RBF-MGN:Solving spatiotemporal PDEs with Physics-informed Graph Neural Network
This work addresses a domain-specific problem in computational physics and engineering by offering a more effective method for PDE solving on complex geometries, though it appears incremental as it builds on existing PINN and GNN techniques.
The authors tackled the challenge of solving spatiotemporal PDEs on irregular domains with unstructured meshes by proposing a novel framework combining graph neural networks and radial basis function finite difference, achieving improved accuracy and efficiency in numerical experiments on Poisson and wave equations.
Physics-informed neural networks (PINNs) have lately received significant attention as a representative deep learning-based technique for solving partial differential equations (PDEs). Most fully connected network-based PINNs use automatic differentiation to construct loss functions that suffer from slow convergence and difficult boundary enforcement. In addition, although convolutional neural network (CNN)-based PINNs can significantly improve training efficiency, CNNs have difficulty in dealing with irregular geometries with unstructured meshes. Therefore, 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 with unstructured meshes. RBF-FD is used to construct a high-precision difference format of the differential equations to guide model training. Finally, we perform numerical experiments on Poisson and wave equations on irregular domains. We illustrate the generalizability, accuracy, and efficiency of the proposed algorithms on different PDE parameters, numbers of collection points, and several types of RBFs.