  Simulations of condensed matter systems often focus on the dynamics of a few
distinguished components but require integrating the dynamics of the full
system. A prime example is a molecular dynamics simulation of a (macro)molecule
in solution, where both the molecules(s) and the solvent dynamics needs to be
integrated. This renders the simulations computationally costly and often
unfeasible for physically or biologically relevant time scales. Standard coarse
graining approaches are capable of reproducing equilibrium distributions and
structural features but do not properly include the dynamics. In this work, we
develop a stochastic data-driven coarse-graining method inspired by the
Mori-Zwanzig formalism. This formalism shows that macroscopic systems with a
large number of degrees of freedom can in principle be well described by a
small number of relevant variables plus additional noise and memory terms. Our
coarse-graining method consists of numerical integrators for the distinguished
components of the system, where the noise and interaction terms with other
system components are substituted by a random variable sampled from a
data-driven model. Applying our methodology on three different systems -- a
distinguished particle under a harmonic potential and under a bistable
potential; and a dimer with two metastable configurations -- we show that the
resulting coarse-grained models are not only capable of reproducing the correct
equilibrium distributions but also the dynamic behavior due to temporal
correlations and memory effects. Our coarse-graining method requires data from
full-scale simulations to be parametrized, and can in principle be extended to
different types of models beyond Langevin dynamics.
