mailitics

Category: math.DS

  • Error Analysis of Generalized Langevin Equations with Approximated Memory Kernels

    Error Analysis of Generalized Langevin Equations with Approximated Memory Kernels arXiv:2512.10256v1 Announce Type: new Abstract: We analyze prediction error in stochastic dynamical systems with memory, focusing on generalized Langevin equations (GLEs) formulated as stochastic Volterra equations. We establish that, under a strongly convex potential, trajectory discrepancies decay at a rate determined by the decay of…

  • How to Tame Your LLM: Semantic Collapse in Continuous Systems

    How to Tame Your LLM: Semantic Collapse in Continuous Systems arXiv:2512.05162v1 Announce Type: new Abstract: We develop a general theory of semantic dynamics for large language models by formalizing them as Continuous State Machines (CSMs): smooth dynamical systems whose latent manifolds evolve under probabilistic transition operators. The associated transfer operator $P: L^2(M,mu) to L^2(M,mu)$ encodes…

  • Extreme Event Aware ($eta$-) Learning

    Extreme Event Aware ($eta$-) Learning arXiv:2510.19161v1 Announce Type: new Abstract: Quantifying and predicting rare and extreme events persists as a crucial yet challenging task in understanding complex dynamical systems. Many practical challenges arise from the infrequency and severity of these events, including the considerable variance of simple sampling methods and the substantial computational cost of…

  • A Bayesian Framework for Symmetry Inference in Chaotic Attractors

    A Bayesian Framework for Symmetry Inference in Chaotic Attractors arXiv:2510.16509v1 Announce Type: new Abstract: Detecting symmetry from data is a fundamental problem in signal analysis, providing insight into underlying structure and constraints. When data emerge as trajectories of dynamical systems, symmetries encode structural properties of the dynamics that enable model reduction, principled comparison across conditions,…

  • Physics-Informed Regression: Parameter Estimation in Parameter-Linear Nonlinear Dynamic Models

    Physics-Informed Regression: Parameter Estimation in Parameter-Linear Nonlinear Dynamic Models arXiv:2508.19249v1 Announce Type: cross Abstract: We present a new efficient hybrid parameter estimation method based on the idea, that if nonlinear dynamic models are stated in terms of a system of equations that is linear in terms of the parameters, then regularized ordinary least squares can…

  • Echoes of the past: A unified perspective on fading memory and echo states

    Echoes of the past: A unified perspective on fading memory and echo states arXiv:2508.19145v1 Announce Type: new Abstract: Recurrent neural networks (RNNs) have become increasingly popular in information processing tasks involving time series and temporal data. A fundamental property of RNNs is their ability to create reliable input/output responses, often linked to how the network…

  • Stochastic dynamics learning with state-space systems

    Stochastic dynamics learning with state-space systems arXiv:2508.07876v1 Announce Type: new Abstract: This work advances the theoretical foundations of reservoir computing (RC) by providing a unified treatment of fading memory and the echo state property (ESP) in both deterministic and stochastic settings. We investigate state-space systems, a central model class in time series learning, and establish…

  • Learning the Simplest Neural ODE

    Learning the Simplest Neural ODE arXiv:2505.02019v1 Announce Type: new Abstract: Since the advent of the “Neural Ordinary Differential Equation (Neural ODE)” paper, learning ODEs with deep learning has been applied to system identification, time-series forecasting, and related areas. Exploiting the diffeomorphic nature of ODE solution maps, neural ODEs has also enabled their use in generative…

  • On the emergence of numerical instabilities in Next Generation Reservoir Computing

    On the emergence of numerical instabilities in Next Generation Reservoir Computing arXiv:2505.00846v1 Announce Type: new Abstract: Next Generation Reservoir Computing (NGRC) is a low-cost machine learning method for forecasting chaotic time series from data. However, ensuring the dynamical stability of NGRC models during autonomous prediction remains a challenge. In this work, we uncover a key…

  • Resonances in reflective Hamiltonian Monte Carlo

    Resonances in reflective Hamiltonian Monte Carlo arXiv:2504.12374v1 Announce Type: new Abstract: In high dimensions, reflective Hamiltonian Monte Carlo with inexact reflections exhibits slow mixing when the particle ensemble is initialised from a Dirac delta distribution and the uniform distribution is targeted. By quantifying the instantaneous non-uniformity of the distribution with the Sinkhorn divergence, we elucidate…

  • Modeling COVID-19 spread in the USA using metapopulation SIR models coupled with graph convolutional neural networks

    Modeling COVID-19 spread in the USA using metapopulation SIR models coupled with graph convolutional neural networks arXiv:2501.02043v1 Announce Type: new Abstract: Graph convolutional neural networks (GCNs) have shown tremendous promise in addressing data-intensive challenges in recent years. In particular, some attempts have been made to improve predictions of Susceptible-Infected-Recovered (SIR) models by incorporating human mobility…