Jacobian analysis of a parameter aware reservoir computer

Abstract

This thesis delves into the intersection of non-linear dynamics and network science to gain deeper insights into parameter-aware reservoir computing—a machine learning algorithm renowned for its ability to predict time series data of non-linear dynamical systems at different parameter values. Despite its success, reservoir computing is often regarded as a black box, with its internal mechanisms and theoretical foundations remaining largely unexplored. This work seeks to shed light on how the algorithm processes information and makes predictions, bridging the gap between empirical success and theoretical understanding. The work conducted thus far focuses on developing a parameter-aware reservoir computing (PARC) model and leveraging it to predict the behaviour of various nonlinear dynamical systems- Logistic map, Higher Order Kuramoto oscillator network and coupled Stuart Landau oscillators. A Bayesian optimisation algorithm has been developed to efficiently determine the optimal hyperparameters for the PARC model to make successful predictions. Additionally, the PARC model has been analysed as a network of interconnected maps. jacobian analysis of the reservoir network is conducted to understand the mechanism utilised by the machine learning model to predict the behaviour of the systems in different parameter regimes, we do this by mapping the bifurcation seen in the RC network to known bifurcations in map networks. Future work may focus on analysing the behaviour of the PARC network when predicting more complex datasets, such as period-2 to period-4 dynamics, and identifying the underlying network dynamics that enable accurate predictions. Additionally, a better understanding of how the machine learning network works internally can help build a stronger theoretical foundation for creating more efficient algorithms. It can also make it easier to find the best hyperparameters to improve its overall performance.