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  1. The information detection of complex systems from data is currently undergoing a revolution, driven by the emergence of big data and machine learning methodology. Discovering governing equations and quantifying the dynamical properties of complex systems are among the central challenges. In this work, we devised a nonparametric approach to learning the relative entropy rate from observations of stochastic differential equations with different drift functions. The estimator corresponding to the relative entropy rate is then presented via the Gaussian process kernel theory. Meanwhile, this approach enables us to extract the governing equations. We illustrate our approach with several examples. Numerical experiments show the proposed approach performs well for rational drift functions, not only polynomial drift functions.

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