Pith Number

pith:NI5DRLRS

pith:2026:NI5DRLRS7XEZLDUE2FQPCE5HFN

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Universal Approximation of Nonlinear Operators and Their Derivatives

Filippo de Feo

Nonlinear k-times differentiable operators between Banach spaces and their derivatives can be universally approximated by operator learning architectures.

arxiv:2605.15285 v1 · 2026-05-14 · cs.LG · cs.AI · cs.NA · math.FA · math.NA · math.OC

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Claims

C1strongest claim

We prove the first UATs of non-linear k-times differentiable operators between Banach spaces and their derivatives, uniformly on compact sets and in weighted Sobolev norms for general finite input measures, via OL architectures. Our results are the first complete generalizations of the corresponding influential classical results in [Hornik, 1991] to infinite-dimensional settings and OL.

C2weakest assumption

The proofs depend on k-times continuous differentiability in the sense of Bastiani together with the construction of novel weighted Sobolev spaces and the choice of natural compact-open topologies; if these specific choices fail to capture the intended class of operators or if the approximation properties of Banach spaces do not hold in the required uniform sense, the claimed UATs would not follow.

C3one line summary

Proves the first universal approximation theorems for k-times differentiable nonlinear operators between Banach spaces and their derivatives uniformly on compact sets in weighted Sobolev norms via encoder-decoder operator learning architectures.

References

132 extracted · 132 resolved · 3 Pith anchors

[1] Robert A Adams and John JF Fournier.Sobolev spaces, volume 140. Elsevier, 2003 2003

[2] On the closability of differential operators.Journal of Functional Analysis, 289(7):111029, 2025 2025

[3] Fernando Albiac and Nigel J Kalton.Topics in Banach space theory. Springer, 2006 2006

[4] Tucker Tensor Train Taylor Series.arXiv preprint arXiv:2603.21141, 2026 2026

[5] A PINN approach for the online identification and control of unknown PDEs.Journal of Optimization Theory and Applications, 206(1):8, 2025 2025

Formal links

2 machine-checked theorem links

Receipt and verification

First computed	2026-05-20T00:00:50.656417Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

6a3a38ae32fdc9958e84d160f113a72b68e9418043bfc58c533fd60e60758bf3

Aliases

arxiv: 2605.15285 · arxiv_version: 2605.15285v1 · doi: 10.48550/arxiv.2605.15285 · pith_short_12: NI5DRLRS7XEZ · pith_short_16: NI5DRLRS7XEZLDUE · pith_short_8: NI5DRLRS

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/NI5DRLRS7XEZLDUE2FQPCE5HFN \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: 6a3a38ae32fdc9958e84d160f113a72b68e9418043bfc58c533fd60e60758bf3

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "e877eb111a1ad47ab660c3667a7722521db2ea754596af55305fc2f0288669a2",
    "cross_cats_sorted": [
      "cs.AI",
      "cs.NA",
      "math.FA",
      "math.NA",
      "math.OC"
    ],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "cs.LG",
    "submitted_at": "2026-05-14T18:00:58Z",
    "title_canon_sha256": "3d09e69b54d193e638a2b92a97433eeb092be5bf31ebf4b93b10669f950cda8c"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.15285",
    "kind": "arxiv",
    "version": 1
  }
}