  In this paper, we discuss information-theoretic tools for obtaining optimized
coarse-grained molecular models for both equilibrium and non-equilibrium
molecular dynamics. The latter are ubiquitous in physicochemical and biological
applications, where they are typically associated with coupling mechanisms,
multi-physics and/or boundary conditions. In general the non-equilibrium steady
states are not known explicitly as they do not necessarily have a Gibbs
structure.
  The presented approach can compare microscopic behavior of molecular systems
to parametric and non-parametric coarse-grained one using the relative entropy
between distributions on the path space and setting up a corresponding path
space variational inference problem. The methods can become entirely
data-driven when the microscopic dynamics are replaced with corresponding
correlated data in the form of time series. Furthermore, we present connections
and generalizations of force matching methods in coarse-graining with
path-space information methods, as well as demonstrate the enhanced
transferability of information-based parameterizations to general observables
due to information inequalities.
  We further discuss methodological connections between information-based
coarse-graining of molecular systems and variational inference methods
primarily developed in the machine learning community. However, we note that
the work presented here addresses variational inference for correlated time
series due to the focus on dynamics. The applicability of the proposed methods
is demonstrated on high-dimensional stochastic processes given by Langevin,
overdamped and driven Langevin dynamics of interacting particles.
