Error Analysis of Generalized Langevin Equations with Approximated Memory Kernels

• URL: http://arxiv.org/abs/2512.10256v1
• Date: Thu, 11 Dec 2025 03:27:58 GMT
• Title: Error Analysis of Generalized Langevin Equations with Approximated Memory Kernels
• Authors: Quanjun Lang, Jianfeng Lu,
• Abstract summary: We analyze prediction error in dynamical systems with memory, focusing on generalized Langevin equations (GLEs)<n>We establish that, under a strongly convex potential, discrepancies decay at a rate determined by the decay of the memory kernel and are quantitatively bounded by the estimation error of the kernel in a weighted norm.<n>This framework accommodates non-translation-invariant kernels and white-noise forcing, explicitly linking improved kernel estimation to enhanced trajectory prediction.
• Score: 5.726854405157353
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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 the memory kernel and are quantitatively bounded by the estimation error of the kernel in a weighted norm. Our analysis integrates synchronized noise coupling with a Volterra comparison theorem, encompassing both subexponential and exponential kernel classes. For first-order models, we derive moment and perturbation bounds using resolvent estimates in weighted spaces. For second-order models with confining potentials, we prove contraction and stability under kernel perturbations using a hypocoercive Lyapunov-type distance. This framework accommodates non-translation-invariant kernels and white-noise forcing, explicitly linking improved kernel estimation to enhanced trajectory prediction. Numerical examples validate these theoretical findings.

Related papers

• Preconditioned Langevin Dynamics with Score-Based Generative Models for Infinite-Dimensional Linear Bayesian Inverse Problems [4.2223436389469144]
  Langevin dynamics driven by score-based generative models (SGMs) acting as priors, formulated directly in function space.<n>We derive, for the first time, error estimates that explicitly depend on the approximation error of the score.<n>As a consequence, we obtain sufficient conditions for global convergence in Kullback-Leibler divergence on the underlying function space.
  arXiv  Detail & Related papers  (2025-05-23T18:12:04Z)
• High-Dimensional Kernel Methods under Covariate Shift: Data-Dependent Implicit Regularization [83.06112052443233]
  This paper studies kernel ridge regression in high dimensions under covariate shifts. By a bias-variance decomposition, we theoretically demonstrate that the re-weighting strategy allows for decreasing the variance. For bias, we analyze the regularization of the arbitrary or well-chosen scale, showing that the bias can behave very differently under different regularization scales.
  arXiv  Detail & Related papers  (2024-06-05T12:03:27Z)
• Learning Analysis of Kernel Ridgeless Regression with Asymmetric Kernel Learning [33.34053480377887]
  This paper enhances kernel ridgeless regression with Locally-Adaptive-Bandwidths (LAB) RBF kernels. For the first time, we demonstrate that functions learned from LAB RBF kernels belong to an integral space of Reproducible Kernel Hilbert Spaces (RKHSs)
  arXiv  Detail & Related papers  (2024-06-03T15:28:12Z)
• Learning Memory Kernels in Generalized Langevin Equations [5.266892492931388]
  We introduce a novel approach for learning memory kernels in Generalized Langevin Equations.<n>This approach initially utilizes a regularized Prony method to estimate correlation functions from trajectory data, followed by regression over a Sobolev norm-based loss function with RKHS regularization.
  arXiv  Detail & Related papers  (2024-02-18T21:01:49Z)
• Sampling and estimation on manifolds using the Langevin diffusion [45.57801520690309]
  Two estimators of linear functionals of $mu_phi $ based on the discretized Markov process are considered.<n>Error bounds are derived for sampling and estimation using a discretization of an intrinsically defined Langevin diffusion.
  arXiv  Detail & Related papers  (2023-12-22T18:01:11Z)
• Non-Parametric Learning of Stochastic Differential Equations with Non-asymptotic Fast Rates of Convergence [65.63201894457404]
  We propose a novel non-parametric learning paradigm for the identification of drift and diffusion coefficients of non-linear differential equations.<n>The key idea essentially consists of fitting a RKHS-based approximation of the corresponding Fokker-Planck equation to such observations.
  arXiv  Detail & Related papers  (2023-05-24T20:43:47Z)
• On the Benefits of Large Learning Rates for Kernel Methods [110.03020563291788]
  We show that a phenomenon can be precisely characterized in the context of kernel methods. We consider the minimization of a quadratic objective in a separable Hilbert space, and show that with early stopping, the choice of learning rate influences the spectral decomposition of the obtained solution.
  arXiv  Detail & Related papers  (2022-02-28T13:01:04Z)
• Convex Analysis of the Mean Field Langevin Dynamics [49.66486092259375]
  convergence rate analysis of the mean field Langevin dynamics is presented. $p_q$ associated with the dynamics allows us to develop a convergence theory parallel to classical results in convex optimization.
  arXiv  Detail & Related papers  (2022-01-25T17:13:56Z)
• Nonconvex Stochastic Scaled-Gradient Descent and Generalized Eigenvector Problems [98.34292831923335]
  Motivated by the problem of online correlation analysis, we propose the emphStochastic Scaled-Gradient Descent (SSD) algorithm. We bring these ideas together in an application to online correlation analysis, deriving for the first time an optimal one-time-scale algorithm with an explicit rate of local convergence to normality.
  arXiv  Detail & Related papers  (2021-12-29T18:46:52Z)
• The Connection between Discrete- and Continuous-Time Descriptions of Gaussian Continuous Processes [60.35125735474386]
  We show that discretizations yielding consistent estimators have the property of invariance under coarse-graining' This result explains why combining differencing schemes for derivatives reconstruction and local-in-time inference approaches does not work for time series analysis of second or higher order differential equations.
  arXiv  Detail & Related papers  (2021-01-16T17:11:02Z)
• Learning interaction kernels in stochastic systems of interacting particles from multiple trajectories [13.3638879601361]
  We consider systems of interacting particles or agents, with dynamics determined by an interaction kernel. We introduce a nonparametric inference approach to this inverse problem, based on a regularized maximum likelihood estimator. We show that a coercivity condition enables us to control the condition number of this problem and prove the consistency of our estimator.
  arXiv  Detail & Related papers  (2020-07-30T01:28:06Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.