arXiv：cs.LG· Filippo de Feo·· 2 天前AI 评分40
非线性算子及其导数的通用逼近：DIOL 在一般 Banach 空间上的理论基础
Universal Approximation of Nonlinear Operators and Their Derivatives
AI 导读
研究证明非线性算子及其导数的通用逼近（UA）在 C^k_F 紧开拓扑和 Fréchet–Sobolev 范数下不成立，并通过 C^k_B（Bastiani）紧开拓扑与新型加权 Bastiani–Sobolev 空间恢复了 UA，这是 Hornik 1991 经典结果首次完整推广到无穷维空间与算子学习。
正文
Abstract:We show that Universal Approximation (UA) of nonlinear operators and their derivatives via Operator Learning (OL) architectures fails in ${C^k_F}$ (Fréchet) compact-open topologies and in Fréchet--Sobolev norms (i.e. under operator norms). We solve this obstruction by restoring UA in natural weaker topologies: $C^k_B$ (Bastiani) compact-open topologies and (novel) 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 results in [Hornik, 1991] to infinite-dimensional spaces and OL. Based on our UATs, we formulate Bastiani--Sobolev training in DIOL. These results launch Derivative-Informed Operator Learning (DIOL) (i.e. learning nonlinear operators and their derivatives) on general Banach spaces. We parameterize nonlinear operators via Encoder-Decoder Architectures, classical OL architectures available in general Banach spaces; these include DeepONets, Deep-H-ONets, and PCA-Nets, which our UATs cover.
A key mathematical result is that our new weighted Bastiani--Sobolev spaces generalize classical Gaussian (Malliavin) Sobolev spaces on Banach spaces.
Open frontiers where DIOL and our UATs find applications are: 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).
| Comments: | The presentation of the results has been streamlined and improved |
| Subjects: | Machine Learning (cs.LG); Artificial Intelligence (cs.AI); Functional Analysis (math.FA); Numerical Analysis (math.NA); Optimization and Control (math.OC) |
| Cite as: | arXiv:2605.15285 [cs.LG] |
| (or arXiv:2605.15285v4 [cs.LG] for this version) | |
| https://doi.org/10.48550/arXiv.2605.15285 arXiv-issued DOI via DataCite |
Submission history
From: Filippo De Feo [view email]
[v1]
Thu, 14 May 2026 18:00:58 UTC (77 KB)
[v2]
Mon, 22 Jun 2026 09:14:29 UTC (85 KB)
[v3]
Tue, 1 Sep 2026 17:56:44 UTC (92 KB)
[v4]
Thu, 1 Oct 2026 17:23:26 UTC (85 KB)
来源：arXiv：cs.LG · arxiv.org