Convergence analysis of a quasi-Monte Carlo-based deep learning algorithm for solving partial differential equations
This work addresses the accuracy and efficiency of deep learning algorithms for PDEs, specifically for Neumann problems in Poisson and Schrödinger equations, representing an incremental improvement by integrating QMC methods.
The authors tackled the problem of solving partial differential equations (PDEs) using deep learning by proposing a quasi-Monte Carlo (QMC)-based Deep Ritz Method, which achieved faster convergence and smaller gradient estimator variances compared to standard methods, as shown in numerical experiments.
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 Schrödinger 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.