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

Manuel Santos Gutiérrez1, Valerio Lucarini1, Mickaël D Chekroun2

  • 1Department of Mathematics and Statistics, University of Reading, Reading RG6 6AX, United Kingdom.

Chaos (Woodbury, N.Y.)
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Summary
This summary is machine-generated.

This study links data-driven and theoretical methods for efficient model reduction. It derives general stochastic parameterizations for dynamical systems, offering a robust foundation for empirical model reduction techniques.

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Area of Science:

  • Dynamical systems theory
  • Statistical physics
  • Computational science

Background:

  • Model reduction is crucial for understanding complex systems.
  • Parameterization of unresolved variables remains a challenge.
  • Bridging data-driven and theoretical approaches is highly desirable.

Purpose of the Study:

  • To establish a rigorous link between data-driven and theoretical methods for model reduction.
  • To derive general stochastic parameterizations for weakly coupled dynamical systems.
  • To provide a theoretical foundation for empirical model reduction (EMR).

Main Methods:

  • Formal perturbation expansions of the Koopman operator.
  • Derivation of stochastic integrodifferential equations with explicit noise and memory kernels.
  • Recasting integrodifferential equations into multilevel Markovian models.
  • Establishing connections with generalized Langevin equations.

Main Results:

  • General stochastic parameterizations for weakly coupled dynamical systems were derived.
  • Perturbation expansions do not require truncation for additive coupling.
  • A connection between theoretical derivations and empirical model reduction was established.
  • The physical basis and robustness of EMR were supported.

Conclusions:

  • The study provides a theoretical framework for data-driven model reduction.
  • The findings validate and enhance the empirical model reduction methodology.
  • The work highlights the practical relevance of perturbative expansions in parameterization.