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Area of Science:

  • Dynamical Systems Theory
  • Machine Learning
  • Control Theory

Background:

  • Koopman operator theory offers a data-driven method to linearize nonlinear dynamics.
  • Identifying Koopman eigenfunctions, which provide these linearizing coordinates, is computationally challenging.
  • Existing methods struggle with representing eigenfunctions, especially for systems with continuous spectra.

Purpose of the Study:

  • To leverage deep learning for discovering parsimonious and interpretable representations of Koopman eigenfunctions.
  • To identify nonlinear coordinate transformations that globally linearize complex dynamics.
  • To generalize Koopman representations for systems with continuous spectra.

Main Methods:

  • A modified auto-encoder network is employed to discover low-dimensional embeddings of dynamics.
  • Deep learning models are utilized to learn representations of Koopman eigenfunctions from data.
  • An auxiliary network is introduced to parameterize continuous frequencies for systems with continuous spectra.

Main Results:

  • The developed deep learning framework successfully identifies nonlinear coordinates that linearize dynamics.
  • The approach provides a compact, efficient, and interpretable embedding of dynamical systems.
  • The generalization to continuous spectra connects deep learning models with established asymptotic theories.

Conclusions:

  • Deep learning offers a powerful tool for discovering Koopman eigenfunctions and linearizing nonlinear dynamics.
  • The proposed method enhances the practical application of Koopman operator theory in prediction, estimation, and control.
  • This work bridges the gap between data-driven methods and physical interpretability in dynamical systems analysis.