Regularity of Second-Order Elliptic PDEs in Spectral Barron Spaces

TL;DR

This paper proves a regularity theorem for second-order elliptic PDEs in spectral Barron spaces, showing solutions can be approximated by neural networks with width independent of dimension under certain conditions.

Contribution

It introduces a regularity result in spectral Barron spaces and links PDE solutions to neural network approximation with dimension-independent width.

Findings

Solutions gain two orders of Barron regularity

PDE solutions can be approximated by shallow neural networks

Neural network width is independent of spatial dimension

Abstract

We establish a regularity theorem for second-order elliptic PDEs on $\mathbb{R}^{d}$ in spectral Barron spaces. Under mild ellipticity and smallness assumptions, the solution gains two additional orders of Barron regularity. As a corollary, we identify a class of PDEs whose solutions can be approximated by two-layer neural networks with cosine activation functions, where the width of the neural network is independent of the spatial dimension.