Solving PDEs on Spheres with Physics-Informed Convolutional Neural Networks
Not on the map. Why?
Abstract (arXiv)
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.
Key facts. Cited by 3 papers in the map (data through 7 Oct 2026). Builds on 21 of its 59 references that are in the map. First posted on arXiv 18 Aug 2023.
Builds on
21 of 59 references.
References · 21
- 2014Adam: A Method for Stochastic Optimizationcited by 36,960
- 2016Gaussian Error Linear Units (GELUs)cited by 3,431
- 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,806
- 2019DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operatorscited by 1,965
- 2020Neural Operator: Graph Kernel Network for Partial Differential Equationscited by 722
- 2017DGM: A deep learning algorithm for solving partial differential equationscited by 674
- 2017The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problemscited by 581
- 2018Universality of Deep Convolutional Neural Networkscited by 176
- 2018Adaptivity of deep ReLU network for learning in Besov and mixed smooth Besov spaces: optimal rate and curse of dimensionalitycited by 160
- 20191D Convolutional Neural Networks and Applications: A Surveycited by 152
- 2019Deep ReLU network approximation of functions on a manifoldcited by 73
- 2019Nonparametric Regression on Low-Dimensional Manifolds using Deep ReLU Networks : Function Approximation and Statistical Recoverycited by 73
9 more
- 2019Deep learning is adaptive to intrinsic dimensionality of model smoothness in anisotropic Besov spacecited by 46
- 2021Machine Learning For Elliptic PDEs: Fast Rate Generalization Bound, Neural Scaling Law and Minimax Optimalitycited by 40
- 2021Besov Function Approximation and Binary Classification on Low-Dimensional Manifolds Using Convolutional Residual Networkscited by 32
- 2020Theory of Deep Convolutional Neural Networks II: Spherical Analysiscited by 22
- 2021Theory of Deep Convolutional Neural Networks III: Approximating Radial Functionscited by 22
- 2022Physics-Informed Neural Networks for Shell Structurescited by 12
- 2022Learning Ability of Interpolating Deep Convolutional Neural Networkscited by 9
- 2023Pairwise Ranking with Gaussian Kernelscited by 2
- 2023Numerical Methods For PDEs Over Manifolds Using Spectral Physics Informed Neural Networkscited by 1
Built on by
Citing papers per year: 2024: 1; 2025: 1; 2026: 1.
3 papers.