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Area of Science:

  • Nonlinear dynamics
  • Data-driven modeling
  • Time series analysis

Background:

  • Delay embeddings offer a powerful coordinate basis for approximating nonlinear dynamics using linear methods.
  • Dynamic Mode Decomposition (DMD) is effective for finite-dimensional Koopman approximations in delay coordinates.
  • Existing methods like DMD with control (DMDc) require prior knowledge of control signals.

Purpose of the Study:

  • To demonstrate nonlinear dynamics modeling in delay coordinates using DMD.
  • To develop a method for unsupervised discovery of exogenous forcing signals.
  • To extend DMDc to scenarios with unknown control inputs.

Main Methods:

  • Representing nonlinear dynamics with sparse Fourier spectra as principal component trajectories.
  • Applying DMD in delay coordinate space for modeling.
  • Augmenting DMD with an external forcing term for continuous or mixed spectra.
  • Developing an unsupervised method for learning linear control models and discovering forcing signals simultaneously.

Main Results:

  • Nonlinear dynamics with sparse Fourier spectra can be modeled by DMD in delay coordinates.
  • A novel unsupervised method successfully learns linear control models and identifies exogenous forcing signals.
  • The learned forcing signals are validated against ground truth and show statistical similarity.
  • The method is demonstrated on power grid load data, highlighting its diagnostic utility.

Conclusions:

  • The proposed method effectively models nonlinear dynamics and identifies unknown forcing in complex systems.
  • This unsupervised approach extends DMDc capabilities to scenarios lacking prior control signal information.
  • The application to power grid data showcases the method's practical utility for system diagnostics and interpretation.