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.
arXiv:2607. 29038v1 Announce Type: new Abstract: Fractional scientific machine learning requires numerical operators that can be differentiated, batched, accelerated, and composed with neural networks.
arXiv:2606. 09929v1 Announce Type: cross Abstract: Physical reservoir computing harnesses nonlinear mechanical dynamics but, by convention, freezes the substrate and trains only a linear readout, presuming the substrate is not usefully trainable.
arXiv:2608. 12624v1 Announce Type: new Abstract: Structure-preserving machine learning embeds physical structure directly into model architectures, yet uncertainty quantification (UQ) for such hard-constrained models remains limited because standard UQ methods may violate the encoded admissibility conditions, require architectural modifications, or impose substantial computational costs.
arXiv:2507. 05164v2 Announce Type: replace-cross Abstract: In this chapter, we utilize dynamical systems to analyze several aspects of machine learning algorithms.
arXiv:2608.27883v1 Announce Type: new Abstract: Physical systems are often modeled by solution operators that map input fields, parameters, geometries, or past states to steady or future physical sta...
arXiv:2606. 08343v1 Announce Type: new Abstract: We introduce GENERIC-FNO, the first neural operator to embed the full GENERIC (metriplectic) structure of nonequilibrium thermodynamics -- reversible, energy-conserving dynamics and irreversible, entropy-producing dynamics coupled through the degeneracy conditions -- directly in function space.
arXiv:2606. 03067v1 Announce Type: cross Abstract: A recurring data mining task in complex networks is to determine how individual nodes contribute to system behavior.
arXiv:2602. 15649v2 Announce Type: replace Abstract: In dynamical systems reconstruction (DSR) we aim to recover the dynamical system (DS) underlying observed time series.
arXiv:2609.24021v1 Announce Type: cross Abstract: Neural operators are data-driven models that learn mappings from inputs that parameterize partial differential equations, such as spatially varying c...
The paper investigates how machine learning models can regress scale‑free processes, such as earthquakes or avalanches, focusing on predicting rare, large events that require extrapolation. It studies two self‑similar systems: a 2‑dimensional fractional Gaussian field and the Abelian sandpile model. Experiments compare existing architectures (U‑net, Riesz network) with new proposals (wavelet‑based Graph Neural Network, Fourier embedding, Fourier‑Mellin Neural Operator) to identify spectral bias and coarse‑graining challenges and suggest inductive biases to address them.
arXiv:2606. 00716v1 Announce Type: new Abstract: Inference and control in engineered physical systems pay a heavy physics cost at deployment: state estimators, inverse-problem solvers, model-predictive controllers, schedulers, and observers are often not closed-form and must re-solve a numerical optimization per instance, with the operator re-supplied each time.
arXiv:2608.22636v1 Announce Type: cross Abstract: Q-learning with linear function approximation can be unstable because an arbitrary approximation architecture need not preserve the Bellman contracti...