Learning local Dirichlet-to-Neumann maps of nonlinear elliptic PDEs with rough coefficients

Partial Differential Equations (PDEs) involving high-contrast and oscillating coefficients are common in scientific and industrial applications. Numerically approximating these PDEs is a challenging task, which can be addressed, for example, through Multi-scale Finite Element analysis. For linear problems, the Multi-scale Finite Element Method (MsFEM) is well-established, and several viable extensions to nonlinear PDEs have been proposed. However, some aspects of the method seem to be inherently linear. In particular, traditional MsFEM heavily relies on reusing computations. For example, the multi-scale basis and the stiffness matrix can be calculated once and used for multiple right-hand sides or as part of a multi-level iterative algorithm. In this contribution, we present preliminary numerical results of combining MsFEM with Machine Learning tools. The extension of MsFEM to nonlinear problems is achieved by learning local nonlinear approximate Dirichlet-to-Neumann maps. The resulting learning-based multi-scale method is tested on a set of model nonlinear PDEs involving the p–Laplacian and degenerate nonlinear diffusion.