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

TL;DR

This work tackles the bottleneck of physics-informed neural networks (PINNs) in industrial PDE problems by highlighting limitations in residual computation via autodifferentiation and invariance handling. It introduces PiGMeN, a physics-informed graph-mesh network that combines an encoder–processor–decoder graph architecture with differentiable finite-element gradient kernels and a Lagrangian-based hybrid loss to enforce physics and boundary constraints. The approach preserves physical invariances through relative edge coordinates and leverages a differentiable FE loop to backpropagate through numerical kernels, enabling robust generalization to complex 2D and 3D geometries. Experiments in 2D and 3D demonstrate strong interpolation and extrapolation capabilities, supported by ablation studies that underscore the benefits of graph structure and the hybrid loss for learning the underlying physical operator while reducing reliance on dense training data.

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.

Figures (14)

• Figure 1: Model architecture. First, an encoder processes the physical input data. Next, this enriched input is fed through $L$ graph network blocks. Finally, a node decoder is applied to predict the target physical field. The black elements refer to the physical input, the red elements are the encoded data, the blue objects are the processed data, and the green nodes represent the output field, i.e the model's prediction.
• Figure 2: Fully connected vs locally biased architectures. The output formula show a translation invariance for the locally biased architecture, as opposed to the fully connected model.
• Figure 3: Workflow of the proposed process. The red quantities are computed by the numerical gradient kernel, while the blue quantities are computed inside the deep learning framework. Plain arrows trace the operations made during the forward call, while dashed arrows represent the operations made during the backward pass.
• Figure 4: Geometry and meshes used for the 2D numerical experiments. The boundary condition nodes are in red.
• Figure 5: (Left) Target potential, given as input to the model. (Right) Predicted potential.