Fractional Laplace Neural Operators: Exact Architectures, an Expressivity Frontier at Criticality, and Certified Stability for Memory-Driven Network Dynamics
Abstract
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.
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BibTeX
@misc{herreramarin2026fractional,
title = {{Fractional Laplace Neural Operators: Exact Architectures, an Expressivity Frontier at Criticality, and Certified Stability for Memory-Driven Network Dynamics}},
author = {Mauricio Herrera-Marín},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.00515},
eprint = {2610.00515},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.00515},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Fractional Laplace Neural Operators: Exact Architectures, an Expressivity Frontier at Criticality, and Certified Stability for Memory-Driven Network Dynamics
AU - Mauricio Herrera-Marín
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.00515
UR - https://arxiv.org/abs/2610.00515
AB - 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.
ER -