mlsanityData to 14 Oct 2026

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

arXiv 2607.25100 ↗First posted 27 Jul 2026math.AP cs.NA math.NAcited by 1

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Abstract (arXiv)

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 H2. 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 ~ varepsilon by Barron functions of low norm ~ |logvarepsilon| 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.

Key facts. Cited by 1 paper in the map (data through 14 Oct 2026). Builds on 28 of its 57 references that are in the map. First posted on arXiv 27 Jul 2026.

Builds on

28 of 57 references.

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Cited by year (2026 to 14 Oct)

Citing papers per year: 2026: 1.

1 paper.

  1. 2026Fractional Parabolic Partial Differential Equations in Anisotropic Spectral Barron Spaces: Regularity and Neural Approximationcited by 0