  Deep learning methods have achieved great success in solving partial
differential equations (PDEs), where the loss is often defined as an integral.
The accuracy and efficiency of these algorithms depend greatly on the
quadrature method. We propose to apply quasi-Monte Carlo (QMC) methods to the
Deep Ritz Method (DRM) for solving the Neumann problems for the Poisson
equation and the static Schr\"{o}dinger equation. For error estimation, we
decompose the error of using the deep learning algorithm to solve PDEs into the
generalization error, the approximation error and the training error. We
establish the upper bounds and prove that QMC-based DRM achieves an
asymptotically smaller error bound than DRM. Numerical experiments show that
the proposed method converges faster in all cases and the variances of the
gradient estimators of randomized QMC-based DRM are much smaller than those of
DRM, which illustrates the superiority of QMC in deep learning over MC.
