Statistical Error Bounds in the Deep Ritz Method for the p- Laplacian

This thesis investigates the statistical error of the Deep Ritz Method (DRM) when applied to the p-Laplace partial differential equation (PDE), a nonlinear generalization of the classical Laplace equation. The DRM, a variational approach to solving PDEs leveraging deep neural networks, has shown promise in solving high dimensional and nonlinear PDEs, yet its theoretical error characteristics remain insufficiently understood. We focus on the statistical error introduced by the empirical approximation of the energy functional used during training. In this work, we derive a theoretical upper bound on this for the case p ≥ 2, quantifying its dependence on sample size, network capacity, and the properties of the underlying PDE. This analysis provides insight into the reliability and convergence behavior of deep learning-based variational solvers in nonlinear PDE settings.