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Bifurcation diagrams in estimated parameter space using a pruned extreme learning machine.

Yoshitaka Itoh1, Masaharu Adachi1

  • 1Department of Electrical and Electronic Engineering, Tokyo Denki University, 5 Senju-Asahicho Adachi-ku, Tokyo, Japan.

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Summary

We developed a new algorithm using a pruned extreme learning machine to estimate system parameter spaces, enabling prediction of behavior changes and visualization via bifurcation diagrams.

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Area of Science:

  • Computational Neuroscience
  • Dynamical Systems Theory
  • Machine Learning

Background:

  • Estimating parameter spaces is crucial for predicting system behavior and optimizing unknown systems.
  • Existing methods, like Bagarinao et al.'s, are limited to linear-in-parameter maps.
  • Principal Component Analysis (PCA) is often used but can be computationally intensive or introduce limitations.

Purpose of the Study:

  • To propose and validate a novel algorithm for estimating parameter spaces using a pruned extreme learning machine (PELM).
  • To extend parameter space estimation to nonlinear-in-parameter maps, overcoming limitations of prior work.
  • To visualize system dynamics by plotting bifurcation diagrams within these estimated parameter spaces.

Main Methods:

  • Utilizing a pruned extreme learning machine (PELM) for parameter space estimation without PCA.
  • Estimating the dimension of parameter spaces via singular values of trained synaptic weights.
  • Applying the PELM-based method to nonlinear-in-parameter maps, including real-world models.

Main Results:

  • Successfully estimated parameter spaces for various systems, including nonlinear ones.
  • Demonstrated the ability to plot accurate bifurcation diagrams in the estimated parameter spaces.
  • Validated the method on a mathematical model of induction motor drives and an ecosystem vegetation biomass model.

Conclusions:

  • The proposed PELM-based algorithm effectively estimates parameter spaces for both linear and nonlinear systems.
  • This method provides a powerful tool for predicting system behavior and visualizing complex dynamics.
  • The approach extends the applicability of parameter space estimation to a wider range of real-world problems.