{
  "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).",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2605.15285",
  "author_total": 1,
  "authors": [
    {
      "name": "Filippo de Feo"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/aghvhjccgr59j.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2605.15285",
  "fulltext_url": null,
  "id": "aghvhjccgr59j",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2605.15285",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.AI"
    ],
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      "arxiv",
      "cs.LG"
    ],
    [
      "arxiv",
      "cs.NA"
    ],
    [
      "arxiv",
      "math.FA"
    ],
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      "arxiv",
      "math.NA"
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    [
      "arxiv",
      "math.OC"
    ],
    [
      "openalex",
      "3109|Model Reduction and Neural Networks"
    ]
  ],
  "openalex": "W7161540349",
  "pdf_url": "https://arxiv.org/pdf/2605.15285",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/aghvhjccgr59j.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:07Z",
      "native_id": "2605.15285",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2605.15285"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Universal Approximation of Nonlinear Operators and Their Derivatives",
  "title_text": "Universal Approximation of Nonlinear Operators and Their Derivatives",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:07Z",
  "url": "https://research.draconitic.com/paper/aghvhjccgr59j",
  "venue": "arXiv",
  "versions": []
}