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Generic and isometric embeddings in reservoir computers.

Allen G Hart1

  • 1Department of Mathematics, University of Warwick, Coventry CV4 7AL, United Kingdom.

Chaos (Woodbury, N.Y.)
|November 3, 2025
PubMed
Summary

We show that reservoir systems can achieve generalized synchronization, embedding the input system

Area of Science:

  • Dynamical Systems and Control Theory
  • Nonlinear Dynamics
  • Chaos Theory

Background:

  • Reservoir computing systems offer a powerful framework for modeling complex dynamics.
  • Generalized synchronization is a phenomenon where the dynamics of one system become constrained by another.
  • Understanding synchronization in high-dimensional systems is crucial for applications in signal processing and control.

Purpose of the Study:

  • To investigate the existence and nature of generalized synchronization in generic reservoir systems.
  • To determine conditions under which generalized synchronization corresponds to topological or isometric embeddings.
  • To provide explicit constructions for isometric embeddings in linear reservoir systems.

Main Methods:

  • Utilizing concepts from topological dynamics and embedding theorems.

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  • Applying Nash's embedding theorem to establish conditions for isometric embeddings.
  • Developing analytical methods for constructing embeddings in linear systems.

    • Main Results:

    • A generic reservoir system admits generalized synchronization as a topological embedding of the input system's attractor.
    • For sufficiently high reservoir dimensions, an isometric embedding generalized synchronization exists.
    • Explicit construction of the isometric embedding is possible for linear reservoir and source dynamics.

    Conclusions:

    • Generalized synchronization in reservoir systems can be understood as a form of embedding.
    • High dimensionality in reservoir systems facilitates stronger forms of synchronization (isometric embeddings).
    • The findings provide theoretical foundations for designing reservoir systems with predictable synchronization properties.