Using machine-learning modeling to understand macroscopic dynamics in a system of coupled maps

Machine-learning techniques not only offer efficient tools for modeling dynamical systems from data but can also be employed as frontline investigative instruments for the underlying physics. Nontrivial information about the original dynamics, which would otherwise require sophisticated ad hoc techniques, can be obtained by a careful usage of such methods. To illustrate this point, we consider as a case study the macroscopic motion emerging from a system of globally coupled maps. We build a coarse-grained Markov process for the macroscopic dynamics both with a machine-learning approach and with a direct numerical computation of the transition probability of the coarse-grained process, and we compare the outcomes of the two analyses. Our purpose is twofold: on the one hand, we want to test the ability of the stochastic machine-learning approach to describe nontrivial evolution laws as the one considered in our study. On the other hand, we aim to gain some insight into the physics of the macroscopic dynamics. By modulating the information available to the network, we are able to infer important information about the effective dimension of the attractor, the persistence of memory effects, and the multiscale structure of the dynamics.