Emergence of transient chaos and intermittency in machine learning

An emerging paradigm for predicting the state evolution of chaotic systems is machine learning with reservoir computing, the core of which is a dynamical network of artificial neurons. Through training with measured time series, a reservoir machine can be harnessed to replicate the evolution of the target chaotic system for some amount of time, typically about half dozen Lyapunov times. Recently, we developed a reservoir computing framework with an additional parameter channel for predicting system collapse and chaotic transients associated with crisis. It was found that the crisis point after which transient chaos emerges can be accurately predicted. The idea of adding a parameter channel to reservoir computing has also been used by others to predict bifurcation points and distinct asymptotic behaviors. In this paper, we address three issues associated with machine-generated transient chaos. First, we report the results from a detailed study of the statistical behaviors of transient chaos generated by our parameter-aware reservoir computing machine. When multiple time series from a small number of distinct values of the bifurcation parameter, all in the regime of attracting chaos, are deployed to train the reservoir machine, it can generate the correct dynamical behavior in the regime of transient chaos of the target system in the sense that the basic statistical features of the machine generated transient chaos agree with those of the real system. Second, we demonstrate that our machine learning framework can reproduce intermittency of the target system. Third, we consider a system for which the known methods of sparse optimization fail to predict crisis and demonstrate that our reservoir computing scheme can solve this problem. These findings have potential applications in anticipating system collapse as induced by, e.g., a parameter drift that places the system in a transient regime.