \kappa \left(g ^ {- 1} x\right) \cdot v (x) \mathrm {d} \mu (x), \tag {8}
+$$
+
+where  $g, x \in G$  and  $\mathrm{d}\mu(x)$  is the invariant Haar measure on  $G$ .
+
+Remark B.2. In some cases, signals are not defined on a group but rather on a quotient space  $G / H$ , where  $H$  is a subgroup of  $G$ . In such cases, a convolution may still be defined by taking  $g \in G / H$ . For an example, see spherical convolutions (Driscoll & Healy, 1994; Ocampo et al., 2022).
+
+While group convolutions can typically be computed by generalized Fourier transforms on the corresponding manifolds, their usage is generally preferred if the convolutions are non-local operators, i.e., the convolution kernel  $\kappa$  is not compactly supported. On the periodic domain  $\mathbb{T}^d$  (i.e., Euclidean space with periodic boundaries), such convolutions are typically computed discretely by directly sliding the kernel.
+
+Definition B.3 (DISCO convolutions). Given a quadrature rule with quadrature points  $x_{j} \in G$  and quadrature weights  $q_{j}$ , we approximate the group convolution (8) with the discrete sum
+
+$$
+(\kappa \star v) (g) = \int \kappa \left(g ^ {- 1} x\right) \cdot v (x) \mathrm {d} \mu (x) \approx \sum_ {j = 1} ^ {m} \kappa \left(g ^ {- 1} x _ {j}\right) \cdot v \left(x _ {j}\right) q _ {j}. \tag {28}
+$$
+
+In particular, the group action  $g$  is applied analytically to the kernel function  $\kappa$ , whereas the integral is approximated using the quadrature rule.
+
+For a discrete set of output locations  $g_{i}$ , this becomes a straight-forward matrix-vector multiplication
+
+$$
+\sum_ {j = 1} ^ {m} \kappa \left(g _ {i} ^ {- 1} x _ {j}\right) \cdot v \left(x _ {j}\right) q _ {j} = \sum_ {j = 1} ^ {m} K _ {i j} \cdot v \left(x _ {j}\right) q _ {j} \tag {29}
+$$
+
+with  $K_{ij} = \kappa (g_i^{-1}x_j)$ . In the case where  $\kappa$  is compactly supported,  $K_{ij}$  is a sparse matrix with the number of non-zero entries per row depending on the resolution of the grid  $x_j$  and the support of  $\kappa$ . To obtain a learnable filter,  $\kappa$  is parametrized as a linear combination of a chosen set of basis functions.
+
+# B.2. DISCO convolutions in one dimension
+
+For the sake of simplicity, we discuss the simple one-dimensional case on  $D = [0,1]$  with periodic boundary conditions. We note that this corresponds to the circle group (or the torus)  $\mathbb{T}$ . For any element  $y \in D$ , the corresponding group operation  $T_y$  is the translation  $T_y: D \to D$ ,  $x \mapsto x \oplus y$ , where we denote by  $x \oplus y$  a modular shift such that the result remains in  $D$ .
+
+Then, the DISCO convolution in one dimension, for Lebesgue square-integrable functions  $v$  and  $\kappa$  becomes
+
+$$
+(\kappa \star v) (y) = \int_ {[ 0, 1 ]} \kappa \left(T _ {y} ^ {- 1} x\right) v (x) \mathrm {d} x = \int_ {[ 0, 1 ]} \kappa (x - y) v (x) \mathrm {d} x \approx \sum_ {j = 1} ^ {m} \kappa \left(x _ {j} - y\right) v \left(x _ {j}\right) q _ {j}, \tag {30}
+$$
+
+for suitable quadrature points  $D^{h} = \{x_{j}\}_{j = 1}^{m}$  with corresponding quadrature weights  $\{q_j\}_{j = 1}^m$ .
