2021-07-21-lu21a.md

title	abstract	layout	series	publisher	issn	id	month	tex_title	firstpage	lastpage	page	order	cycles	bibtex_author	author	date	address	container-title	volume	genre	issued	pdf	extras
A Priori Generalization Analysis of the Deep Ritz Method for Solving High Dimensional Elliptic Partial Differential Equations	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.	inproceedings	Proceedings of Machine Learning Research	PMLR	2640-3498	lu21a	0	A Priori Generalization Analysis of the Deep Ritz Method for Solving High Dimensional Elliptic Partial Differential Equations	3196	3241	3196-3241	3196	false	Lu, Yulong and Lu, Jianfeng and Wang, Min	given family Yulong Lu given family Jianfeng Lu given family Min Wang	2021-07-21		Proceedings of Thirty Fourth Conference on Learning Theory	134	inproceedings	date-parts 2021 7 21	http://proceedings.mlr.press/v134/lu21a/lu21a.pdf