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The spatiotemporal coupling in delay-coordinates dynamic mode decomposition.

Emil Bronstein1, Aviad Wiegner2, Doron Shilo1

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Dynamic mode decomposition (DMD) with delay-coordinates enhances dynamical systems analysis. A novel method addresses spatiotemporal coupling in DMD components for superior selection and reduced-order modeling.

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Area of Science:

  • Dynamical Systems Analysis
  • Data-Driven Science
  • Fluid Dynamics

Background:

  • Dynamic Mode Decomposition (DMD) is crucial for analyzing complex, high-dimensional dynamical systems without explicit equations.
  • Standard DMD separates temporal and spatial information, providing a spectral representation via eigenvalues and modes.
  • Delay-coordinates embedding augments observations with past time steps, expanding DMD's applicability.

Purpose of the Study:

  • To investigate the spectral structure arising from applying DMD to delay-coordinates embeddings.
  • To introduce and analyze the phenomenon of spatiotemporal coupling in delay-coordinates DMD.
  • To develop an improved method for selecting DMD components in delay-coordinates analysis.

Main Methods:

  • Augmenting observational data using delay-coordinates embedding.
  • Applying Dynamic Mode Decomposition to the augmented data.
  • Formulating and analyzing the spatiotemporal coupling in the resulting DMD components.
  • Developing a novel component selection strategy based on the identified spatiotemporal coupling.

Main Results:

  • Identified and analyzed a novel spectral structure termed 'spatiotemporal coupling' in delay-coordinates DMD.
  • Proposed a new method for selecting essential DMD components from redundant sets.
  • Demonstrated superior component selection performance compared to amplitude-based methods.
  • Validated the method on noisy simulated signals and diverse dynamical systems.

Conclusions:

  • Spatiotemporal coupling is an inherent characteristic of delay-coordinates DMD that impacts component interpretation.
  • The proposed component selection method effectively extracts meaningful dynamics from delay-coordinates DMD.
  • This approach facilitates more accurate and compact reduced-order modeling of complex systems.