TY - JOUR
T1 - A Reduced Order Schwarz Method for Nonlinear Multiscale Elliptic Equations Based on Two-Layer Neural Networks
AU - Chen , Shi
AU - Ding , Zhiyan
AU - Li , Qin
AU - Wright , Stephen J.
JO - Journal of Computational Mathematics
VL - 2
SP - 570
EP - 596
PY - 2024
DA - 2024/01
SN - 42
DO - http://doi.org/10.4208/jcm.2204-m2021-0311
UR - https://global-sci.org/intro/article_detail/jcm/22892.html
KW - Nonlinear homogenization, Multiscale elliptic problem, Neural networks, Domain decomposition.
AB - <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>