# Transition manifolds of complex metastable systems: Theory and data-driven computation of effective dynamics

## TL;DR

This paper develops a theoretical framework and a local data-driven algorithm to identify low-dimensional reaction coordinates that capture the dominant dynamics of complex metastable systems without clear fast-slow scale separation.

## Contribution

It introduces a new theoretical criterion for the existence of effective reaction coordinates and proposes a local algorithm for their computation, suitable for high-dimensional systems.

## Key findings

- Theoretical guarantee for the existence of reaction coordinates under certain reducibility conditions.
- A novel local numerical method for computing reaction coordinates from data.
- Potential application to molecular dynamics and other complex systems.

## Abstract

We consider complex dynamical systems showing metastable behavior but no local separation of fast and slow time scales. The article raises the question of whether such systems exhibit a low-dimensional manifold supporting its effective dynamics. For answering this question, we aim at finding nonlinear coordinates, called reaction coordinates, such that the projection of the dynamics onto these coordinates preserves the dominant time scales of the dynamics. We show that, based on a specific reducibility property, the existence of good low-dimensional reaction coordinates preserving the dominant time scales is guaranteed. Based on this theoretical framework, we develop and test a novel numerical approach for computing good reaction coordinates. The proposed algorithmic approach is fully local and thus not prone to the curse of dimension with respect to the state space of the dynamics. Hence, it is a promising method for data-based model reduction of complex dynamical systems such as molecular dynamics.