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 capture both fine scale posture dynamics and large scale effective diffusion in the worms centroid trajectories and introduce a top-down, operator-based clustering which reveals subtle subdivisions of the "run" behavior. POPULAR SUMMARYComplex structure is often composed from a limited set of relatively simple building blocks; such as novels from letters or proteins from amino acids. In musical composition, e.g., sounds and silences combine to form longer time scale structures; motifs form passages which in turn form movements. The challenge we address is how to identify collective variables which distinguish structures across such disparate time scales. We introduce a principled framework for learning effective descriptions directly from observations. Just as a musical piece transitions from one movement to the next, the collective dynamics we infer consists of transitions between macroscopic states, like jumps between metastable states in an effective potential landscape. The statistics of these transitions are captured compactly by transfer operators. These operators play a central role, guiding the construction of maximally-predictive short-time states from incomplete measurements and identifying collective modes via eigenvalue decomposition. We demonstrate our analysis in both stochastic and deterministic systems, and with an application to the movement dynamics of an entire organism, unravelling new insight in long time scale behavioral states directly from measurements of posture dynamics. We can, in principle, also make connections to both longer or shorter timescales. Microscopically, postural dynamics result from the fine scale interactions of actin and myosin in the muscles, and from electrical impulses in the brain and nervous system. Macroscopically, behavioral dynamics may be extended to longer time scales, to moods or dispositions, including changes during aging, or over generations due to ecological or evolutionary adaptation. The generality of our approach provides opportunity for insights on long term dynamics within a wide variety of complex systems.