TY - JOUR
T1 - Convergence Analysis of a Quasi-Monte Carlo-Based Deep Learning Algorithm for Solving Partial Differential Equations
AU - Fu , Fengjiang
AU - Wang , Xiaoqun
JO - Numerical Mathematics: Theory, Methods and Applications
VL - 3
SP - 668
EP - 700
PY - 2023
DA - 2023/08
SN - 16
DO - http://doi.org/10.4208/nmtma.OA-2022-0166
UR - https://global-sci.org/intro/article_detail/nmtma/21962.html
KW - Deep Ritz method, quasi-Monte Carlo, Poisson equation, static Schrödinger equation, error bound.
AB - <p style="text-align: justify;">Deep learning has 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.</p>