Error Analysis of Deep Ritz Methods for Elliptic Equations
Numerical Analysis
2021-09-07 v2 Numerical Analysis
Abstract
Using deep neural networks to solve PDEs has attracted a lot of attentions recently. However, why the deep learning method works is falling far behind its empirical success. In this paper, we provide a rigorous numerical analysis on deep Ritz method (DRM) \cite{Weinan2017The} for second order elliptic equations with Drichilet, Neumann and Robin boundary condition, respectively. We establish the first nonasymptotic convergence rate in norm for DRM using deep networks with smooth activation functions including logistic and hyperbolic tangent functions. Our results show how to set the hyper-parameter of depth and width to achieve the desired convergence rate in terms of number of training samples.
Cite
@article{arxiv.2107.14478,
title = {Error Analysis of Deep Ritz Methods for Elliptic Equations},
author = {Yuling Jiao and Yanming Lai and Yisu Lo and Yang Wang and Yunfei Yang},
journal= {arXiv preprint arXiv:2107.14478},
year = {2021}
}