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Constructing polynomial libraries for reservoir computing in nonlinear dynamical system forecasting.

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This study enhances reservoir computing for predicting chaotic systems by creating a linear Hilbert space using polynomial transformations. This method improves the representation of complex nonlinear dynamics for more dependable state-evolution predictions.

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Area of Science:

  • Dynamical Systems and Control Theory
  • Computational Physics
  • Machine Learning

Background:

  • Reservoir computing effectively models nonlinear and chaotic dynamical systems.
  • Predicting the evolution of these systems reliably remains a significant challenge.
  • Koopman operator theory and sparse identification of nonlinear dynamics (SINDy) offer frameworks for analyzing nonlinear systems.

Purpose of the Study:

  • To propose an alternative reservoir computing method for improved prediction of nonlinear and chaotic dynamical systems.
  • To enhance the nonlinear representation of reservoir states by incorporating nonlinear terms into the optimization process.
  • To develop a theoretically and practically applicable method for state-evolution prediction.

Main Methods:

  • Developed a reservoir computing approach based on Koopman operator theory.
  • Integrated nonlinear terms into the reservoir computing optimization to create a linear Hilbert space.
  • Employed polynomial transformations of arbitrary order for fitting the readout matrix.
  • Constructed polynomial libraries using reservoir-state vectors.

Main Results:

  • The proposed method successfully enhances the nonlinear representation of reservoir states.
  • Polynomial libraries effectively capture the complexity of nonlinear systems.
  • Validated effectiveness on the Lorenz-63, Lorenz-96, and Kuramoto-Sivashinsky systems for state-evolution prediction.

Conclusions:

  • The novel reservoir computing approach provides a more dependable method for predicting nonlinear and chaotic dynamical systems.
  • Incorporating polynomial libraries advances the theoretical understanding of reservoir computing.
  • Offers a practical framework for analyzing and predicting complex system dynamics.