Operator Barron Regularity for Elliptic Equations and Application to Neural Operator Approximation
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.
Related
- Local and global C1,β-regularity for uniformly elliptic quasilinear equations of p-Laplace and Orlicz-Laplace type
- Regularity Estimates for Quasilinear Elliptic Equations with Measure Data and Matrix-Valued Weight
- Potential estimates and improved Sobolev regularity for nonlocal fully nonlinear elliptic equations
- Boundary Regularity for Fully Nonlinear Singular Elliptic Equations
- Almost Lipschitz regularity for solutions of elliptic equations with discontinuous coefficients
BibTeX
@misc{cao2026operator,
title = {{Operator Barron Regularity for Elliptic Equations and Application to Neural Operator Approximation}},
author = {Shuhao Cao and Ruchi Guo and Daniel Zhengyu Huang and Na Liu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.37101},
eprint = {2610.37101},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.37101},
}
RIS
TY - UNPB
TI - Operator Barron Regularity for Elliptic Equations and Application to Neural Operator Approximation
AU - Shuhao Cao
AU - Ruchi Guo
AU - Daniel Zhengyu Huang
AU - Na Liu
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2610.37101
UR - https://arxiv.org/abs/2610.37101
AB - 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.
ER -