Solving Partial Differential Equations Using Point-Based Neural Networks

Abstract

Recently, solving partial differential equations (PDEs) using neural networks (NNs) has been attracting increasing interests with promising potential to be applied in wide areas. In this paper, we propose a theoretical model that approximates the operator from a parametric function space to a solution function space and prove its universal approximation theorem in the operator space. For practical application, by regarding the domain of a parametric (or solution) function as a discrete point cloud, we propose a novel idea that implements the theoretical model by introducing the point-based NN as the backbone. We show that the present model can approximate the solution operator of static PDEs using the training data generated on unstructured meshes while most existing methods work for the data generated on lattice grid meshes. We conduct experiments to demonstrate the performance of our model on different types of PDEs. The numerical results verify that our model possesses a higher precision and a faster inference speed compared with the existing models for data of unstructured meshes; in addition, our model has competitive performance compared with the existing works dealing with data of lattice grid meshes.