A Reduced Order Schwarz Method for Nonlinear Multiscale Elliptic Equations Based on Two-Layer Neural Networks

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 multiscale fully nonlinear elliptic makes use domain decomposition, an accelerated Schwarz framework, and two-layer neural to approximate the boundary-to-boundary map subdomains, which is key step procedure. Conventionally, requires boundary-value problems on each subdomain. By leveraging compressibility problems, our approach trains network offline serve as surrogate usual implementation map. Our method applied semilinear equation $p$-Laplace equation. In both cases we demonstrate significant improvement efficiency well good accuracy generalization performance.