  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 dynamical
properties of complex systems are among central challenges. In this work, we
devise a nonparametric approach to learn the relative entropy rate from
observations of stochastic differential equations with different drift
functions.The estimator corresponding to the relative entropy rate then is
presented via the Gaussian process kernel theory. Meanwhile, this approach
enables to extract the governing equations. We illustrate our approach in
several examples. Numerical experiments show the proposed approach performs
well for rational drift functions, not only polynomial drift functions.
