Reduced-order models for coupled dynamical systems: data-driven methods and the Koopman operator

Santos Gutiérrez, M. ORCID: https://orcid.org/0000-0001-8617-2804, Lucarini, V. ORCID: https://orcid.org/0000-0001-9392-1471, Chekroun, M. D. ORCID: https://orcid.org/0000-0002-4525-5141 and Ghil, M. ORCID: https://orcid.org/0000-0001-5177-7133 (2021) Reduced-order models for coupled dynamical systems: data-driven methods and the Koopman operator. Chaos: An Interdisciplinary Journal of Nonlinear Science, 31 (5). 053116. ISSN 1089-7682 doi: 10.1063/5.0039496

Abstract/Summary

Providing efficient and accurate parameterizations 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 parameterizations of weakly coupled dynamical systems. Such parameterizations yield a set of stochastic integrodifferential 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 integrodifferential 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, equation-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, on the one hand, support 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 parameterizations. Parameterizations aim to reduce the complexity of high-dimensional dynamical systems. Here, a theory-based and a data-driven approach for the parameterization of coupled systems are compared, showing that both yield the same stochastic multilevel structure. The results provide very strong support to the use of empirical methods in model reduction and clarify the practical relevance of the proposed theoretical framework.

Item Type	Article
URI	https://reading-clone.eprints-hosting.org/id/eprint/98672
Identification Number/DOI	10.1063/5.0039496
Refereed	Yes
Divisions	Interdisciplinary Research Centres (IDRCs) > Centre for the Mathematics of Planet Earth (CMPE) Science > School of Mathematical, Physical and Computational Sciences > Department of Mathematics and Statistics
Publisher	American Institute of Physics
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