Year: 2021
Author: Jingrun Chen, Panchi Li, Rui Du, Liyao Lyu, Panchi Li, Liyao Lyu
Numerical Mathematics: Theory, Methods and Applications, Vol. 14 (2021), Iss. 2 : pp. 377–404
Abstract
Solving partial differential equations in high dimensions by deep neural networks has brought significant attentions in recent years. In many scenarios, the loss function is defined as an integral over a high-dimensional domain. Monte-Carlo method, together with a deep neural network, is used to overcome the curse of dimensionality, while classical methods fail. Often, a neural network outperforms classical numerical methods in terms of both accuracy and efficiency. In this paper, we propose to use quasi-Monte Carlo sampling, instead of Monte-Carlo method to approximate the loss function. To demonstrate the idea, we conduct numerical experiments in the framework of deep Ritz method. For the same accuracy requirement, it is observed that quasi-Monte Carlo sampling reduces the size of training data set by more than two orders of magnitude compared to that of Monte-Carlo method. Under some assumptions, we can prove that quasi-Monte Carlo sampling together with the deep neural network generates a convergent series with rate proportional to the approximation accuracy of quasi-Monte Carlo method for numerical integration. Numerically the fitted convergence rate is a bit smaller, but the proposed approach always outperforms Monte Carlo method.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/nmtma.OA-2020-0062
Numerical Mathematics: Theory, Methods and Applications, Vol. 14 (2021), Iss. 2 : pp. 377–404
Published online: 2021-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 28
Keywords: Quasi-Monte Carlo sampling deep Ritz method loss function convergence analysis.
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