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Enhancing predictive capabilities in data-driven dynamical modeling with automatic differentiation: Koopman and

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Area of Science:

  • Dynamical Systems and Control Theory
  • Machine Learning for Scientific Discovery
  • Computational Physics and Engineering

Background:

  • Predicting the time evolution of complex dynamical systems is crucial in science and engineering.
  • Data-driven Koopman operator approximations offer a powerful framework for analyzing nonlinear dynamics.
  • Extended Dynamic Mode Decomposition with Dictionary Learning (EDMD-DL) is a prominent but improvable method.

Purpose of the Study:

  • To develop a modified EDMD-DL approach that simultaneously optimizes the dictionary of observables and the Koopman operator approximation.
  • To evaluate the performance of the proposed method against various Koopman-based and state-space approaches.
  • To assess predictive accuracy across diverse dynamical systems, including ODEs and PDEs with complex attractors.

Main Methods:

  • Introduced a novel EDMD-DL variant utilizing automatic differentiation for gradient-based optimization via the pseudoinverse.
  • Compared the proposed method with a 'pure' Koopman approach (time-integration in observable space).
  • Evaluated an alternating state-observable space Koopman approach and a neural ordinary differential equation (state-space) method.

Main Results:

  • The proposed modified EDMD-DL framework significantly outperformed the standard EDMD-DL.
  • The state-space approach demonstrated superior predictive performance over the 'pure' Koopman method.
  • The alternating state-observable space Koopman approach achieved prediction accuracy comparable to the state-space method.

Conclusions:

  • The developed data-driven framework enhances Koopman operator approximation for complex system dynamics.
  • The study highlights the trade-offs between different Koopman operator implementations and state-space models.
  • This work provides a more robust and accurate tool for predicting the time evolution of intricate dynamical systems.