The paper introduces the fractional Laplace neural operator (fLNO), a neural operator that embeds Volterra resolvent structures with non‑rational Laplace symbols into learned maps. It demonstrates that a single graph‑spectral layer can exactly represent the full linear Volterra solution for commuting excitation–Laplacian pairs, and establishes limits on the expressivity of finite rational realizations, showing they cannot capture non‑integer critical asymptotics. The authors also provide trainable parametrizations that enforce stability margins, a graphon‑transfer theorem, and empirical results on benchmark data, Chilean aftershock sequences, and renewal models, highlighting the fLNO’s ability to recover branching coordinates with few parameters while maintaining stability.
By Mauricio Herrera-Mar\'in
arXiv:2607. 00162v1 Announce Type: new Abstract: Parameter-efficient fine-tuning (PEFT) reparameterizes weight updates in a fixed basis: low-rank adapters operate in the spatial domain, while a recent line of spectral methods operates in a fixed Fourier domain.
By Tom Saliencro, Maya Lindqvist, Rohan Desai, Priya Nair, Daniel Whitmore
arXiv:2608. 14636v1 Announce Type: cross Abstract: Fractional optimization methods and fractal activation functions are two independent directions for improving neural network training.
By Sebastian Raubitzek, Georg Goldenits, Sebastian Schrittwieser, Philip K\"onig, Kevin Mallinger
arXiv:2608. 12879v1 Announce Type: new Abstract: Fractional partial differential equations describe nonlocal dynamics, but discovering them from noisy data is difficult because fractional differentiation amplifies high-frequency measurement noise and the derivative orders are unknown.
By Pongpisit Thanasutives, Yoshinobu Kawahara
The paper introduces Spectral-like Neural Discretisation (SpeND), a mesh‑free method that learns stencil weights via a neural network to approximate the modal response of a spectral operator across a specified band of wavenumbers. By projecting the network output onto the space of polynomial‑consistent weights, SpeND ensures exact consistency while minimizing dispersion and dissipation errors in a self‑supervised, physics‑agnostic manner. Experiments on disordered 2‑D node sets demonstrate that the learned fourth‑order operator matches the exact spectral response over a wider band than traditional LABFM or structured‑grid finite differences, and retains fourth‑order convergence upon refinement.
By Lucas Gerken Starepravo, Henry Broadley, Steven Lind, Jack R. C. King
We develop the fTNN, a deterministic tensor neural network subspace method for problems involving the fractional Laplacian on bounded domains, taking the fractional Poisson equation and time-dependent fractional advection-diffusion equation as typical representatives. The work employs a geometry-adapted integration split featuring a spatially dependent near-field radius, which decomposes the fractional Laplacian into three contributions: a singular near field, a regular interior far field, and an analytical exterior far field.