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Reservoir computing on manifolds.

Masato Hara1, Hiroshi Kokubu2

  • 1School of Business Administration, Hitotsubashi University, 2-1 Naka, Kunitachi, Tokyo 186-8601, Japan.

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This study introduces a novel reservoir computing method for chaotic time series analysis on complex manifolds. The approach effectively learns dynamical systems beyond traditional Euclidean spaces.

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Area of Science:

  • Dynamical systems theory
  • Machine learning
  • Nonlinear dynamics

Background:

  • Reservoir computing is a powerful technique for analyzing chaotic time series.
  • Current methods are primarily suited for dynamical systems on Euclidean spaces.
  • Generalizing reservoir computing to manifolds is crucial for broader applications.

Purpose of the Study:

  • To propose a novel reservoir computing approach for dynamical systems on general manifolds.
  • To extend existing reservoir computing methodologies to non-Euclidean geometries.
  • To demonstrate the efficacy of the proposed method on complex dynamical systems.

Main Methods:

  • Developed a new reservoir computing framework adapted for manifold-based dynamical systems.
  • Applied the method to learn the hyperbolic toral automorphism.
  • Tested the approach on the tripling map dynamical system on a circle.

Main Results:

  • The proposed reservoir computing method effectively learns chaotic time series on general manifolds.
  • Numerical results confirm the method's performance on the hyperbolic toral automorphism.
  • The tripling map on the circle was successfully learned, validating the approach's versatility.

Conclusions:

  • The novel reservoir computing approach offers a significant advancement for analyzing complex dynamical systems.
  • This method provides a robust framework for extending reservoir computing to non-Euclidean settings.
  • The demonstrated effectiveness on manifold-based systems opens new avenues in time series analysis.