DiffGLE: Differentiable Coarse-Grained Dynamics using Generalized Langevin Equation

Abstract

Capturing dynamical fidelity at the coarse-grained scale remains a central challenge in systematic molecular modelling. The generalized Langevin equation, rooted in the Mori--Zwanzig formalism, provides a principled framework for representing the memory friction and stochastic forces induced by eliminated microscopic degrees of freedom. In practice, however, parameterising its memory kernel is difficult because the exact kernel is a history-dependent projected-dynamics object coupled by the fluctuation--dissipation theorem to coloured random forces that are not directly observable from ordinary trajectories. Here we combine differentiable simulation with a coloured-noise ansatz to learn non-Markovian memory kernels in a top-down manner. The random force is represented by a trainable filter whose autocorrelation defines the friction memory, enforcing fluctuation--dissipation consistency by construction. The filter is optimised by backpropagating through coarse-grained generalized Langevin trajectories to match reference velocity autocorrelation functions, avoiding explicit projected-force reconstruction. We demonstrate the approach on bulk water, bulk carbon dioxide, and a single particle star-polymer memory benchmark. Across all these systems, the proposed framework accurately reproduces the target dynamical correlations, demonstrating that differentiable simulation enables direct optimisation of non-Markovian memory kernels from time-correlation observables.