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Fractional Laplace Neural Operators: Exact Architectures, an Expressivity Frontier at Criticality, and Certified Stability for Memory-Driven Network Dynamics — arXiv:2610.00515

Source: arXiv (en)
Original: https://arxiv.org/abs/2610.00515
License (as archived): CC0-1.0
Archived: 2026-10-06T15:32:42.255Z
SHA-256 of the archived text: b65d83a0f2f34a110614994068dbbe53c901ec1d25560ea86d72b629c794d72a
Stable URL of this page: https://metraipolis.com/documento/dc_cb15c8317c4d

How to cite

Fractional Laplace Neural Operators: Exact Architectures, an Expressivity Frontier at Criticality, and Certified Stability for Memory-Driven Network Dynamics — arXiv:2610.00515 — arXiv (en), https://arxiv.org/abs/2610.00515 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_cb15c8317c4d (Metraipolis archive dc_cb15c8317c4d, sha256 b65d83a0f2f34a110614994068dbbe53c901ec1d25560ea86d72b629c794d72a)

BibTeX

@misc{dccb15c8317c4d,
  title = {Fractional Laplace Neural Operators: Exact Architectures, an Expressivity Frontier at Criticality, and Certified Stability for Memory-Driven Network Dynamics — arXiv:2610.00515},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2610.00515},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_cb15c8317c4d; sha256 b65d83a0f2f34a110614994068dbbe53c901ec1d25560ea86d72b629c794d72a},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2610.00515. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

Neural operators learn maps between function spaces, while hereditary network dynamics are described by Volterra resolvents with non-rational Laplace symbols. We introduce a fractional Laplace neural operator (fLNO) that embeds this structure in the learned map. For commuting excitation--Laplacian pairs, one block graph-spectral layer represents the full linear Volterra solution operator exactly. We establish an expressivity frontier for finite rational realizations: they approximate fractional memory geometrically on compact frequency windows, but cannot reproduce the non-integer critical asymptotics generated by a branch point, and on the half-line the best rational rate is root-exponential. The same theory yields trainable parametrizations that enforce a prescribed stability margin by construction, and a graphon-transfer theorem separates genuine operator consistency from parameter sharing. In a common-data benchmark, positive rational operators can match or exceed fLNO accuracy on finite horizons, whereas in controlled near-critical experiments fLNO recovers the branching coordinate more faithfully with far fewer parameters; unconstrained rational fits can cross the stability boundary, while certified parametrizations cannot. A four-parameter spectral law transfers without retraining from graphs of size 48 to 192 with 0.51--0.62% relative error. Applications to Chilean aftershock sequences and to renewal models for Chile and 21 Italian regions illustrate structured inference with explicit uncertainty. The contribution is an operator-learning architecture in which exact memory structure, physical coordinates and stability guarantees coexist with competitive accuracy.

Tema degli esploratori: statistica e scienza dei dati (retro-catalogo arXiv: annunci del 2026-10-02 in stat.ML).
Categorie arXiv: Machine Learning (stat.ML) ; Machine Learning (cs.LG).
Pagina del lavoro: https://arxiv.org/abs/2610.00515 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-10-06T15:12:55.864Z dalla pagina https://arxiv.org/catchup/stat.ML/2026-10-02?abs=True (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).