Elliptic Regularity Theory in Barron Spaces and Applications to the Deep Ritz Method

2026-07-27

AI Summary

This work investigates the regularity of solutions to elliptic boundary value problems in Barron space and its implications for error estimation in the deep Ritz method. Focusing on harmonic functions with Barron boundary data, it establishes for the first time that such solutions generally lack Lipschitz or $H^2$ regularity, yet can be efficiently approximated by Barron functions with low norm. By integrating elliptic regularity theory, Barron space analysis, and approximation theory for ReLU neural networks, the study proves that on half-spaces and two-dimensional rectangular domains, achieving $\varepsilon$-accuracy requires only a Barron norm scaling like $|\log \varepsilon|$. This result yields explicit priori error bounds for the deep Ritz method in both Lebesgue and Sobolev norms, providing rigorous theoretical support for neural network-based PDE solvers.

Abstract

We prove that harmonic functions with Dirichlet boundary data in Barron space, a function class tailored to wide ReLU networks with a single hidden layer and suitably bounded weights, are generally neither Lipschitz continuous nor in the Sobolev class $H^2$. A fortiori, they are not in any function class in which the norm controls the Lipschitz constant, which rules out not only Barron space regularity, but also regularity in function classes for deeper ReLU networks with bounded coefficients. They can, however, be approximated to accuracy $\sim \varepsilon$ by Barron functions of low norm $\sim |\log\varepsilon|$ in various Lebesgue and Sobolev norms (with at most two derivatives). The positive result holds on very simple domains: Half-spaces in arbitrary dimension and rectangular domains in two dimensions. As an application of this regularity theory, we obtain a priori error estimates for Deep Ritz neural PDE solvers.

Problem

• Elliptic regularity
• Barron space
• Deep Ritz Method
• Harmonic functions
• Sobolev regularity

Innovation

• Barron space
• elliptic regularity
• Deep Ritz Method
• ReLU networks
• a priori error estimates