Dynamical analysis of a parameter-aware reservoir computer

TL;DR

This paper investigates how reservoir computers predict critical transitions in dynamical systems, revealing the underlying bifurcation mechanisms and phase space structures that influence their accuracy.

Contribution

It demonstrates that the reservoir's learned map undergoes bifurcations similar to the original system, linking machine learning predictions to dynamical system theory.

Findings

The reservoir map exhibits Neimark-Sacker bifurcation near the critical point.

The study compares phase space structures to understand prediction mechanisms.

The approach provides theoretical insight into machine learning-based critical transition prediction.

Abstract

Reservoir computing has been shown to be a useful framework for predicting critical transitions of a dynamical system if the bifurcation parameter is also provided as an input. Its utility is significant because in real-world scenarios, the exact model equations are unknown. This Letter shows how the theory of dynamical system provides the underlying mechanism behind the prediction. Using numerical methods, by considering dynamical systems which show Hopf bifurcation, we demonstrate that the map produced by the reservoir after a successful training undergoes a Neimark-Sacker bifurcation such that the critical point of the map is in immediate proximity to that of the original dynamical system. In addition, we have compared and analyzed different structures in the phase space. Our findings provide insight into the functioning of machine learning algorithms for predicting critical transitions.