发表机构
School of Mathematical Sciences and LPMC, Nankai University（南开大学数学科学学院和LPMC）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对巴拿赫空间紧子集上的算子学习，刻画了其一致逼近性质，推导了恢复维度、网络参数数量及误差收敛速率的相关界，为算子学习的理论分析提供了关键支撑。
AI 中文摘要
我们研究巴拿赫空间紧子集上算子学习的一致逼近问题。我们根据紧集复杂度、目标规范以及恢复稳定性，对恢复维度和网络参数数量进行了界定。对于具有可分定义域的有界线性算子，我们通过具有公共消失规范的坐标分解，刻画了其规范可逼近性，且可实现任意小的加性误差；这些坐标分解的部分和构成了一类固定的非线性编码器-解码器对。我们证明，具有公共消失规范且秩界仅依赖于模型维度的非线性恢复，等价于一致逼近性质（UAP）。我们还得到了具有相同秩界的线性恢复估计，以及针对非线性目标线性因子的算子版本。对于取值于p型巴拿赫空间Y（p∈(1,2]）且具有博赫纳可积密度的Barron算子，在输入紧集上的概率测度λ下，采样可得到期望L^p(λ;Y)误差的收敛速率为N^{-(1-1/p)}。对于固定的有限维因子和一类赫尔德映射，我们在固定足够大的深度下得到了匹配的上下参数界。经典采样和截断示例提供了显式恢复映射，一个非线性积分算子则说明了参数界的情况。
英文摘要
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 $p\in(1,2]$ 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.
