Reduced-Order Models for Coupled Dynamical Systems: Data-driven Methods and the Koopman Operator

TL;DR

This paper links data-driven and theoretical methods for model reduction in coupled dynamical systems using the Koopman operator, deriving stochastic parametrizations and simplifying complex equations into Markovian models.

Contribution

It introduces a novel connection between perturbation expansions of the Koopman operator and empirical model reduction techniques for coupled systems.

Findings

Derivation of stochastic integro-differential equations with explicit noise and memory kernels.

Recasting complex equations as simpler multilevel Markovian models.

Establishing a connection with the generalized Langevin equation.

Abstract

Providing efficient and accurate parametrizations for model reduction is a key goal in many areas of science and technology. Here we present a strong link between data-driven and theoretical approaches to achieving this goal. Formal perturbation expansions of the Koopman operator allow us to derive general stochastic parametrizations of weakly coupled dynamical systems. Such parametrizations yield a set of stochastic integro-differential equations with explicit noise and memory kernel formulas to describe the effects of unresolved variables. We show that the perturbation expansions involved need not be truncated when the coupling is additive. The unwieldy integro-differential equations can be recast as a simpler multilevel Markovian model, and we establish an intuitive connection with a generalized Langevin equation. This connection helps setting up a parallelism between the top-down, equations-based methodology herein and the well-established empirical model reduction (EMR) methodology that has been shown to provide efficient dynamical closures to partially observed systems. Hence, our findings support, on the one hand, the physical basis and robustness of the EMR methodology and, on the other hand, illustrate the practical relevance of the perturbative expansion used for deriving the parametrizations.