Solving PDEs on Spheres with Physics-Informed Convolutional Neural Networks
This work addresses a theoretical gap for researchers in computational mathematics and machine learning, providing foundational insights for solving PDEs on manifolds, though it is incremental in building upon existing PINN methods.
The paper tackles the lack of theoretical understanding for physics-informed neural networks (PINNs) on surfaces by establishing rigorous analysis for solving PDEs on spheres, proving an upper bound for approximation error and fast convergence rates, with experimental confirmation.
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.