融合图神经网络与有限元法，提升复杂几何下PDE求解的精度与泛化能力。
Physics-Informed Graph-Mesh Networks for PDEs: A hybrid approach for complex problems
- 用图神经网络建模物理规律，结合有限元数值核处理复杂几何。
- 在2D/3D复杂域上实现高精度求解，优于传统物理信息网络。
- 适合需要高鲁棒性与可扩展性的工业级PDE求解场景。
深度学习在求解偏微分方程方面取得进展，尤其依赖物理信息神经网络（PINN）的方法在学术研究中表现良好。然而，其缺乏物理不变性、难以处理复杂几何结构且泛化能力不足，使其在工业场景中难以超越经典数值求解器。本文指出物理信息学习中自动微分应用的局限性，并提出一种混合方法：将物理信息图神经网络与有限元法的数值核相结合。通过理论分析与消融实验验证模型设计，该方法在二维和三维复杂几何上均表现出色，显著提升了求解精度与泛化能力。
原文摘要 · Abstract (English)
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.
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