Volume 14, Issue 2
Quasi-Monte Carlo Sampling for Solving Partial Differential Equations by Deep Neural Networks

Jingrun Chen, Rui Du, Panchi LiLiyao Lyu

Numer. Math. Theor. Meth. Appl., 14 (2021), pp. 377-404.

Published online: 2021-01

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  • 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.

  • Keywords

Quasi-Monte Carlo sampling, deep Ritz method, loss function, convergence analysis.

  • AMS Subject Headings

11K50, 35J20, 65N99

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COPYRIGHT: © Global Science Press

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@Article{NMTMA-14-377, author = {Chen , Jingrun and Du , Rui and Li , Panchi and Lyu , Liyao}, title = {Quasi-Monte Carlo Sampling for Solving Partial Differential Equations by Deep Neural Networks}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2021}, volume = {14}, number = {2}, pages = {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.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.OA-2020-0062}, url = {http://global-sci.org/intro/article_detail/nmtma/18604.html} }
TY - JOUR T1 - Quasi-Monte Carlo Sampling for Solving Partial Differential Equations by Deep Neural Networks AU - Chen , Jingrun AU - Du , Rui AU - Li , Panchi AU - Lyu , Liyao JO - Numerical Mathematics: Theory, Methods and Applications VL - 2 SP - 377 EP - 404 PY - 2021 DA - 2021/01 SN - 14 DO - http://doi.org/10.4208/nmtma.OA-2020-0062 UR - https://global-sci.org/intro/article_detail/nmtma/18604.html KW - Quasi-Monte Carlo sampling, deep Ritz method, loss function, convergence analysis. AB -

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.

Jingrun Chen, Rui Du, Panchi Li & Liyao Lyu. (2021). Quasi-Monte Carlo Sampling for Solving Partial Differential Equations by Deep Neural Networks. Numerical Mathematics: Theory, Methods and Applications. 14 (2). 377-404. doi:10.4208/nmtma.OA-2020-0062
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