Uniform Approximation for Operator Learning on Compact Sets
Abstract
We study uniform approximation for operator learning on compact subsets of Banach spaces. We bound the recovery dimensions and the number of network parameters in terms of compact-set complexity, the target gauge, and recovery stability. For bounded linear operators with a separable domain, we characterize gauge approximability by coordinate decompositions with a common vanishing gauge and arbitrarily small additive errors. Their partial sums give a fixed family of nonlinear encoder--decoder pairs. We prove that nonlinear recovery with a common vanishing gauge and a rank bound depending only on the model dimension is equivalent to the uniform approximation property (UAP). We also obtain linear recovery estimates with the same rank bound and an operator version for the linear factors of nonlinear targets. For Barron operators taking values in a Banach space Y of type p∈(1,2] and admitting a Bochner-integrable density, sampling gives the rate N−(1−1/p) for the expected Lp(λ;Y) error, where λ is a probability measure on the input compact set. For fixed finite-dimensional factors and a class of Hölder maps, we obtain matching upper and lower parameter bounds at fixed sufficiently large depth. Classical sampling and truncation examples provide explicit recovery maps, and a nonlinear integral operator illustrates the parameter bounds.
Related
- Weak Approximation Properties Induced by Unconditionally p-Compact Sets
- Oka-Weil approximation theorem for polynomials with exponents in compact convex sets in ℝn
- Physics-informed random neural architectures on compact boundaryless multiply connected manifolds for random-operator models in machine learning and uncertainty quantification
- Physics-informed random neural architectures on compact boundaryless multiply connected manifolds for random-operator models in machine learning and uncertainty quantification
- Uniform Continuity of Boundary Crossing for Convex Sets
BibTeX
@misc{liu2026uniform,
title = {{Uniform Approximation for Operator Learning on Compact Sets}},
author = {Rui Liu and Jie Shen},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.11709},
eprint = {2610.11709},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.11709},
}
RIS
TY - UNPB
TI - Uniform Approximation for Operator Learning on Compact Sets
AU - Rui Liu
AU - Jie Shen
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.11709
UR - https://arxiv.org/abs/2610.11709
AB - We study uniform approximation for operator learning on compact subsets of Banach spaces. We bound the recovery dimensions and the number of network parameters in terms of compact-set complexity, the target gauge, and recovery stability. For bounded linear operators with a separable domain, we characterize gauge approximability by coordinate decompositions with a common vanishing gauge and arbitrarily small additive errors. Their partial sums give a fixed family of nonlinear encoder--decoder pairs. We prove that nonlinear recovery with a common vanishing gauge and a rank bound depending only on the model dimension is equivalent to the uniform approximation property (UAP). We also obtain linear recovery estimates with the same rank bound and an operator version for the linear factors of nonlinear targets. For Barron operators taking values in a Banach space Y of type p∈(1,2] and admitting a Bochner-integrable density, sampling gives the rate N^−(1−1/p) for the expected L^p(λ;Y) error, where λ is a probability measure on the input compact set. For fixed finite-dimensional factors and a class of Hölder maps, we obtain matching upper and lower parameter bounds at fixed sufficiently large depth. Classical sampling and truncation examples provide explicit recovery maps, and a nonlinear integral operator illustrates the parameter bounds.
ER -