Learning slow manifolds from time order

Time series are ubiquitous, from climate science, healthcare to economics. The high dimensionality, complex correlations, and multiscale dynamics in rich time-series data challenge expert intuition and drive the development of machine-learning approaches. Here, we ask what information is encoded in time order and how to retrieve it. Within the Fokker-Planck formalism, we show that time order encodes the eigenfunctions and eigenvalues of the time-evolution operator. Time-order information is concentrated in the slowest nontrivial modes and blind to the steady-state distribution. Machine learning of time order thus efficiently extracts the slow manifold in high-dimensional spaces. Applied to molecular dynamics simulations, agnostic time-order learning identifies translational and rotational diffusion, and the slow modes of the protein folding dynamics. Time-order learning is general and scalable. From a grayscale video of a protein, the folding rate and mechanism is recovered. For α-hemolysin nanopores, the learned complex eigenvalues and eigenvectors quantify the non-equilibrium ion conductance and encode the conduction mechanism. Order learning extends to non-Markovian dynamics and to order beyond time. Extracting the rich information encoded in order enables agnostic, scalable learning of core mechanisms as a basis for understanding, enhanced sampling, and prediction of time-series data.