arXiv：cs.LG（机器学习，全量分类）· Mauricio Herrera-Mar\'in·· 13 小时前AI 评分42
分数阶 Laplace 神经算子 fLNO：精确架构、临界可表达性边界与可认证稳定性
Fractional Laplace Neural Operators: Exact Architectures, an Expressivity Frontier at Criticality, and Certified Stability for Memory-Driven Network Dynamics
AI 导读
研究者提出分数阶 Laplace 神经算子（fLNO），将 Volterra 记忆结构嵌入学习映射，对可交换激励—Laplace 对，单个图谱层即可精确表示完整线性 Volterra 解算子。
正文
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.
| Subjects: | Machine Learning (stat.ML); Machine Learning (cs.LG) |
| Cite as: | arXiv:2610.00515 [stat.ML] |
| (or arXiv:2610.00515v1 [stat.ML] for this version) | |
| https://doi.org/10.48550/arXiv.2610.00515 arXiv-issued DOI via DataCite (pending registration) |
Submission history
From: Mauricio Herrera [view email]
[v1]
Wed, 30 Sep 2026 18:08:40 UTC (3,162 KB)
来源：arXiv：cs.LG（机器学习，全量分类） · arxiv.org