Fractional laplace neural operators capture exact memory and guarantee stability
Fractional Laplace Neural Operators: Exact Architectures, an Expressivity Frontier at Criticality, and Certified Stability for Memory-Driven Network Dynamics
Machine Learning
Summary
Modeling systems with memory or long-lasting effects is challenging because they need special math to handle past influences exactly. The authors introduce a neural operator that can represent these memory effects perfectly when certain conditions hold, and they provide ways to ensure the model remains stable and accurate. Their approach also works well across different network sizes and real-world seismic data, showing promise for physical and dynamic system modeling. They balance exact memory structure, interpretability, and competitive accuracy in a new neural architecture.
What this means in practice
- •For seismology data analysts: Model aftershock sequences with explicit memory effects and stable predictions using fractional Laplace neural operators trained on seismic data.
- •For epidemic modeling teams: Implement renewal models with guaranteed stability and interpretable memory structure to better simulate disease spread dynamics across regions.
Authors
Mauricio Herrera-Marín
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.