Solving PDEs on Spheres with Physics-Informed Convolutional Neural Networks

TL;DR

This paper provides a rigorous theoretical analysis of physics-informed convolutional neural networks (PICNNs) for solving PDEs on spheres, establishing error bounds and convergence rates, supported by experiments and exploring high-dimensional PDE solutions.

Contribution

It offers the first rigorous error and convergence analysis of PICNNs on spherical surfaces, advancing understanding of their numerical performance.

Findings

Established upper bounds for approximation errors on the sphere.

Proved fast convergence rates for PICNNs.

Validated theoretical results with experiments.

Abstract

Physics-informed neural networks (PINNs) have been demonstrated to be efficient in solving partial differential equations (PDEs) from a variety of experimental perspectives. Some recent studies have also proposed PINN algorithms for PDEs on surfaces, including spheres. However, theoretical understanding of the numerical performance of PINNs, especially PINNs on surfaces or manifolds, is still lacking. In this paper, we establish rigorous analysis of the physics-informed convolutional neural network (PICNN) for solving PDEs on the sphere. By using and improving the latest approximation results of deep convolutional neural networks and spherical harmonic analysis, we prove an upper bound for the approximation error with respect to the Sobolev norm. Subsequently, we integrate this with innovative localization complexity analysis to establish fast convergence rates for PICNN. Our theoretical results are also confirmed and supplemented by our experiments. In light of these findings, we explore potential strategies for circumventing the curse of dimensionality that arises when solving high-dimensional PDEs.