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The information detection of complex systems from data is currentlyundergoing a revolution, driven by the emergence of big data and machinelearning methodology. Discovering governing equations and quantifying dynamicalproperties of complex systems are among central challenges. In this work, wedevise a nonparametric approach to learn the relative entropy rate fromobservations of stochastic differential equations with different driftfunctions.The estimator corresponding to the relative entropy rate then ispresented via the Gaussian process kernel theory. Meanwhile, this approachenables to extract the governing equations. We illustrate our approach inseveral examples. Numerical experiments show the proposed approach performswell for rational drift functions, not only polynomial drift functions.

Nonparametric inference of stochastic differential equations based on the relative entropy rate