A Finite Element based Deep Learning solver for parametric PDEs

Abstract

We introduce a dynamic Deep Learning (DL) architecture based on the Finite Element Method (FEM) to solve linear parametric Partial Differential Equations (PDEs). The connections between neurons in the architecture mimic the Finite Element connectivity graph when applying mesh refinements. We select and discuss several losses employing preconditioners and different norms to enhance convergence. For simplicity, we implement the resulting Deep-FEM in one spatial domain (1D), although its extension to 2D and 3D problems is straightforward. Extensive numerical experiments show in general good approximations for both symmetric positive definite (SPD) and indefinite problems in parametric and non-parametric problems. However, in some cases, lack of convexity prevents us from obtaining high-accuracy solutions.