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 governingequations and quantifying
dynamical properties of complex systems are among central challenges.
Inthis work, we devise a nonparametric approach to learn the relative
entropy rate from observationsof stochastic differential equations with
different drift functions. The estimator corresponding tothe relative
entropy rate then is presented via the Gaussian process kernel theory.
Meanwhile, thisapproach 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,
notonly polynomial drift functions.