  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.
