Maximally predictive ensemble dynamics from data

We leverage the interplay between microscopic variability and macroscopic order, fundamental to statistical physics, to extract predictive coarse-grained dynamics from data. We define a dynamical state as a sequence of measurements, partition the resulting space, and choose the sequence length to maximize predictive information. We approximate the dynamics of densities in the partitioned space through transfer operators, providing simple, yet accurate, models on multiple scales. The operator spectrum provides a principled means of timescale separation and coarse-graining. Applicable to both deterministic and stochastic systems, we illustrate our approach in the Langevin dynamics of a particle in a double-well potential and the Lorenz system. As an example where the fundamental dynamics are unknown, we consider high-resolution posture measurements of the nematode C. elegans. We show that a long-time (10's of s) ``run’' and ``pirouette’' description of navigation naturally emerges from short-time (10's of ms) posture samples.