+
+To parameterize the filter  $\kappa$ , we pick a finite support  $[0, x_{\mathrm{cutoff}}]$  with  $L$  equidistant collocation points  $\xi^{(\ell)} \in [0, x_{\mathrm{cutoff}}]$  and corresponding hat functions. The  $\ell$ -th hat function is then defined as
+
+$$
+\kappa^ {(\ell)} (x) = \left\{ \begin{array}{l l} \frac {x - \xi^ {(\ell - 1)}}{\xi^ {(\ell)} - \xi^ {(\ell - 1)}} & \text {f o r} \xi^ {(\ell - 1)} \leq x <   \xi^ {(\ell)} \\ \frac {\xi^ {(\ell + 1)} - x}{\xi^ {(\ell + 1)} - \xi^ {(\ell)}} & \text {f o r} \xi^ {(\ell)} \leq x <   \xi^ {(\ell + 1)} \\ 0 & \text {e l s e ,} \end{array} \right. \tag {31}
+$$
+
+where  $\xi^{(0)},\xi^{(L + 1)}\in [0,x_{\mathrm{cutoff}}]$  are suitable boundary points. The resulting filter is obtained as a linear combination  $\kappa = \sum_{\ell = 1}^{L}\theta^{(\ell)}k^{(\ell)}$  with trainable parameters  $\theta^{(\ell)}$ . Plugging this into (29), we obtain the trainable DISCO convolution
+
+$$
+\sum_ {j = 1} ^ {m} \kappa \left(x _ {j} - y _ {i}\right) v \left(x _ {j}\right) q _ {j} = \sum_ {\ell = 1} ^ {L} \sum_ {j = 1} ^ {m} \theta^ {(\ell)} \kappa^ {(\ell)} \left(x _ {j} - y _ {i}\right) v \left(x _ {j}\right) q _ {j} = \sum_ {\ell = 1} ^ {L} \sum_ {j = 1} ^ {m} \theta^ {(\ell)} K _ {i j} ^ {(\ell)} v \left(x _ {j}\right) q _ {j}, \tag {32}
+$$
+
+where  $K_{ij}^{(\ell)} = \kappa^{(\ell)}(x_j - y_i)$  are the shifted filter functions.
+
+Let us now consider the special case of an equidistant grid  $D^h$ , i.e.,  $x_{j + 1} - x_j = h$ , and a trapezoidal quadrature rule  $q_{j} = h$ . Let us further assume that the output points  $y_{i}$  coincide with the grid points and that the collocation points  $\xi^{(\ell)}$  are given as the first  $L$  grid points. Then, due to the property of the hat functions,  $K_{ij}^{(\ell)}$  can only contain either 0 or 1 and we obtain the circulant convolution matrices given by
+
+$$
+K ^ {(1)} = (e _ {1}, e _ {2}, e _ {3}, \ldots , e _ {m}), \qquad K ^ {(2)} = (e _ {2}, e _ {3}, \ldots , e _ {m}, e _ {1}), \qquad K ^ {(3)} = (e _ {3}, \ldots , e _ {m}, e _ {1}, e _ {2}), \quad \ldots
+$$
+
+where  $e_i \in \mathbb{R}^m$  is the  $i$ -th standard basis vector. For regular grids on planar geometries, we can thus efficiently implement the DISCO convolution in (32) in terms of highly-optimized CUDA kernels based on common-place convolutional layers. This observation also generalizes to higher dimensions, where the same discrete kernel is obtained when the kernel is continuously shifted on the grid.
+
+# B.3. DISCO convolutions on the sphere
+
+For general group actions  $g \in SO(3)$ , the outcome of the group convolution (8) will be a function defined on  $SO(3)$ . This is due to  $\mathbb{S}^2$  not being a group but rather a manifold on which  $SO(3)$  acts. We can see this by fixing the north-pole  $n = [0,0,1]^\top$  and applying any rotation  $g \in SO(3)$  to it. This will trace out the whole sphere despite the north pole eliminating one of the Eulerian rotation angles. Therefore, to ensure that the result of the convolution is still a function defined on  $\mathbb{S}^2$ , we can simply restrict  $g$  in (8) to rotations in  $SO(3)/SO(2)$ , which is isomorphic to  $\mathbb{S}^2$ . Formally, this can be achieved by fixing the first of the three Euler angles parameterizing  $g$  to 0.
