DiffGLE: Differentiable coarse-grained dynamics using generalized Langevin equation

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 a useful reduced model is difficult because the exact projected dynamics may involve state-dependent memory, non-stationary fluctuations, and generalized fluctuation relations that are not directly accessible from ordinary trajectories. Here we combine differentiable simulation with a coloured-noise ansatz to learn non-Markovian memory kernels in a top-down manner. A state-independent generalized langevin equation, with linear memory kernel friction and stationary random force, is employed as a tractable dynamical closure, rather than presumed to provide an exact description of the evolution of the coarse-grained variables. The random force is generated by a trainable coloured-noise filter, whose autocorrelation defines the friction memory enforcing fluctuation--dissipation consistency within the chosen GLE model. The filter is optimised by differentiating through coarse-grained 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 these systems, differentiable memory learning improves dynamical correlation agreement while preserving the fixed conservative model and its structural properties.