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.

中文翻译：

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使用物理信息卷积神经网络求解球体上的 PDE


物理信息神经网络 （PINN） 已被证明可以从各种实验角度有效地求解偏微分方程 （PDE）。最近的一些研究还提出了表面（包括球体）上偏微分方程的 PINN 算法。然而，对 PINN 的数值性能，尤其是表面或流形上的 PINN 的理论理解仍然缺乏。在本文中，我们对物理信息卷积神经网络 （PICNN） 进行了严格的分析，用于求解球体上的 PDE。通过使用和改进深度卷积神经网络和球谐分析的最新近似结果，我们证明了相对于 Sobolev 范数的近似误差的上限。随后，我们将其与创新的定位复杂性分析相结合，为 PICNN 建立快速收敛率。我们的理论结果也得到了实验的证实和补充。鉴于这些发现，我们探索了规避求解高维偏微分方程时出现的维数诅咒的潜在策略。