الملخص
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.
الكلمات المفتاحية
الموضوع
بيانات النشر
- المجلة
- غير متاح
- وصول مفتوح
- وصول مفتوح أخضر
اقتبس هذه المقالة
APA 7
Herrera-Marín, M. (2026). Fractional Laplace Neural Operators: Exact Architectures, an Expressivity Frontier at Criticality, and Certified Stability for Memory-Driven Network Dynamics. https://omanscience.com/ar/articles/fractional-laplace-neural-operators-exact-architectures-an-expressivity-frontier-at-criticality-and-certified-stability-for-memory-driven-network-dyna
MLA 9
Herrera-Marín, Mauricio. "Fractional Laplace Neural Operators: Exact Architectures, an Expressivity Frontier at Criticality, and Certified Stability for Memory-Driven Network Dynamics." https://omanscience.com/ar/articles/fractional-laplace-neural-operators-exact-architectures-an-expressivity-frontier-at-criticality-and-certified-stability-for-memory-driven-network-dyna.
شيكاغو (المؤلف–التاريخ)
Herrera-Marín, Mauricio. 2026. "Fractional Laplace Neural Operators: Exact Architectures, an Expressivity Frontier at Criticality, and Certified Stability for Memory-Driven Network Dynamics." https://omanscience.com/ar/articles/fractional-laplace-neural-operators-exact-architectures-an-expressivity-frontier-at-criticality-and-certified-stability-for-memory-driven-network-dyna.
هارفارد
Herrera-Marín, M. (2026) 'Fractional Laplace Neural Operators: Exact Architectures, an Expressivity Frontier at Criticality, and Certified Stability for Memory-Driven Network Dynamics', Available at: https://omanscience.com/ar/articles/fractional-laplace-neural-operators-exact-architectures-an-expressivity-frontier-at-criticality-and-certified-stability-for-memory-driven-network-dyna.
فانكوفر
Herrera-Marín M. Fractional Laplace Neural Operators: Exact Architectures, an Expressivity Frontier at Criticality, and Certified Stability for Memory-Driven Network Dynamics. https://omanscience.com/ar/articles/fractional-laplace-neural-operators-exact-architectures-an-expressivity-frontier-at-criticality-and-certified-stability-for-memory-driven-network-dyna
IEEE
M. Herrera-Marín, "Fractional Laplace Neural Operators: Exact Architectures, an Expressivity Frontier at Criticality, and Certified Stability for Memory-Driven Network Dynamics," https://omanscience.com/ar/articles/fractional-laplace-neural-operators-exact-architectures-an-expressivity-frontier-at-criticality-and-certified-stability-for-memory-driven-network-dyna.