Reduced Markovian Models of Dynamical Systems

TL;DR

This paper introduces a data-driven approach to create reduced Markov models of dynamical systems, capturing essential dynamics across multiple timescales and applicable to both low- and high-dimensional systems.

Contribution

It develops a novel clustering algorithm for extracting salient reduced-order dynamics and an alternative method for constructing the infinitesimal generator that preserves statistical features.

Findings

Successfully applied to low-dimensional systems with stochastic and chaotic behavior.

Demonstrated robustness on high-dimensional systems like Kuramoto-Sivashinsky equations.

Effective in capturing system statistics over various timescales.

Abstract

Leveraging recent work on data-driven methods for constructing a finite state space Markov process from dynamical systems, we address two problems for obtaining further reduced statistical representations. The first problem is to extract the most salient reduced-order dynamics for a given timescale by using a modified clustering algorithm from network theory. The second problem is to provide an alternative construction for the infinitesimal generator of a Markov process that respects statistical features over a large range of timescales. We demonstrate the methodology on three low-dimensional dynamical systems with stochastic and chaotic dynamics. We then apply the method to two high-dimensional dynamical systems, the Kuramoto-Sivashinky equations and data sampled from fluid-flow experiments via Particle-Image Velocimetry. We show that the methodology presented herein provides a robust reduced-order statistical representation of the underlying system.