🤖 AI Summary
To address low modeling accuracy and poor physical consistency in spatiotemporal partial differential equation (PDE) learning under irregular meshes and limited data, this paper proposes a novel graph neural network (GNN) framework integrated with numerical integration. The method explicitly encodes the discrete Laplace–Beltrami operator via a learnable Laplacian module and incorporates boundary-condition-aware padding to embed geometric and physical priors directly into the network architecture. This design enables high-fidelity dynamic prediction of diffusion and reaction–diffusion PDE systems on coarse, unstructured grids. Experiments demonstrate that the approach significantly outperforms existing purely data-driven and physics-informed hybrid methods under scarce training data, achieving state-of-the-art accuracy. Moreover, it exhibits strong generalization across mesh topologies and robust physical consistency, ensuring reliable long-term simulation and interpretability grounded in underlying PDE principles.
📝 Abstract
Solving partial differential equations (PDEs) serves as a cornerstone for modeling complex dynamical systems. Recent progresses have demonstrated grand benefits of data-driven neural-based models for predicting spatiotemporal dynamics (e.g., tremendous speedup gain compared with classical numerical methods). However, most existing neural models rely on rich training data, have limited extrapolation and generalization abilities, and suffer to produce precise or reliable physical prediction under intricate conditions (e.g., irregular mesh or geometry, complex boundary conditions, diverse PDE parameters, etc.). To this end, we propose a new graph learning approach, namely, Physics-encoded Message Passing Graph Network (PhyMPGN), to model spatiotemporal PDE systems on irregular meshes given small training datasets. Specifically, we incorporate a GNN into a numerical integrator to approximate the temporal marching of spatiotemporal dynamics for a given PDE system. Considering that many physical phenomena are governed by diffusion processes, we further design a learnable Laplace block, which encodes the discrete Laplace-Beltrami operator, to aid and guide the GNN learning in a physically feasible solution space. A boundary condition padding strategy is also designed to improve the model convergence and accuracy. Extensive experiments demonstrate that PhyMPGN is capable of accurately predicting various types of spatiotemporal dynamics on coarse unstructured meshes, consistently achieves the state-of-the-art results, and outperforms other baselines with considerable gains.