+
+As basis functions, we pick a set of piecewise linear basis functions as in (31). To accommodate anisotropic kernels, collocation points are distributed in an equidistant manner along both radius and circumference. More precisely, the first collocation point lies at the center, and for each consecutive ring, a fixed amount of collocation points is distributed along the circumference. The resulting basis functions are illustrated in Figure 6, for a cutoff radius of  $r_{\mathrm{cutoff}} = 0.1\pi$
+
+![](images/0bffa083e3c56999e48f98394af716fcf57d7347ed573bd77a74d047ec5c1494.jpg)  
+(a)  $\ell = 1$
+
+![](images/ec0f383ef5d608e71b0e9e4a8bd610346548355226299100e215eca6dbffa411.jpg)  
+(b)  $\ell = 2$
+
+![](images/c508761ad851160f4aec020c6fc762a434ac107374a5b90da9a973efe1ecf504.jpg)  
+(c)  $\ell = 3$  
+Figure 6: Radial, piecewise linear basis functions for the approximation of anisotropic filters on the sphere.
+
+![](images/5f941db6ba1139da87de2ae0858c1e6adeccd77a1d25d70c23c9b8067c352079.jpg)  
+(d)  $\ell = 4$
+
+![](images/65bf4aed49feec670ad67b13dbc83f2923cf792d76f3074dc5a077c0a11f8eac.jpg)  
+$(e)$ $\ell = 5$
+
+# C. Implementation details
+
+A numerical comparison of our methods with baseline FNO and U-Net architectures can be found in Table 4. In this section, we outline the implementation details for our numerical experiments.
+
+For Darcy flow, Navier-Stokes, and the shallow water equations, training is conducted by minimizing the squared  $L^2$ -loss until convergence is reached. Our U-Net baseline is adapted from the model and code of the PDE Arena benchmark (Gupta & Brandstetter, 2022). For the FNOs (Li et al., 2020a) and SFNOs (Bonev et al., 2023), we use the implementation in the neuraloperator and torch-harmonics libraries. Moreover, for all experiments, GELU activation functions and the Adam optimizer are used.
+
+For these three problems, we trained the FNO/SFNO-based models with varying widths and modes, while keeping the overall number of parameters approximately constant. We present the best results for each problem and macro-architecture in Table 4. For the models with local layers, we found that a larger embedding dimension and fewer modes can often improve performance. We conjecture that the increased embedding dimension confers additional expressivity to the local kernels. This also suggests that local operators are an important inductive bias for these problems. When supplementing the FNO/SFNO blocks with additional branches, we also observed improved convergence by scaling the initial parameters or the output by  $n^{-1/2}$ , where  $n$  is the number of branches. A detailed experimental setup is outlined for these three datasets in the
+
+following subsections. We also present results for the 2D diffusion-reaction equation and for super-resolution experiments on Darcy flow and the shallow water equations.
+
+# C.1. 2D Darcy flow equation
+
+In the 2D Darcy flow setting, we generate our data as described in Section 4.1. For the input functions, we consider random linear combinations of eigenfunctions of the Laplace operator with zero Dirichlet boundary conditions, i.e.,
+
+$$
+u (x) = \sum_ {i, j = 1} ^ {2 0} \frac {c _ {i j}}{\sqrt {(i \pi) ^ {2} + (j \pi) ^ {2}}} \sin (i \pi x _ {1}) \sin (j \pi x _ {2}), \quad x \in D,
+$$
+
+with i.i.d.  $c_{ij} \sim \mathcal{N}(0,1 / (i + j))$ . We train and test our models and baselines with data discretized onto a  $256 \times 256$  regular grid. We use 10000 training samples and 2000 testing samples. As a baseline, we compare our proposed models with FNO (Li et al., 2020a) and the U-Net architecture of Gupta & Brandstetter (2022). We note that this U-Net architecture is not agnostic to the discretization (Kovachki et al., 2021), see also Section 3.1. We compare these baselines against our proposed models: the architecture using convolutions with Fourier (i.e., FNO), differential, and integral kernels (Figure 2), as well as architectures using only FNO and differential kernels or only FNO with the proposed integral kernels. Figure 7 compares the predictions of each model.
