  Modeling a high-dimensional Hamiltonian system in reduced dimensions with
respect to coarse-grained (CG) variables can greatly reduce computational cost
and enable efficient bottom-up prediction of main features of the system for
many applications. However, it usually experiences significantly altered
dynamics due to loss of degrees of freedom upon coarse-graining. To establish
CG models that can faithfully preserve dynamics, previous efforts mainly
focused on equilibrium systems. In contrast, various soft matter systems are
known out of equilibrium. Therefore, the present work concerns non-equilibrium
systems and enables accurate and efficient CG modeling that preserves
non-equilibrium dynamics and is generally applicable to any non-equilibrium
process and any observable of interest. To this end, the dynamic equation of a
CG variable is built in the form of the non-stationary generalized Langevin
equation (nsGLE) to account for the dependence of non-equilibrium processes on
the initial conditions, where the two-time memory kernel is determined from the
data of the two-time auto-correlation function of the non-equilibrium
trajectory-averaged observable of interest. By embedding the non-stationary
non-Markovian process in an extended stochastic framework, an explicit form of
the non-stationary random noise in the nsGLE is introduced, and the cost is
significantly reduced for solving the nsGLE to predict the non-equilibrium
dynamics of the CG variable. To prove and exploit the equivalence of the nsGLE
and extended dynamics, the memory kernel is parameterized in a two-time
exponential expansion. A data-driven hybrid optimization process is proposed
for the parameterization, a non-convex and high-dimensional optimization
problem.
