In this paper, we study the partial differential equation models of neural
networks. Neural network can be viewed as a map from a simple base model to a
complicate function. Based on solid analysis, we show that this map can be
formulated by a convection-diffusion equation. This theoretically certified
framework gives mathematical foundation and more understanding of neural
networks. Moreover, based on the convection-diffusion equation model, we design
a novel network structure, which incorporates diffusion mechanism into network
architecture. Extensive experiments on both benchmark datasets and real-world
applications validate the performance of the proposed model.

本文研究神经网络的偏微分方程模型，通过对其进行分析，证明了这些模型可以由对流 - 扩散方程表达，从而为神经网络提供了数学基础和更深入的理解。基于这个模型，我们设计了一种新的网络结构，将扩散机制融入网络体系结构，并通过广泛实验验证了该模型的性能。

对流扩散方程：神经网络的理论认证框架

Convection-Diffusion Equation: A Theoretically Certified Framework for  Neural Networks

Inspired by the relation between deep neural network (DNN) and partial
differential equations (PDEs), we study the general form of the PDE models of
deep neural networks. To achieve this goal, we formulate DNN as an evolution
operator from a simple base model. Based on several reasonable assumptions, we
prove that the evolution operator is actually determined by
convection-diffusion equation. This convection-diffusion equation model gives
mathematical explanation for several effective networks. Moreover, we show that
the convection-diffusion model improves the robustness and reduces the
Rademacher complexity. Based on the convection-diffusion equation, we design a
new training method for ResNets. Experiments validate the performance of the
proposed method.

深度神经网络与偏微分方程之间的关系启发我们研究了深度神经网络的偏微分方程模型的一般形式。在一些合理的假设下，我们证明演化算子实际上由对流 - 扩散方程决定。这个模型为几个有效的网络提供了数学解释。此外，我们还展示了对流 - 扩散模型提高了鲁棒性并减小了 Rademacher 复杂度。基于对流 - 扩散方程，我们设计了一种适用于 ResNets 的新训练方法。实验证实了所提出方法的性能。

一种公理化的深度神经网络 PDE 模型

An axiomatized PDE model of deep neural networks

Graph neural networks (GNNs) have shown promising results across various
graph learning tasks, but they often assume homophily, which can result in poor
performance on heterophilic graphs. The connected nodes are likely to be from
different classes or have dissimilar features on heterophilic graphs. In this
paper, we propose a novel GNN that incorporates the principle of heterophily by
modeling the flow of information on nodes using the convection-diffusion
equation (CDE). This allows the CDE to take into account both the diffusion of
information due to homophily and the ``convection'' of information due to
heterophily. We conduct extensive experiments, which suggest that our framework
can achieve competitive performance on node classification tasks for
heterophilic graphs, compared to the state-of-the-art methods. The code is
available at https://github.com/zknus/Graph-Diffusion-CDE.

本文提出了一种新颖的图神经网络，利用卷积扩散方程（CDE）建模节点上的信息流，考虑到异质性和同质性对信息的影响，通过在异质性图上的节点分类任务的实验证明，我们的框架可以取得竞争性的性能。