+
+We choose all hyperparameters such that the overall number of parameters of all compared models is similar. The number of layers, number of Fourier modes, and embedding dimension are shown in Table 4. Models using convolutions with differential and local integral kernels use these layers in parallel to the Fourier layers and pointwise skip connection on all layers. We use reflective padding for the convolutional operations in the differential and local integral kernels. For all relevant models, the local integral kernels use a radius cutoff of 0.007 on  $[-1,1]^2$ , and they are parameterized by five radial, piecewise linear basis functions for the approximation of anisotropic filters (analogous to Figure 6 on the plane). The differential kernels are parameterized as  $3\times 3$  convolutional kernels over the regular grid. For our U-Net baseline, we use  $3\times 3$  convolutional kernels with 2 residual blocks for each resolution (two downsampling and two upsampling) and three layers within each block, with channel multipliers of 1, 2, 4 for each layer within a block.
+
+Training is conducted by minimizing the squared  $L^2$ -loss for 70 epochs on a single NVIDIA P100 GPU, which is sufficient to achieve convergence on all models. We use a step learning rate decay scheduler, starting at  $10^{-3}$  and halving every 10 epochs. The results are shown in Table 4.
+
+![](images/52d0bf43243512ef75685d963d3b22bdfa616b5070e35545ab96d6dc817c65f3.jpg)  
+(a) Input
+
+![](images/3cf3bf5905b75cc452c10c1af64b262be05daa1d0a1f0b97b374091b7d1a091f.jpg)  
+Figure 7: Comparison of models from Table 4: The outputs of our best-performing model, FNO, and U-Net on a randomly selected input pressure function from the Darcy flow problem. Edge artifacts are very prevalent in the FNO predictions, and they are less dominant in the U-Net predictions.
+
+![](images/f5aa1e79eb830019d677b6bcba1624272b8e58dd978f8887cf5b9eae5dace6a7.jpg)  
+(b) Ground truth  
+(c) Ours
+
+![](images/a04520724f0939785704adaeb54fffb4af3e716faa2e9480e320b6cbc3246736.jpg)  
+(d) FNO
+
+![](images/cce0414660fff41f41c937d678ad3d8c1d57c5e1e5526f8a5065b581a4c512da.jpg)  
+(e) U-Net
+
+# C.2. 2D Navier-Stokes Equations
+
+For the 2D Navier-Stokes equations (Kolmogorov flows), we use the same experimental setup and dataset as Li et al. (2022a), which sets  $m = 4$  and  $\mathrm{Re} = 5000$  in (12) and uses a temporal discretization of  $\tau = 1$  on a  $128 \times 128$  regular grid. The initial conditions are sampled from a Gaussian measure as described in Li et al. (2022a), and the equation is solved with a pseudo-spectral solver. We compare the same five models as in the Darcy experiment.
+
+As in the Darcy setting, we choose all hyperparameters such that the overall number of parameters for all the models is similar. The number of layers, number of Fourier modes, and embedding dimension are shown in Table 4. Models using convolutions with differential and local integral kernels use these layers in parallel to the Fourier layers and pointwise skip
+
+connection on all layers. We enforce periodic boundary conditions during padding for the convolutional operations in the differential and local integral kernels. For all relevant models, the local integral kernels use a radius cutoff of  $0.05\pi$  on the torus, and they are parameterized by five radial, piecewise linear basis functions for the approximation of anisotropic filters (analogous to Figure 6 on the plane). The differential kernels are parameterized as  $3 \times 3$  convolutional kernels over the regular grid. Our U-Net baseline is set up in the same way as in the Darcy experiment.
+
+Training was performed for 136 epochs on a single NVIDIA RTX 4090 GPU with an exponentially decaying learning rate, starting at  $10^{-3}$  and halving every 33 epochs. The results are shown in Table 4.
