  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.
