Since the seminal work of [9] and their Physics-Informed neural networks (PINNs), many efforts have been conducted towards solving partial differential equations (PDEs) with Deep Learning models. However, some challenges remain, for instance the extension of such models to complex three-dimensional geometries, and a study on how such approaches could be combined to classical numerical solvers. In this work, we justify the use of graph neural networks for these problems, based on the similarity between these architectures and the meshes used in traditional numerical techniques for solving partial differential equations. After proving an issue with the Physics-Informed framework for complex geometries, during the computation of PDE residuals, an alternative procedure is proposed, by combining classical numerical solvers and the Physics-Informed framework. Finally, we propose an implementation of this approach, that we test on a three-dimensional problem on an irregular geometry.

通过结合经典数值解法和物理感知框架的方法，本研究在三维非规则几何体上测试了基于图神经网络的解决偏微分方程问题的可行性。

基于物理信息的图卷积网络：面向复杂几何结构的通用框架