分数阶拉普拉斯神经算子：精确架构、临界状态的表达力前沿以及记忆驱动网络动力学的认证稳定性

中文摘要 AI 辅助

神经算子学习函数空间之间的映射，而遗传性网络动力学由具有非有理拉普拉斯符号的沃尔泰拉预解式描述。我们引入了一种分数阶拉普拉斯神经算子（fLNO），将这一结构嵌入到学习映射中。对于可交换的激励-拉普拉斯对，一个块图谱层即可精确表示完整的线性沃尔泰拉解算子。我们为有限有理实现建立了表达力前沿：它们在紧致频率窗口上以几何方式逼近分数阶记忆，但无法再现由分支点产生的非整数临界渐近行为，且在半直线上最佳有理逼近率为根指数级。同一理论产生了可训练的参数化方法，通过构造强制满足规定的稳定裕度，并且一个图传递定理将真正的算子一致性与参数共享区分开来。在公共数据基准上，正有理算子可在有限时间范围内匹配或超过fLNO的精度，而在受控的近临界实验中，fLNO以更少的参数更忠实地恢复分支坐标；无约束的有理拟合可能跨越稳定性边界，而认证参数化则不会。一个四参数谱律无需重新训练即可从规模为48的图迁移到规模为192的图，相对误差为0.51%–0.62%。对智利余震序列以及智利和21个意大利地区的更新模型的应用程序展示了具有显式不确定性的结构化推断。该贡献是一种算子学习架构，其中精确的记忆结构、物理坐标和稳定性保证与竞争性精度共存。

英文摘要

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.

发表机构

• Universidad del Desarrollo（发展大学）

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