Physics-Informed Graph-Mesh Networks for PDEs: A hybrid approach for complex problems

TL;DR

This paper introduces a hybrid physics-informed graph-mesh network that combines neural networks with finite element methods to better solve complex PDEs, addressing limitations of existing neural approaches.

Contribution

It presents a novel hybrid model integrating graph neural networks with numerical kernels, improving handling of complex geometries and generalization in PDE solutions.

Findings

Effective in complex 2D and 3D geometries

Shows improved generalization over traditional neural methods

Supported by ablation studies confirming model components' importance

Abstract

The recent rise of deep learning has led to numerous applications, including solving partial differential equations using Physics-Informed Neural Networks. This approach has proven highly effective in several academic cases. However, their lack of physical invariances, coupled with other significant weaknesses, such as an inability to handle complex geometries or their lack of generalization capabilities, make them unable to compete with classical numerical solvers in industrial settings. In this work, a limitation regarding the use of automatic differentiation in the context of physics-informed learning is highlighted. A hybrid approach combining physics-informed graph neural networks with numerical kernels from finite elements is introduced. After studying the theoretical properties of our model, we apply it to complex geometries, in two and three dimensions. Our choices are supported by an ablation study, and we evaluate the generalisation capacity of the proposed approach.