A Finite Element based Deep Learning solver for parametric PDEs

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 mimic connectivity graph when applying mesh refinements. select and discuss several losses employing preconditioners different norms enhance convergence. For simplicity, we implement resulting Deep-FEM one spatial domain (1D), although its extension 2D 3D problems is straightforward. Extensive numerical experiments show general good approximations for both symmetric positive definite (SPD) indefinite non-parametric problems. However, some cases, lack of convexity prevents us from obtaining high-accuracy solutions.