Data-driven parameterization of the generalized Langevin equation

TL;DR

This paper introduces a data-driven method to accurately determine the memory kernel and noise in generalized Langevin equations, enabling efficient simulation of complex stochastic systems with exact fluctuation-dissipation compliance.

Contribution

It proposes a rational function parameterization of the memory kernel in the Laplace domain linked to equilibrium statistics, allowing high-order approximation and embedding in extended stochastic models.

Findings

Effective kernel approximation demonstrated through numerical tests.

Exact satisfaction of the second fluctuation-dissipation theorem achieved.

Method enables practical and accurate coarse-grained stochastic modeling.

Abstract

We present a data-driven approach to determine the memory kernel and random noise in generalized Langevin equations. To facilitate practical implementations, we parameterize the kernel function in the Laplace domain by a rational function, with coefficients directly linked to the equilibrium statistics of the coarse-grain variables. We show that such an approximation can be constructed to arbitrarily high order and the resulting generalized Langevin dynamics can be embedded in an extended stochastic model without explicit memory. We demonstrate how to introduce the stochastic noise so that the second fluctuation-dissipation theorem is exactly satisfied. Results from several numerical tests are presented to demonstrate the effectiveness of the proposed method.