摘要
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-boundarymap 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.
基金
supported in part by National Science Foundation via grant 1934612
a DOE Subcontract 8F-30039 from Argonne National Laboratory
an AFOSR subcontract UTA20-001224 from UT-Austin.The work of SC
supported in part by NSF-DMS-1750488 and ONR-N00014-21-1-2140.