+
+# C.3. Diffusion-Reaction equation
+
+For the 2D Diffusion-Reaction equation, we use the same experimental setup and dataset as Takamoto et al. (2022). The reference solution is computed using a finite-volume method in space and a fourth-order Runge-Kutta method in time. The dataset consists of 900 training samples and 100 validation samples discretized on a  $128 \times 128$  regular grid with 100 equidistant time-steps in the interval [0, 5] and Gaussian initial conditions. The task is to predict the state of the variables  $(u, v)$  at the next time-step from the states at the previous 10 steps. The errors are measured for the full autoregressive roll-out as in the implementation of PDEBench at github.com/pdebench/PDEBench.
+
+We use the same experimental setup and implementation as Takamoto et al. (2022) and only adapt the learning rate, the number of modes, and the embedding dimension. We then add our proposed layers to (a subset of) the FNO blocks and experiment with both reflective and replicate padding. In particular, we parametrize the local integral kernels by five radial, piecewise linear basis functions and use  $3 \times 3$  convolutional kernels for the differential kernels.
+
+We train on a single NVIDIA RTX 4090 GPU for 500 epochs with early stopping (using the same criteria as Takamoto et al. (2022)). Moreover, we use an exponentially decaying learning rate, starting at  $10^{-4}$  and halving every 100 epochs. We present our best results in Table 2 and refer to Takamoto et al. (2022) for details on the metrics. We compare against the baselines by Takamoto et al. (2022) and note that we also improve their FNO baseline results. However, our local integral and differential kernels still provide a significant improvement with fewer parameters. Specifically, we reduced the modes from 24 to 16 and increased the embedding dimension from 20 to 29.
+
+# C.4. Shallow water Equations
+
+For the shallow water equations, we use the dataset presented by Bonev et al. (2023), which uses a Gaussian random field to generate initial conditions on an equiangular lat-lon-grid on the sphere at a resolution of  $256 \times 512$  and solves for  $\varphi$ ,  $u$  at a lead time of one hour. The target solution is computed using a spectral solver, which takes 24 Euler steps<sup>8</sup>. Physical constants such as the sphere's radius or the Coriolis force are set to match those of Earth. The numerical solver uses 150 explicit Euler steps to advance the solution 1 hour in time. The right-hand side is discretized using the spectral basis provided by the Spherical Harmonics. For a detailed description of the dataset, we refer the reader to Bonev et al. (2023) and the corresponding implementation in the torch-harmonics package.
+
+As a baseline for our experiments, we use the SFNO architecture as presented by Bonev et al. (2023), where the embedding dimension is adjusted to 32 to obtain a manageable parameter count. Moreover, we adapt the U-Net architecture by Gupta & Brandstetter (2022) to the spherical domain by replacing all spatial (i.e., not the  $1 \times 1$ ) convolutions with DISCO convolutions on the sphere. Therefore, the resulting architecture is a spherical U-Net similar to the architecture presented by Ocampo et al. (2022). Moreover, due to the DISCO convolutions' discretization-agnostic nature, this architecture can be interpreted as a neural operator. Finally, we augment the SFNO architecture with local DISCO convolutions to obtain the proposed architecture; see Section 3.5.
+
+For all three architectures, hyperparameters were chosen to achieve roughly similar parameter counts. The learning rates for each architecture were determined with a quick parameter sweep, resulting in  $3 \cdot 10^{-4}$  for the spherical U-Net and  $3 \cdot 10^{-3}$  for both neural operators. As a learning rate scheduler, we use the policy of halving the learning rate upon a plateauing of the loss. Training was performed for 100 epochs on a single NVIDIA RTX A6000 GPU, which was sufficient to achieve convergence on all considered models.
+
+![](images/9efe73ba3444a98c11ed8f8ecc6912dc90a5ad0f0074e6fdbc2a9a9617cf3bc2.jpg)  
+(a) Input pressure
+
+![](images/8eb309cc835be94368eed5a03e7200000c8a450669dd70c3f029475198796a2a.jpg)  
+(b) Ground truth forcing
+
+![](images/3c4b349b053dc1e62ce89ed2dd76d55c5665849bda60de08bbea583d0869587b.jpg)  
+(c) Predicted forcing
+
+![](images/5580952d864023bec856aeae730d32c1bd699fed0a173631380f8247ee36a7e7.jpg)  
+(d) Initial geopotential height
+
+![](images/13c38d41af8ade37069d4d453665318dbf20b52dcb44726d10d13c2213ec6a3b.jpg)  
+(e) Ground truth at 1h
+
+![](images/c2e4e3f61f07bca2fec025650d07171f9f0e88ea93f39516d9de034ea2f5644d.jpg)  
+$(f)$  Prediction at 1h  
+Figure 8: Randomly selected super-resolution samples for the Darcy flow (top row) and shallow water (bottom row) problems.
