证明了非线性算子及其导数的通用逼近定理，为算子学习提供理论基础。
Universal Approximation of Nonlinear Operators and Their Derivatives
- 基于巴斯坦尼微分与加权索伯列夫空间，构建算子逼近框架
- 首次在巴拿赫空间上实现高阶导数的统一逼近，覆盖DeepONet等模型
- 适用于高精度算子学习、偏微分方程最优控制等前沿问题
在算子学习（OL）和非线性泛函分析中，建立非线性算子及其导数的通用逼近定理（UAT）是一个基础性开放问题。本文首次在一般有限输入测度下，于紧集上及加权巴斯坦尼-索伯列夫空间中，证明了k次可微非线性算子及其导数的UAT。在全巴拿赫空间广义性下，这是对[霍尼克, 1991]经典结果向无限维空间和算子学习的首次完整推广，并由此提出导数感知算子学习（DIOL），即学习算子及其导数。基于此，我们提出了巴斯坦尼-索伯列夫训练方案。该理论可应用于高阶精度算子学习、巴拿赫空间中的快速约束优化（如偏微分方程最优控制、反问题）的“先学后优”策略，以及无限维偏微分方程（如来自无限维最优控制的哈密顿-雅可比-贝尔曼方程）的数值方法，包括随机偏微分方程、路径依赖系统、部分观测系统、平均场控制等。我们采用编码器-解码器架构参数化算子，涵盖DeepONets、Deep-H-ONets和PCA-Nets等经典算子学习模型。理论核心基于：(i) 巴拿赫空间的逼近性质；(ii) 连续巴斯坦尼可微性（弱于连续弗雷歇可微性）；(iii) $C^k_B$（巴斯坦尼）紧开拓扑；事实上，在弗雷歇$C^k$紧开拓扑（由算子范数诱导）下通用逼近不成立；(iv) 加权巴斯坦尼-索伯列夫空间的构造，推广了经典巴拿赫空间上的高斯索伯列夫空间。
原文摘要 · Abstract (English)
Establishing Universal Approximation Theorems (UATs) for nonlinear operators and their derivatives is a foundational open problem in Operator Learning (OL) and raises delicate questions in Nonlinear Functional Analysis. We prove the first UATs for $k$-times differentiable nonlinear operators and their derivatives via OL architectures, uniformly on compact sets and in weighted Bastiani--Sobolev spaces for general finite input measures. In full Banach-space generality, these are the first complete generalizations of the corresponding influential classical UATs in [Hornik, 1991] to infinite-dimensional spaces and OL, {and launch Derivative-Informed Operator Learning (DIOL) (i.e. learning nonlinear operators and their derivatives)} on general Banach spaces. Based on our UATs, we formulate Bastiani--Sobolev training in DIOL. We present open frontiers where DIOL and our UATs find applications: high-order accuracy in OL; fast constrained optimization in Banach spaces (e.g. optimal control of PDEs, inverse problems) via Learn-Then-Optimize; numerical methods for infinite-dimensional PDEs (e.g. HJB PDEs on Banach spaces from infinite-dimensional optimal control via Optimize-Then-Learn, such as optimal control of PDEs, SPDEs, path-dependent systems, partially observed systems, mean-field control). We parameterize nonlinear operators via Encoder-Decoder Architectures, classical OL architectures. These include DeepONets, Deep-H-ONets, and PCA-Nets, which our UATs cover. Our UATs are based on (i) Approximation Properties of Banach spaces; (ii) continuous Bastiani differentiability (weaker than continuous Fréchet differentiability); (iii) $C^k_B$ (Bastiani) compact-open topologies; indeed, UA in $C^k$ (Fréchet) compact-open topologies (induced by operator norms) fails; (iv) construction of weighted Bastiani--Sobolev spaces, generalizing classical Gaussian Sobolev spaces on Banach spaces.
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