A Priori Generalization Analysis of the Deep Ritz Method for Solving High Dimensional Elliptic Partial Differential Equations

Yulong Lu, Jianfeng Lu, Min Wang
Proceedings of Thirty Fourth Conference on Learning Theory, PMLR 134:3196-3241, 2021.

Abstract

This paper concerns the a priori generalization analysis of the Deep Ritz Method (DRM) [W. E and B. Yu, 2017], a popular neural-network-based method for solving high dimensional partial differential equations. We derive the generalization error bounds of two-layer neural networks in the framework of the DRM for solving two prototype elliptic PDEs: Poisson equation and static Schrödinger equation on the $d$-dimensional unit hypercube. Specifically, we prove that the convergence rates of generalization errors are independent of the dimension $d$, under the a priori assumption that the exact solutions of the PDEs lie in a suitable low-complexity space called spectral Barron space. Moreover, we give sufficient conditions on the forcing term and the potential function which guarantee that the solutions are spectral Barron functions. We achieve this by developing a new solution theory for the PDEs on the spectral Barron space, which can be viewed as an analog of the classical Sobolev regularity theory for PDEs.

Cite this Paper

BibTeX
@InProceedings{pmlr-v134-lu21a, title = {A Priori Generalization Analysis of the Deep Ritz Method for Solving High Dimensional Elliptic Partial Differential Equations}, author = {Lu, Yulong and Lu, Jianfeng and Wang, Min}, booktitle = {Proceedings of Thirty Fourth Conference on Learning Theory}, pages = {3196--3241}, year = {2021}, editor = {Belkin, Mikhail and Kpotufe, Samory}, volume = {134}, series = {Proceedings of Machine Learning Research}, month = {15--19 Aug}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v134/lu21a/lu21a.pdf}, url = {https://proceedings.mlr.press/v134/lu21a.html}, abstract = {This paper concerns the a priori generalization analysis of the Deep Ritz Method (DRM) [W. E and B. Yu, 2017], a popular neural-network-based method for solving high dimensional partial differential equations. We derive the generalization error bounds of two-layer neural networks in the framework of the DRM for solving two prototype elliptic PDEs: Poisson equation and static Schrödinger equation on the $d$-dimensional unit hypercube. Specifically, we prove that the convergence rates of generalization errors are independent of the dimension $d$, under the a priori assumption that the exact solutions of the PDEs lie in a suitable low-complexity space called spectral Barron space. Moreover, we give sufficient conditions on the forcing term and the potential function which guarantee that the solutions are spectral Barron functions. We achieve this by developing a new solution theory for the PDEs on the spectral Barron space, which can be viewed as an analog of the classical Sobolev regularity theory for PDEs.} }
Endnote
%0 Conference Paper %T A Priori Generalization Analysis of the Deep Ritz Method for Solving High Dimensional Elliptic Partial Differential Equations %A Yulong Lu %A Jianfeng Lu %A Min Wang %B Proceedings of Thirty Fourth Conference on Learning Theory %C Proceedings of Machine Learning Research %D 2021 %E Mikhail Belkin %E Samory Kpotufe %F pmlr-v134-lu21a %I PMLR %P 3196--3241 %U https://proceedings.mlr.press/v134/lu21a.html %V 134 %X This paper concerns the a priori generalization analysis of the Deep Ritz Method (DRM) [W. E and B. Yu, 2017], a popular neural-network-based method for solving high dimensional partial differential equations. We derive the generalization error bounds of two-layer neural networks in the framework of the DRM for solving two prototype elliptic PDEs: Poisson equation and static Schrödinger equation on the $d$-dimensional unit hypercube. Specifically, we prove that the convergence rates of generalization errors are independent of the dimension $d$, under the a priori assumption that the exact solutions of the PDEs lie in a suitable low-complexity space called spectral Barron space. Moreover, we give sufficient conditions on the forcing term and the potential function which guarantee that the solutions are spectral Barron functions. We achieve this by developing a new solution theory for the PDEs on the spectral Barron space, which can be viewed as an analog of the classical Sobolev regularity theory for PDEs.
APA
Lu, Y., Lu, J. & Wang, M.. (2021). A Priori Generalization Analysis of the Deep Ritz Method for Solving High Dimensional Elliptic Partial Differential Equations. Proceedings of Thirty Fourth Conference on Learning Theory, in Proceedings of Machine Learning Research 134:3196-3241 Available from https://proceedings.mlr.press/v134/lu21a.html.

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