  Systematic discovery of reduced-order closure models for multi-scale
processes remains an important open problem in complex dynamical systems. Even
when an effective lower-dimensional representation exists, reduced models are
difficult to obtain using solely analytical methods. Rigorous methodologies for
finding such coarse-grained representations of multi-scale phenomena would
enable accelerated computational simulations and provide fundamental insights
into the complex dynamics of interest. We focus on a heterogeneous population
of oscillators of Kuramoto type as a canonical model of complex dynamics, and
develop a data-driven approach for inferring its coarse-grained description.
Our method is based on a numerical optimization of the coefficients in a
general equation of motion informed by analytical derivations in the
thermodynamic limit. We show that certain assumptions are required to obtain an
autonomous coarse-grained equation of motion. However, optimizing coefficient
values enables coarse-grained models with conceptually disparate functional
forms, yet comparable quality of representation, to provide accurate
reduced-order descriptions of the underlying system.
