Anticipating synchronization with machine learning

In realistic systems of coupled oscillators, it is desired to predict the onset of synchronization where the system equations are unknown, raising the need to develop a prediction framework that is model free and fully data driven. We show that this challenging problem can be addressed with machine learning. In particular, exploiting reservoir computing or echo state networks, we employ a "parameter-aware"scheme to train the neural machine using time series acquired from a small number of distinct asynchronous states in the parameter regime prior to the onset of synchronization. The trained machine can then be used to predict the synchronization transition through tuning the control parameter. We demonstrate the power of the machine learning-based framework using two types of synchronization behaviors: Complete synchronization in coupled identical chaotic oscillators and the phase synchronization in coupled nonidentical phase oscillators, which are representative of the collective dynamics in coupled systems. In addition, we design our numerical experiments such that two transition scenarios are covered: Smooth (second-order) and explosive (first-order) transitions that represent the generic types of phase transition in nonlinear physical systems. A remarkable feature is that, for the network systems exhibiting explosive (first-order) transition, the machine learning scheme is capable of predicting not only the locations of the transition points associated with the forward and backward transition paths but also the hysteresis between the two paths.