Operator Barron Regularity for Elliptic Equations and Application to Neural Operator Approximation

Shuhao Cao; Ruchi Guo; Daniel Zhengyu Huang; Na Liu

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arXiv · 2610.37101

Operator Barron Regularity for Elliptic Equations and Application to Neural Operator Approximation

Shuhao Cao·Ruchi Guo·Daniel Zhengyu Huang·Na Liu

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Abstract

We develop a spectral Barron regularity framework for nonlinear operators between infinite-dimensional function spaces. Using Hilbert-valued Fourier representations of finite-dimensional input truncations, we establish bounds that are uniform in the input truncation dimension and prove arbitrary-order Barron regularity for a class of elliptic solution operators. Second, we use this operator Barron regularity to derive approximation error bounds for an FNO-type neural operator, with explicit dependence on the input truncation dimension, output reconstruction dimension, and network width. For a fixed input truncation dimension, optimizing the neural approximation bound over the Barron regularity order yields stretched-exponential decay of the approximation error in the network width. We also show some numerical experiments to support the predicted rate.

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Shuhao Cao, Ruchi Guo, Daniel Zhengyu Huang, Na Liu. 2026-10-27. Operator Barron Regularity for Elliptic Equations and Application to Neural Operator Approximation. https://arxiv.org/abs/2610.37101

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