Entropic transfer operators for data-driven analysis of dynamical systems

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The analysis of complex dynamical systems plays a central role in many fields, e.g. atmospheric flows in meteorology, turbulent flows in engineering, and large molecules in biochemistry. Systems of interest are typically chaotic, stochastic, and high-dimensional, but sometimes an approximate low-dimensional description exists, e.g. by metastable conformations of a molecule and their transition rates. Methods for systematic identification of such low-dimensional approximations are therefore highly relevant. Mathematically dynamical systems can be modelled via their transfer- and Koopman operators. They provide information on time-scale separation, metastable states, and for dimensionality reduction. In practice the operators must be approximated numerically and estimated from data. A broad range of corresponding methods has been developed and successfully applied in the natural sciences. But discretization and choice of basis functions remain challenging, in particular in high dimensions. We have recently introduced entropic transfer operators, a novel way to estimate an explicit discrete Markov matrix from Lagrangian data. Discretization artefacts and finiteness of the available data are remedied by blurring with entropic optimal transport. The latter is becoming an increasingly popular and numerically mature tool for data analysis. We have shown that in the limit of increasing available data the discrete operator converges to a blurred version of the true operator and allows to recover features of the true operator above the blur length scale. The method is mesh free and the only parameter, the blur length scale, allows for a trade-off of available data and resolution scale of the analysis. Consequently we believe that it will be a valuable addition to the toolbox of data-driven dynamic system analysis. We will generalize the method in various directions, derive convergence rates, and investigate its potential for combination with other recent methods from dynamic system analysis.