Forecasting turbulence in a passive resonator with supervised machine learning

Predicting complex nonlinear dynamical systems has been even more urgent because of the emergence of extreme events such as earthquakes, volcanic eruptions, extreme weather events (lightning, hurricanes/cyclones, blizzards, tornadoes), and giant oceanic rogue waves, to mention a few. The recent milestones in the machine learning framework offer a new prospect in this area [1] , [2] . For a high dimensional chaotic system, increasing the system’s size causes an augmentation of the complexity and, finally, more nodes in the network. Here, we propose a new supervised machine learning strategy to forecast bursts occurring in the turbulent regime of a fiber ring cavity. Indeed, we have recently demonstrated [3] that this system can continuously transform a stable periodic pattern with a well-defined frequency comb into a turbulent state via a spatiotemporal intermittency mechanism. Figure 1a ) shows an illustration of this transition. Even though a turbulent evolutions, an interesting feature is the persistent long-range correlation in the dynamics [3] . The system can then be seen as adjacent subdomains. Owing to this feature, instead of predicting the whole system, we use appropriate tools of chaos theory to identify recurrent picture in the past at the same location of the bursts. Figure 1b ) shows the statistics on the bursts (red) and the profiles detected at a previously computed time lag (blue). We have taken advantage of the apparent causality to make an association precursors-pulses. Knowing when and where the bursts may emerge, a fair question is " What is coming? ". On the other hand, as shown in Figure 1c ), no correlation can be found between the profile (amplitude and size) of the precursors of the corresponding pulse. However, using the pulses/precursors pairs to perform supervised learning with a recurrent neural network, we have obtained a high correlated map between the actual observation and the prediction, as can be seen in Figure 1d ). As our strategy deals with intrinsic characteristic quantities of the dynamical behavior, the pre-trained network can trigger the forecasting even for a more extensive system, provided that all the other parameters are the same.