  Solving partial differential equations in high dimensions by deep neural
network 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 the deep neural network, is used to overcome
the curse of dimensionality, while classical methods fail. Often, a deep 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
proposed by Weinan E and Bing Yu. 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 MC method. Under some
assumptions, we 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. It is worth mentioning that the
convergence analysis is generic whenever a loss function is approximated by the
quasi-Monte Carlo method, although observations here are based on deep Ritz
method.
