@Article{JCM-42-2,
author = {Shi, Chen and Zhiyan, Ding and Qin, Li and Wright, Stephen, J.},
title = {A Reduced Order Schwarz Method for Nonlinear Multiscale Elliptic Equations Based on Two-Layer Neural Networks},
journal = {Journal of Computational Mathematics},
year = {2024},
volume = {42},
number = {2},
pages = {570--596},
abstract = {<p style="text-align: justify;">Neural networks are powerful tools for approximating high dimensional data that have
been used in many contexts, including solution of partial differential equations (PDEs).
We describe a solver for multiscale fully nonlinear elliptic equations that makes use of
domain decomposition, an accelerated Schwarz framework, and two-layer neural networks
to approximate the boundary-to-boundary map for the subdomains, which is the key step in
the Schwarz procedure. Conventionally, the boundary-to-boundary map requires solution
of boundary-value elliptic problems on each subdomain. By leveraging the compressibility
of multiscale problems, our approach trains the neural network offline to serve as a surrogate
for the usual implementation of the boundary-to-boundary map. Our method is applied
to a multiscale semilinear elliptic equation and a multiscale $p$-Laplace equation. In both
cases we demonstrate significant improvement in efficiency as well as good accuracy and
generalization performance.</p>},
issn = {1991-7139},
doi = {https://doi.org/10.4208/jcm.2204-m2021-0311},
url = {https://global-sci.com/article/84099/a-reduced-order-schwarz-method-for-nonlinear-multiscale-elliptic-equations-based-on-two-layer-neural-networks}
}