# Coarse-graining Molecular Systems by Spectral Matching

## TL;DR

This paper introduces a spectral matching framework for coarse-graining molecular systems that preserves kinetic properties by aligning the low-lying spectrum of the system's generator, enabling more accurate long-time simulations.

## Contribution

It presents a novel data-driven spectral matching approach that directly targets generator eigenvalues to improve coarse-grained models' kinetic accuracy.

## Key findings

- Spectral matching effectively preserves slow dynamics in coarse-grained models.
- The method can correct existing coarse-graining techniques like force matching.
- Demonstrated improved retention of kinetic properties in molecular simulations.

## Abstract

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.