Neural Network Approximation of Solutions to Fractional Parabolic Partial Differential Equations

2026-07-30

AI Summary

This work addresses the lack of dimension-independent regularity theory and neural network approximation guarantees for high-dimensional fractional parabolic partial differential equations featuring lower-order drift and potential terms. The authors introduce anisotropic spectral Barron spaces to characterize the regularity of solutions in the space-time frequency domain, and establish a maximal regularity theory via dimension-free product estimates combined with a continuity method. A novel use of Vandermonde matrices enables a finite-time global extension of the fractional heat semigroup. Building on this framework, the study provides the first proof that two-layer neural networks with non-periodic activation functions achieve a dimension-independent approximation rate of $n^{-1/2}$ in mixed Sobolev norms, while also demonstrating that time-uniform spectral Barron regularity generally fails to hold.

Abstract

We establish a dimension-efficient neural network approximation theory for solutions to fractional parabolic equations with lower-order drift and potential terms. By introducing anisotropic spectral Barron spaces, which measure temporal and spatial regularity separately in frequency space, we first develop a dimension-independent maximal regularity theory for these equations, using dimension-independent multiplication estimates and the method of continuity to incorporate the lower-order terms. A key technical novelty is the application of the Vandermonde matrix to the global-in-time extension of the finite-time fractional heat semigroup with sufficient regularity at the initial time, thereby enabling analysis of the forward-in-time evolution via the global space-time Fourier structure of anisotropic Barron norms. We also show that a corresponding uniform-in-time estimate of the spectral Barron regularity generally fails. Finally, we derive $n^{-1/2}$ two-layer approximation bounds in mixed Sobolev norms for non-constant periodic activations and, under additional anisotropic Barron regularity, for non-periodic activations satisfying a polynomial-decay condition.

Problem

• fractional parabolic equations
• neural network approximation
• dimension efficiency
• anisotropic regularity
• spectral Barron spaces

Innovation

• anisotropic spectral Barron spaces
• fractional parabolic equations
• dimension-independent regularity
• Vandermonde matrix extension
• neural network approximation