Abstract
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.
Highlights
AND MOTIVATIONMultiscale systems are typically characterized by the presence of significant variability over a large range of spatial and temporal scales
The unwieldy integrodifferential equations can be recast as a simpler multilevel Markovian model, and we establish an intuitive connection with a generalized Langevin equation
Following Ref. 19, we briefly review in Appendix B a criterion based on Koopman operators—and, more generally, Markov operators—that enables one to decide whether memory effects can help explain the dynamics and statistics in reduced phase spaces
Summary
Multiscale systems are typically characterized by the presence of significant variability over a large range of spatial and temporal scales. This multiscale character is due to a combination of the following factors: the nature of the external forcings; the inhomogeneity of the properties of the system’s various components; the complexity of the coupling mechanisms between the components; and the variety of instabilities, dissipative processes, and feedback acting at different scales In many cases, both the theoretical understanding of such systems and the formulation of numerical models for simulating their properties are based on focusing upon a reduced range of large spatial and long temporal scales of interest, and upon devising an efficient way to effectively capture the impact of the faster dynamical processes acting predominantly in the neglected smaller spatial scales.. Parameterization schemes have been traditionally formulated in such a way that one expresses the net impact on the scales of interest of processes occurring within the unresolved scales via deterministic functions of the resolved variables, as in the pioneering work on the parameterization of convective processes by Arakawa and Schubert. More recently, it has been recognized, mostly on empirical grounds, that parameterizations should involve stochastic and non-Markovian components. Machine learning methods have been proposed as the frontier of data-driven parameterizations, able to deliver a new generation of Earth system models; see, though, the caveats discussed by Refs. 26 and 39
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