A Reduced Order Schwarz Method for Nonlinear Multiscale Elliptic Equations Based on Two-Layer Neural Networks (Journal of Computational Mathematics, 2024)
Neural networks are powerful tools for approximating high dimensional data that havebeen used in many contexts, including solution of partial differential equations (PDEs).We describe a solver for multiscale fully nonlinear elliptic equations that makes use ofdomain decomposition, an accelerated Schwarz framework, and two-layer neural networksto approximate the boundary-to-boundary map for the subdomains, which is the key step inthe Schwarz procedure. Conventionally, the boundary-to-boundary map requires solutionof boundary-value elliptic problems on each subdomain. By leveraging the compressibilityof multiscale problems, our approach trains the neural network offline to serve as a surrogatefor the usual implementation of the boundary-to-boundary map. Our method is appliedto a multiscale semilinear elliptic equation and a multiscale $p$-Laplace equation. In bothcases we demonstrate significant improvement in efficiency as well as good accuracy andgeneralization performance.