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

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.