Learning Memory Kernels in Generalized Langevin Equations

TL;DR

This paper presents a new method for accurately learning memory kernels in generalized Langevin equations by combining correlation function estimation with Sobolev norm-based regression, outperforming existing techniques.

Contribution

It introduces a novel approach that integrates a regularized Prony method with RKHS-based regression to improve kernel estimation in GLEs.

Findings

Outperforms existing L^2 loss-based estimators

Demonstrates consistent advantage across various parameters

Validates approach with numerical examples including force and drift terms

Abstract

We introduce a novel approach for learning memory kernels in Generalized Langevin Equations. 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. Our method guarantees improved performance within an exponentially weighted L^2 space, with the kernel estimation error controlled by the error in estimated correlation functions. We demonstrate the superiority of our estimator compared to other regression estimators that rely on L^2 loss functions and also an estimator derived from the inverse Laplace transform, using numerical examples that highlight its consistent advantage across various weight parameter selections. Additionally, we provide examples that include the application of force and drift terms in the equation.