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arXiv · 2111.02280

A reduced order Schwarz method for nonlinear multiscale elliptic equations based on two-layer neural networks

Shi Chen·Zhiyan Ding·Qin Li·Stephen J. Wright
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Abstract

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.

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Shi Chen, Zhiyan Ding, Qin Li, Stephen J. Wright. 2021-11-03. A reduced order Schwarz method for nonlinear multiscale elliptic equations based on two-layer neural networks. https://doi.org/10.4208/jcm.2204-m2021-0311

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