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Regularity of Second-Order Elliptic PDEs in Spectral Barron Spaces

Ziang Chen, Liqiang Huang, Mengxuan Yang, Shengxuan Zhou

TL;DR

A regularity theorem is established for second-order elliptic PDEs on $\mathbb{R}^{d}$ in spectral Barron spaces under mild ellipticity and smallness assumptions, and 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.

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.

Regularity of Second-Order Elliptic PDEs in Spectral Barron Spaces

TL;DR

A regularity theorem is established for second-order elliptic PDEs on in spectral Barron spaces under mild ellipticity and smallness assumptions, and 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.

Abstract

We establish a regularity theorem for second-order elliptic PDEs on 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.
Paper Structure (7 sections, 10 theorems, 103 equations)

This paper contains 7 sections, 10 theorems, 103 equations.

Table of Contents

  1. Introduction
  2. Main Results and Outline of the Proof
  3. Proof of the Regularity Theorem
  4. Basic Properties of Spectral Barron Spaces
  5. Proof of Theorem \ref{['thm:regularitythm']} and Corollary \ref{['cor:dimind']}
  6. An Alternative Proof of Lemma \ref{['lem:mainlem']}
  7. Examples of Dimension-Independent Approximation

Key Result

Theorem 1

Suppose that the coefficients of eq:PDE satisfy assumptions (A1)--(A3) in Assumption assum. Then for any source term $f \in \mathcal{B}^{s}$, the unique solution $u^*$ to eq:PDE gains two additional orders of Barron regularity, and satisfies the estimate where the constant $C$ depends on the Barron norms of the coefficients, as well as on the dimension $d$ and the order $s$. Additionally, if Assu

Theorems & Definitions (19)

  • Theorem : Informal version
  • Example 1.1
  • Definition 2.1
  • Remark 2.3
  • Theorem 2.4
  • Corollary 2.5
  • Theorem 2.6: FL25
  • Corollary 2.7
  • Proposition 3.1
  • proof
  • ...and 9 more