Physics-Informed Graph-Mesh Networks for PDEs: A hybrid approach for complex problems
This work addresses the problem of applying physics-informed learning to industrial PDE problems with complex geometries, representing an incremental improvement over existing methods.
The paper tackled the limitations of Physics-Informed Neural Networks in handling complex geometries and generalization for solving partial differential equations, by introducing a hybrid approach combining graph neural networks with finite element kernels, achieving improved performance on 2D and 3D complex geometries as validated through ablation studies.
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.