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This study introduces an algebraic approach to delay embedding, enabling accurate approximation of errors in complex nonlinear dynamics. This method can be directly implemented using recurrent neural networks (RNNs), enhancing interpretability and structure incorporation.

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Area of Science:

  • Dynamical Systems and Control Theory
  • Machine Learning and Artificial Intelligence
  • Nonlinear Dynamics

Background:

  • Complex nonlinear dynamics are prevalent across scientific disciplines.
  • Often, not all state variables governing these dynamics are observable.
  • Delay embedding is a technique to reconstruct dynamics from limited observations.

Purpose of the Study:

  • To develop an algebraic method for delay embedding that quantifies approximation error.
  • To explore the relationship between approximation error and system size.
  • To integrate delay embedding with recurrent neural networks (RNNs) for enhanced analysis.

Main Methods:

  • An algebraic formulation of delay embedding was developed.
  • The asymptotic dependence of first-order approximation error on system size was derived.
  • The proposed delay embedding method was implemented using RNNs.

Main Results:

  • An explicit approximation of error in delay embedding was achieved.
  • The study provides the asymptotic dependence of approximation error on system size.
  • A direct implementation of delay embedding using RNNs was demonstrated.

Conclusions:

  • The algebraic approach offers a principled way to handle unobserved variables in nonlinear dynamics.
  • The integration with RNNs enhances the interpretability of both delay embedding and neural networks.
  • This framework allows for the incorporation of structural information and constraints into dynamic system analysis.