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.

翻译：通过数据对复杂系统进行信息探测目前正在经历一场革命,其动力是大数据和机器学习方法的出现。发现管理方程式和量化复杂系统的动态特性是中心挑战之一。在这项工作中,我们设计了一种非参数方法,从观测具有不同漂移功能的随机差异方程式的观测中学习相对的倍增率。与当时相对的倍增率相应的估计值通过高西亚进程内核理论提出。与此同时,这一方法可以提取调节方程式。我们用几个例子来说明我们的方法。数字实验表明,拟议方法在合理漂移功能方面运行良好,而不仅仅是多数值漂移功能。