Data-driven approach to the deep learning of the dynamics of a non-integrable Hamiltonian system

The dynamics of non-integrable Hamiltonian systems, described by area-preserving mappings, are regulated by the KAM theorem. This states that the phase space of the system is made up of interwoven sets of regular and chaotic dynamics, whose extent depends on a chaoticity parameter k. The chaoticity parameter measures the degree of non-integrability of the Hamiltonian; the extent of regular orbits decreases as the non-integrable contribution increases. Deep learning is proving increasingly useful in predicting natural phenomena from a set of data, even for chaotic time series forecasting. In this paper we investigate numerical simulations of the standard map with different values of k, as a learning process of a typical non-integrable Hamiltonian system. Our aim is to investigate to what extent a deep learning process is able to recognize the degree of non-integrability of a Hamiltonian system, namely to forecast the actual value of the chaoticity parameter, using data obtained from the same system used in the learning process. Results show that, in general, forecasting the chaoticity parameter is far from being guaranteed, because the KAM theorem is at work. However, the accuracy of the forecasting process depends on both the number of initial conditions and the length of the trajectories used in the learning process. The maximum of the forecasting accuracy is obtained for intermediate values of k, when the phase space is formed by roughly equally spaced regular and irregular trajectories. On the contrary, both relatively low values of k (prevalence of regular orbits) and high values of k (prevalence of irregular orbits), are more difficult to predict. According to our results, a standard deep learning process has difficulty distinguishing between regular and slightly irregular dynamics, and between a purely stochastic system and a system with residual regular orbits.