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A data-driven framework for Koopman semigroup estimation in stochastic dynamical systems.

Yuanchao Xu1, Kaidi Shao2, Isao Ishikawa3

  • 1Department of Mathematical and Statistical Sciences, University of Alberta, University Commons 5-140 Edmonton, Alberta T6G 2N8, Canada.

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Stochastic Dynamic Mode Decomposition (SDMD) offers a stable and precise method for analyzing stochastic systems by approximating the Koopman semigroup. This data-driven framework enhances efficiency and accuracy in understanding complex dynamics.

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Area of Science:

  • Dynamical Systems Theory
  • Data-Driven Modeling
  • Computational Mathematics

Background:

  • Analyzing stochastic dynamical systems is crucial for understanding complex phenomena.
  • Existing methods for Koopman semigroup approximation face challenges with noise and computational cost.
  • A need exists for numerically stable and efficient frameworks.

Purpose of the Study:

  • Introduce Stochastic Dynamic Mode Decomposition (SDMD), a novel data-driven framework.
  • To approximate the Koopman semigroup in stochastic dynamical systems accurately and efficiently.
  • To address limitations of existing methods regarding numerical stability and computational expense.

Main Methods:

  • SDMD formulation explicitly incorporates sampling time for enhanced numerical stability and precision.
  • Direct approximation of the Koopman semigroup, bypassing computationally intensive matrix exponential calculations.
  • Integration of neural networks for automated basis selection, reducing manual intervention.

Main Results:

  • Theoretical convergence guarantees established across large data, infinitesimal sampling time, and increasing dictionary size limits.
  • Demonstrated effectiveness in capturing Koopman semigroup spectral properties in various canonical stochastic systems.
  • Successful application to oscillatory systems, mean-reverting processes, metastable systems, and neural mass models.

Conclusions:

  • SDMD provides a computationally efficient and numerically stable approach for analyzing stochastic dynamics.
  • The framework effectively handles complex random behavior in dynamical systems.
  • SDMD offers a practical pathway for advancing the understanding of stochastic systems through data-driven methods.