Machine tells you how many variables are at least needed to describe space-time chaos you see

The Lyapunov exponents and the associated Lyapunov vectors serve as central quantities to characterize various aspects of spatiotemporal chaos, such as the instability, the fractal dimension, and even the inertial manifold dimension, which indicates the smallest possible number of variables needed to describe the trajectory. Nevertheless, such Lyapunov analysis has remained an immense challenge for large experimental systems, even after Pathak et al. (2017) numerically demonstrated that a machine learning technique, specifically the reservoir computing, may be used to obtain the Lyapunov spectrum from space-time data. Here we first show that the reservoir computing can also predict the inertial manifold dimension from data, as we demonstrate numerically for the Kuramoto-Sivashinsky equation. Then we test this approach with real experimental data of collective motion of bacteria, and evaluate the results through comparison with a phenomenological equation used in this context.