Solving spatiotemporal partial differential equations with Physics-informed Graph Neural Network

Abstract

Physics-informed neural networks (PINNs) have recently gained considerable attention as a prominent deep learning technique for solving partial differential equations (PDEs). However, traditional fully connected PINNs often encounter slow convergence issues attributed to automatic differentiation in constructing loss functions. In addition, convolutional neural network (CNN)-based PINNs face challenges when dealing with irregular domains and unstructured meshes. To address these issues, we propose a novel framework based on graph neural networks (GNNs) and radial basis function finite difference (RBF-FD). We introduce GNNs into physics-informed learning to better handle irregular domains. RBF-FD is employed to construct a high-precision difference format of the PDE to guide model training. We perform numerical experiments on various PDEs, including heat, wave, and shallow water equations on irregular domains. The results demonstrate that our method is capable of accurate predictions and exhibits strong generalization, allowing for inference with different initial. We evaluate the generalizability, accuracy, and efficiency of our approach by considering different Gaussian noise, PDE parameters, numbers of collection points, and various types of radial basis functions.