Interpretable and Equation-Free Response Theory for Complex Systems

TL;DR

This paper introduces an interpretable, data-driven response theory for complex systems that leverages Markov models and Koopman-inspired algebraic expansions to predict system responses across multiple timescales.

Contribution

It develops simple, implementable response formulas for Markov chains that are equation-agnostic and applicable even without explicit knowledge of underlying dynamics.

Findings

Provides explicit formulas for system response and correlations.

Demonstrates methodology on a metastable system.

Connects response theory with the Prony method for signal analysis.

Abstract

Response theory provides a pathway for understanding the sensitivity of a system and for predicting how its statistical properties change when a perturbation is applied. In the case of complex and multiscale systems, to achieve enhanced practical applicability, response theory should be interpretable, capable of focusing on relevant timescales, and amenable to data-driven and equation-agnostic implementations. Along these lines, in the spirit of Markov state modelling, we present linear and nonlinear response formulas for Markov chains. We obtain simple and easily implementable expressions that can be used to predict the response of observables as well as of higher-order correlations. The methodology proposed here can be implemented in a purely data-driven setting and even if the underlying evolution equations are unknown. The use of algebraic expansions inspired by Koopmanism allows to elucidate the role of different time scales and modes of variability, and to find explicit and interpretable expressions for the Green's functions at all orders. This is a major advantage of the framework proposed here. We illustrate our methodology in a very simple yet instructive metastable system. Finally, our results provide a dynamical foundation for the Prony method, which is commonly used for the statistical analysis of discrete time signals.