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.

翻译：神经网络是近似高维数据的强大工具,许多情况下都使用了这些数据,包括部分差异方程式(PDEs)的解决方案。我们描述一个使用域分解、加速Schwarz框架和两层神经网络的多级非线性全线性椭圆方程式的解决方案,以近似次域的边界至边界地图,这是Schwarz程序的关键步骤。从边界到边界的地图要求解决每个子域的边界-价值椭圆问题。通过利用多尺度问题的压缩,我们的方法培训线性网络离线,作为通常执行边界至边界地图的替代装置。我们的方法适用于多尺度半线性椭圆方程式和多尺度的美元-Laplace方程式。在这两种情况下,我们显示了效率的显著提高以及良好的准确性和一般性表现。