A deep learning approach to the measurement of long-lived memory kernels from Generalised Langevin Dynamics

TL;DR

This paper introduces a deep neural network-based method to accurately measure memory kernels in complex systems described by the Generalised Langevin Equation, overcoming limitations of traditional inverse Laplace transform techniques.

Contribution

The authors develop KernelLearner, a novel deep learning pipeline that reliably extracts memory kernels from dynamical data, even in noisy conditions and across different systems.

Findings

DNNs outperform conventional methods in noisy environments.

The trained network generalizes well to different systems.

The approach effectively captures long-lived memory effects in glassy systems.

Abstract

Memory effects are ubiquitous in a wide variety of complex physical phenomena, ranging from glassy dynamics and metamaterials to climate models. The Generalised Langevin Equation (GLE) provides a rigorous way to describe memory effects via the so-called memory kernel in an integro-differential equation. However, the memory kernel is often unknown, and accurately predicting or measuring it via e.g. a numerical inverse Laplace transform remains a herculean task. Here we describe a novel method using deep neural networks (DNNs) to measure memory kernels from dynamical data. As proof-of-principle, we focus on the notoriously long-lived memory effects of glassy systems, which have proved a major challenge to existing methods. Specifically, we learn a training set generated with the Mode-Coupling Theory (MCT) of hard spheres. Our DNNs are remarkably robust against noise, in contrast to conventional techniques which require ensemble averaging over many independent trajectories. Finally, we demonstrate that a network trained on data generated from analytic theory (hard-sphere MCT) generalises well to data from simulations of a different system (Brownian Weeks-Chandler-Andersen particles). We provide a general pipeline, KernelLearner, for training networks to extract memory kernels from any non-Markovian system described by a GLE. The success of our DNN method applied to glassy systems suggests deep learning can play an important role in the study of dynamical systems that exhibit memory effects.