RBF-MGN:Solving spatiotemporal PDEs with Physics-informed Graph Neural Network

TL;DR

This work addresses solving spatiotemporal PDEs on irregular domains where standard PINNs struggle with boundary enforcement and unstructured meshes. It introduces RBF-MGN, a physics-informed graph neural network that couples MeshGraphNets with a hard constraint PDE residual computed via radial basis function finite difference ($\text{RBF-FD}$). Key contributions include leveraging GNNs to handle unstructured geometries, employing $\text{RBF-FD}$ to construct high-precision differential operators and enforce boundary conditions, and validating the approach on Poisson and wave equations across diverse domains, time steps, and RBF choices. Results demonstrate accurate solutions on complex domains (e.g., amoeba, butterfly) and robust extrapolation with unstructured meshes, highlighting practical potential for solving challenging PDEs in engineering and physics.

Abstract

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.

Figures (20)

• Figure 1: An example of a GNN, given the input/output graph $G = (V, E)$, where V and E are a set of vertices $V = {1, 2, 3}$ and edges $E \subseteq\left(V2\right)$. And the same adjacency matrix $(N(1) = {2, 3}, N(2) = {1, 3}, N(3) = {1, 2})$. The input and out features are the nodal solution $i$ at time $t$ vector ( $f^{(in)}_i = u^{t+\delta t} (x_i ))$ )and at time $t+\Delta t$ vector ( $f^{(out)}_i = u^{t+\delta t} (x_i ))$.
• Figure 2: Diagram of MeshGraphNets.
• Figure 3: The corresponding local region of (a) The classical difference. (b) The RBF-FD method.
• Figure 4: The Laplace operator $\Delta u$ approximated using RBF-FD \ref{['Delta']} and AD.
• Figure 5: The graph $G = (V, E)$ with nodes $V$ connected by edges $E$, V is a set of 167 points on a two-dimensional domain , including boundary nodes (red triangles), interior nodes (blue pentagons).