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Dynamic mode decomposition in vector-valued reproducing kernel Hilbert spaces for extracting dynamical structure

Keisuke Fujii1, Yoshinobu Kawahara2

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Summary

This study introduces a new method for analyzing complex nonlinear dynamical systems (NLDSs) with structured data, like graphs. The approach extends dynamic mode decomposition (DMD) for better understanding of underlying system dynamics.

Keywords:
Dimensionality reductionDynamical systemsSpectral analysisUnsupervised learning

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Area of Science:

  • Mathematics
  • Physics
  • Computer Science
  • Engineering

Background:

  • Nonlinear dynamical systems (NLDSs) are prevalent in science and engineering.
  • Dynamic mode decomposition (DMD) offers insights into NLDSs by analyzing Koopman operator spectra.
  • Existing DMD methods struggle with data exhibiting dependent structures, such as graph sequences.

Purpose of the Study:

  • To develop a Koopman spectral analysis framework for NLDSs with structured observables.
  • To propose an estimation algorithm applicable to data with inherent dependencies.
  • To enable visualization and understanding of low-dimensional global dynamics in such systems.

Main Methods:

  • Formulation of Koopman spectral analysis in vector-valued reproducing kernel Hilbert spaces.
  • Development of a tensor-based dynamic mode decomposition reformulation.
  • Introduction of Graph DMD for analyzing graph dynamical systems using adjacency matrices.

Main Results:

  • The proposed method effectively extracts and visualizes low-dimensional dynamics from structured NLDS data.
  • Graph DMD, a specialized case, demonstrates applicability to graph-structured data.
  • Empirical validation using both synthetic and real-world datasets confirms method performance.

Conclusions:

  • The novel approach extends DMD capabilities to NLDSs with structured observables.
  • This facilitates a deeper understanding of complex systems with dependent data.
  • The method provides a powerful tool for analyzing graph dynamical systems and similar structures.