Maximally predictive ensemble dynamics from data

We leverage the interplay between microscopic variability and macroscopic order to connect physical descriptions across scales directly from data, without underlying equations. We reconstruct a state space by concatenating measurements in time, building a maximum entropy partition of the resulting sequences, and choosing the sequence length to maximize predictive information. Trading non-linear trajectories for linear, ensemble evolution, we analyze reconstructed dynamics through transfer operators. The evolution is parameterized by a transition time $\tau$: capturing the source entropy rate at small $\tau$ and revealing timescale separation with collective, coherent states through the operator spectrum at larger $\tau$. Applicable to both deterministic and stochastic systems, we illustrate our approach through the Langevin dynamics of a particle in a double-well potential and the Lorenz system. Applied to the behavior of the nematode worm $C.$ $elegans$, we derive a "run-and-pirouette" navigation strategy directly from posture dynamics. We demonstrate how sequences simulated from the ensemble evolution recover effective diffusion in the worm's centroid trajectories and introduce a top-down, operator-based clustering which reveals subtle subdivisions of the "run" behavior.