Graph Neural Networks (GNNs) have established themselves as the preferred
methodology in a multitude of domains, ranging from computer vision to
computational biology, especially in contexts where data inherently conform to
graph structures. While many existing methods have endeavored to model GNNs
using various techniques, a prevalent challenge they grapple with is the issue
of over-smoothing. This paper presents new Graph Neural Network models that
incorporate two first-order Partial Differential Equations (PDEs). These models
do not increase complexity but effectively mitigate the over-smoothing problem.
Our experimental findings highlight the capacity of our new PDE model to
achieve comparable results with higher-order PDE models and fix the
over-smoothing problem up to 64 layers. These results underscore the
adaptability and versatility of GNNs, indicating that unconventional approaches
can yield outcomes on par with established techniques.

新的图神经网络模型利用两个一阶偏微分方程（PDEs）在不增加复杂度的情况下有效缓解了过度平滑问题，并在最多 64 层中取得与高阶 PDE 模型可比较的结果，突显了图神经网络的适应性和多功能性。

图神经网络的一阶偏微分方程：平流和 Burgers 方程模型

First-order PDES for Graph Neural Networks: Advection And Burgers  Equation Models

A wide range of applications in science and engineering involve a PDE model
in a domain with perforations, such as perforated metals or air filters.
Solving such perforated domain problems suffers from computational challenges
related to resolving the scale imposed by the geometries of perforations. We
propose a neural network-based mesh-free approach for perforated domain
problems. The method is robust and efficient in capturing various configuration
scales, including the averaged macroscopic behavior of the solution that
involves a multiscale nature induced by small perforations. The new approach
incorporates the derivative-free loss method that uses a stochastic
representation or the Feynman-Kac formulation. In particular, we implement the
Neumann boundary condition for the derivative-free loss method to handle the
interface between the domain and perforations. A suite of stringent numerical
tests is provided to support the proposed method's efficacy in handling various
perforation scales.

提出了一种基于神经网络的无网格方法来解决具有孔洞领域的 PDE 模型问题，该方法能够捕捉各种尺度的配置，包括由小孔洞引起的多尺度特性，并采用了无导数损失方法处理领域与孔洞之间的界面。通过一系列严格的数值测试支持了该方法在处理不同孔洞尺度问题上的有效性。