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.

翻译：物理信息神经网络已被证明能够高效地从多种实验视角求解偏微分方程。近期研究也提出了针对包括球面在内的曲面上的偏微分方程的物理信息神经网络算法。然而，关于物理信息神经网络数值性能的理论理解，尤其是曲面或流形上的物理信息神经网络，仍然较为匮乏。本文针对球面上的物理信息卷积神经网络求解偏微分方程问题建立了严谨分析。通过利用并改进深度卷积神经网络的最新逼近结果与球谐分析，我们证明了索伯列夫范数下逼近误差的上界。随后，我们将其与创新的局部化复杂度分析相结合，建立了物理信息卷积神经网络的快速收敛率。实验结果也印证并补充了我们的理论成果。基于这些发现，我们进一步探索了规避高维偏微分方程求解中维度灾难问题的潜在策略。