+
+# C.5. Flow past a cylinder
+
+To demonstrate the capability of dealing with unstructured representations, we use the dataset provided by Rahman et al. (2024) to train a model for predicting velocity and pressure fields for a Navier-Stokes problem in a channel with a suspended cylinder and attached membrane (see Figure 4). We use the dataset with a viscosity of  $\mu = 1.0$ , which corresponds to a Reynolds number of  $\mathrm{Re} = 2000$ . To deal with the unstructured data, we employ a local integral convolution layer in both the encoder and decoder with a cutoff radius of  $r_{\mathrm{cutoff}} = 0.052$  and 5 piecewise-linear basis functions. This layer transforms the unstructured data to a regularly spaced grid of  $48 \times 192$  in the internal representation. The processor part of the architecture then consists of 4 FNO blocks with local integral kernels, which have the same filter basis as the convolutions used in the encoder/decoder. The embedding dimension is fixed at 16 and all Fourier modes are kept in the internal representation. Overall, this leads to a parameter count of  $9.6 \cdot 10^{6}$ . The architecture is trained on a dataset of 250 samples for 50 epochs using the Adam optimizer and a learning rate of  $2 \cdot 10^{-3}$ . The results alongside baselines from Rahman et al. (2024) are reported in Table 3.
+
+# C.6. Zero-shot super-resolution results
+
+In this paper, we propose two methods to embed the inductive bias of locality into neural operator architectures. The key distinction between our proposed methods and CNN-based architectures is that our methods are agnostic to the discretization of the input function. In this section, we present and discuss the super-resolution capabilities of our proposed models. In particular, we focus on two examples: (1) the Darcy flow equation to showcase our differential layers on a regular Cartesian grid and (2) the shallow water equation to demonstrate super-resolution for our local integration on the sphere. The experimental setting that we consider is that of zero-shot super-resolution. In particular, suppose that the model has been trained at a particular training resolution (or at multiple different resolutions). Given input at a higher resolution than the training resolution, the task of zero-shot super-resolution is to then predict the output function at this higher resolution and evaluate the resulting model error.
+
+In the Darcy flow setting, we use the same setup and dataset as described in Section 4.1. We train two models on data sampled at a  $256 \times 256$  regular grid and evaluate on a  $128 \times 128$  regular grid ( $\frac{1}{2}$ x resolution and  $\frac{1}{4}$ x the number of points),  $512 \times 512$  grid (2x super-resolution), and  $1024 \times 1024$  grid (4x super-resolution). For the shallow water equations, a similar approach is taken. We take the models in Section 4.4 which were trained at a resolution of  $256 \times 512$  and apply them to data generated at a resolution of  $128 \times 256$ ,  $512 \times 1024$ , and  $1024 \times 2048$ .
+
+Table 5: Zero-shot super-resolution results for Darcy flow and the spherical shallow water problems. The validation error is reported in terms of relative  $L^2$ -error at various resolutions. Autoregressive rollouts are super-resolved in the sense that the rollout is performed at the high resolution.  
