Gradient-Regularized Deep Ritz Networks for Solving Elliptic PDEs: Application to the Poisson Equation

Abstract

In this work we propose a gradient-regularized Deep Ritz Network (GR-DRN) for solving elliptic partial differential equations, which we shall concretely apply to the Poisson equation on the unit square domain. We then propose a method that generalizes the classical Deep Ritz formulation by adding a gradient-penalty term to achieve smoother solution and lower non-physical oscillations during training. The model is evaluated against three benchmark datasets that are analytically generated, representing a smooth solution, a case with high-frequency oscillatory solution, and a variable-coefficient setting that reflects heterogeneous diffusion. Our numerical results show that GR-DRN consistently obtains lower L² and H¹ errors than the standard Deep Ritz method, especially on the difficult problems with high-frequency or variable-coefficient. It also produces more accurate gradient fields and stable convergence shift. Overall this shows that gradient regularization is an easy but effective improvement to neural variational solvers for elliptic PDEs.