Learning transitions to extreme events using reservoir computing

Predicting extreme events is a challenging task due to their occasional appearance at irregular time intervals and with sudden large amplitudes. In particular, accurate forecasting of both the amplitude and timing of occurrence is difficult. We make an attempt to address the challenges using reservoir computing machine learning based on partial or complete information of the system variables in a few paradigmatic dynamical systems, namely, the forced Liénard system, the coupled FitzHugh-Nagumo model, and a model of hidden attractor. The efficacy of the machine learning approach has been tested using numerically generated data using models and a real-time experiment. The machine is able to successfully predict the transition points to extreme events via two well-known nonlinear processes, namely, Pomeau-Manneville intermittency and crisis-induced intermittency (interior crisis) against a system parameter by reconstruction of the bifurcation diagram by training the machine with simulated data. The machine is able to retrace the attractors of the systems quite efficiently and also reproduce the distributions of events and interevent intervals, thereby preserving the statistical properties of extreme events in the model systems. Prediction of time evolution is still limited to the length of a few Lyapunov times before it diverges from the original state as expected due to the complex nature of the extreme event dynamics. The amplitude of extreme events in the time series is preserved or predicted almost accurately. A prediction horizon metric is used for measuring the success level of the machine and deciding the minimal number of inputs necessary for efficient learning and prediction of the forced Liénard system and the hidden attractor model. For the coupled FitzHugh-Nagumo model, we use the mean square error measure to find an appropriate choice of a pair of inputs that is necessary for successful prediction. Using experimental data from an analog forced Liénard circuit, so far the machine could learn the dynamics and the long-term statistics of extreme events originating via Pomeau-Manneville intermittency.