  Coarse-graining has become an area of tremendous importance within many
different research fields. For molecular simulation, coarse-graining bears the
promise of finding simplified models such that long-time simulations of
large-scale systems become computationally tractable. While significant
progress has been made in tuning thermodynamic properties of reduced models, it
remains a key challenge to ensure that relevant kinetic properties are retained
by coarse-grained dynamical systems. In this study, we focus on data-driven
methods to preserve the rare-event kinetics of the original system, and make
use of their close connection to the low-lying spectrum of the system's
generator. Building on work by Crommelin and Vanden-Eijnden, SIAM Multiscale
Model. Simul. (2011), we present a general framework, called spectral matching,
which directly targets the generator's leading eigenvalue equations when
learning parameters for coarse-grained models. We discuss different parametric
models for effective dynamics and derive the resulting data-based regression
problems. We show that spectral matching can be used to learn effective
potentials which retain the slow dynamics, but also to correct the dynamics
induced by existing techniques, such as force matching.
