Neural Network Approximation of Solutions to Fractional Parabolic Partial Differential Equations

• URL: http://arxiv.org/abs/2607.27781v1
• Date: Thu, 30 Jul 2026 07:14:01 GMT
• Title: Neural Network Approximation of Solutions to Fractional Parabolic Partial Differential Equations
• Abstract summary: We introduce anisotropic spectral Barron spaces, which measure temporal and spatial regularity separately in frequency space.<n>We first develop a dimension-independent maximal regularity theory for these equations, using dimension-independent multiplication estimates.<n>A key technical novelty is the application of the Vandermonde matrix to the global-in-time extension of the finite-time fractional heat semigroup.
• Score: 1.7118940953649353
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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