Model-free prediction of multistability using echo state network

TL;DR

This paper demonstrates that echo state networks can accurately predict and analyze multistable system dynamics and bifurcation diagrams across different parameters, even with limited training data.

Contribution

It introduces a data-driven ESN approach capable of predicting multiple attractors and bifurcation diagrams in multistable systems from minimal training.

Findings

ESN accurately predicts dynamics at distant parameters

The method captures entire bifurcation diagrams with high precision

It identifies basins of attraction for co-existing attractors

Abstract

In the field of complex dynamics, multistable attractors have been gaining a significant attention due to its unpredictability in occurrence and extreme sensitivity to initial conditions. Co-existing attractors are abundant in diverse systems ranging from climate to finance, ecological to social systems. In this article, we investigate a data-driven approach to infer different dynamics of a multistable system using echo state network (ESN). We start with a parameter-aware reservoir and predict diverse dynamics for different parameter values. Interestingly, machine is able to reproduce the dynamics almost perfectly even at distant parameters which lie considerably far from the parameter values related to the training dynamics. In continuation, we can predict whole bifurcation diagram significant accuracy as well. We extend this study for exploring various dynamics of multistable attractors at unknown parameter value. While, we train the machine with the dynamics of only one attarctor at parameter $p$, it can capture the dynamics of co-existing attractor at a new parameter value $p+\Delta p$. Continuing the simulation for multiple set of initial conditions, we can identify the basins for different attractors. We generalize the results by applying the scheme on two distinct multistable systems.