Solving PDEs on Spheres with Physics-Informed Convolutional Neural
  Networks
Guanhang Lei Zhen Lei Lei Shi Chenyu Zeng Ding-Xuan Zhou
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.