  Providing efficient and accurate parametrizations for model reduction is a
key goal in many areas of science and technology. Here we present a strong link
between data-driven and theoretical approaches to achieving this goal. Formal
perturbation expansions of the Koopman operator allow us to derive general
stochastic parametrizations of weakly coupled dynamical systems. Such
parametrizations yield a set of stochastic integro-differential equations with
explicit noise and memory kernel formulas to describe the effects of unresolved
variables. We show that the perturbation expansions involved need not be
truncated when the coupling is additive. The unwieldy integro-differential
equations can be recast as a simpler multilevel Markovian model, and we
establish an intuitive connection with a generalized Langevin equation. This
connection helps setting up a parallelism between the top-down, equations-based
methodology herein and the well-established empirical model reduction (EMR)
methodology that has been shown to provide efficient dynamical closures to
partially observed systems. Hence, our findings support, on the one hand, the
physical basis and robustness of the EMR methodology and, on the other hand,
illustrate the practical relevance of the perturbative expansion used for
deriving the parametrizations.