+
+<table><tr><td rowspan="3">Model</td><td colspan="4">Parameters</td><td colspan="4">Relative L2-Error</td></tr><tr><td rowspan="2"># Layers</td><td rowspan="2"># Modes</td><td rowspan="2">Embedding</td><td rowspan="2"># Parameters</td><td colspan="4">resolution (relative to training resolution)</td></tr><tr><td>1/2×</td><td>1×</td><td>2×</td><td>4×</td></tr><tr><td colspan="9">Darcy Flow</td></tr><tr><td>FNO</td><td>4</td><td>20</td><td>41</td><td>2.715·106</td><td>1.475·10-1</td><td>5.867·10-2</td><td>8.646·10-2</td><td>7.731·10-2</td></tr><tr><td>FNO + diff. (ours)</td><td>4</td><td>20</td><td>40</td><td>2.599·106</td><td>1.174·10-1</td><td>5.851·10-2</td><td>7.774·10-2</td><td>6.681·10-2</td></tr><tr><td colspan="9">Spherical Shallow Water Equations</td></tr><tr><td>Spherical U-Net</td><td>17</td><td>-</td><td>32</td><td>1.639·106</td><td>5.586·10-3</td><td>6.160·10-4</td><td>3.386·10-3</td><td>4.102·10-3</td></tr><tr><td>SFNO</td><td>4</td><td>128</td><td>32</td><td>1.066·106</td><td>1.342·10-3</td><td>9.220·10-4</td><td>3.830·10-3</td><td>4.419·10-3</td></tr><tr><td>SFNO + loc. int. (ours)</td><td>4</td><td>128</td><td>31</td><td>1.019·106</td><td>8.673·10-4</td><td>2.624·10-4</td><td>3.341·10-3</td><td>4.097·10-3</td></tr></table>
+
+Table 5 provides the details of the models we used for these experiments as well as the results of our super-resolution experiments with our proposed models. In the Darcy setting, we compare the baseline FNO to the FNO augmented with the differential kernel. On the sphere, we make use of the fact that the spherical U-Net presented in (Ocampo et al., 2022) is already a neural operator due to its discretization-independence. As such, we can use it alongside the SFNO as a baseline for zero-shot super-resolution on the sphere.
+
+As with all neural operator architectures, during training there is the possibility of overfitting to the training resolution. For instance, FNO may learn features in Fourier space that are intrinsically tied to the resolution of the input function. Similarly, it is possible that our proposed differential and integral convolutional operators will learn a function of the training discretization. This effect can be remedied by using high-resolution training data where the local details are fully resolved. This minimum required resolution of the training data is thus a function of the smoothness of the input function. For this reason, we decided to exclude the 2D Navier-Stokes problem from our super-resolution experiments, since training at a resolution that sufficiently resolves the local dynamics would be prohibitively expensive.
+
+In our experiments, we noticed that our differential layers tend to incur some discretization errors when trained on data that is not of sufficiently high resolution. If differential convolutions are present in consecutive layers in the model, this error can propagate quickly. As such, we found that the best-performing model on zero-shot super-resolution for the Darcy flow problem is a model with a differential convolution in only the first layer. Using fewer differential layers in our model reduces the expressivity, but we find that the super-resolution capabilities are still better than the baseline FNO.
+
+Lastly, we would like to note that in our experiments, FNO had a larger error near the boundary in non-periodic problems, as the FFT used in FNO assumes periodic boundary conditions. We note that our proposed differential layer can help reduce the error at the boundary caused by FNO, but some of these effects may still be present (see Figure 8).
+
+# C.7. Computational efficiency of differential and local integral kernels
+
+On equidistant grids, our differential and local integral kernels can be implemented as standard convolutional kernels, which are heavily optimized for GPU performance. The computational complexity of our differential kernels is linear in the number of grid points since the size of the convolutional kernel remains constant regardless of resolution. In contrast, the size of the discretized local integral kernels does increase with input resolution. While this may be expensive for some high-resolution datasets, in our experiments, we found that the heavily-optimized convolutions in deep learning libraries can help reduce this computational burden. As a consequence, local integral kernels are more efficient than GNO (Li et al., 2020b) and other graph-based approaches on equidistant grids.
+
+Moreover, the local integral kernels can also be applied to irregular grids, where they are implemented as sparse matrix multiplications. This tends to outperform GNOs (Li et al., 2020b), which need to evaluate a neural network on the graph. The complexity of the local integral layer is linear with a constant that depends on the sparsity of the matrix. This, in turn, depends on the support of the local kernel; choosing a fixed support will scale quadratically with the resolution, whereas a support that matches the resolution will scale linearly in practice.

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