@Article{NMTMA-14-2,
author = {Jingrun, Chen and Panchi, Li and Rui, Du and Lyu, Liyao and Panchi, Li 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 = {<p style="text-align: justify;">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.</p>},
issn = {2079-7338},
doi = {https://doi.org/10.4208/nmtma.OA-2020-0062},
url = {https://global-sci.com/article/90303/quasi-monte-carlo-sampling-for-solving-partial-differential-equations-by-deep-neural-networks}
}