Full Lyapunov exponents spectrum with Deep Learning from single-variable time series

In this article we study if a Deep Learning technique can be used to obtain an approximate value of the Lyapunov exponents of a dynamical system. Moreover, we want to know if Machine Learning techniques are able, once trained, to provide the full Lyapunov exponents spectrum with just single-variable time series. We train a Convolutional Neural Network and use the resulting network to approximate the full spectrum using the time series of just one variable from the studied systems (Lorenz system and coupled Lorenz system). The results are quite surprising since all the values are well approximated with only partial data. This strategy allows to speed up the complete analysis of the systems and also to study the hyperchaotic dynamics in the coupled Lorenz system.