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

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.