Solving PDEs on Spheres with Physics-Informed Convolutional Neural Networks
Fuente:
arXiv
Saved in:
| Main Authors: | , , , , |
|---|---|
| Format: | Preprint |
| Published: |
2023
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866914898251874304 |
|---|---|
| author | Lei, Guanhang Lei, Zhen Shi, Lei Zeng, Chenyu Zhou, Ding-Xuan |
| author_facet | Lei, Guanhang Lei, Zhen Shi, Lei Zeng, Chenyu Zhou, Ding-Xuan |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_09605 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Solving PDEs on Spheres with Physics-Informed Convolutional Neural Networks Lei, Guanhang Lei, Zhen Shi, Lei Zeng, Chenyu Zhou, Ding-Xuan Numerical Analysis Machine Learning Statistics Theory 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. |
| title | Solving PDEs on Spheres with Physics-Informed Convolutional Neural Networks |
| topic | Numerical Analysis Machine Learning Statistics Theory |
| url | https://arxiv.org/abs/2308.09605 |