Data-driven discovery of slow–fast stochastic differential equations with data binning and sparse bayesian strategy

Slow–fast stochastic differential equations (SDEs) are extensively utilized in the mathematical modeling of dynamical systems exhibiting multiple time scales. Due to a larger sampling interval tailored to the slow time scale, it is challenging to discover the slow–fast SDEs from data, particularly when internal or external noise is present. This paper proposes a data-driven modeling method to learn the slow–fast SDEs from observation data. The proposed method first introduces a data binning strategy to preprocess the data, which are measured in different time scales, while the Kramers–Moyal expansion theory is leveraged to construct the time sequence describing the drift and diffusion terms in SDEs. Then, a large candidate library is created to represent the slow–fast dynamics of SDEs and a sparse Bayesian learning algorithm is proposed to solve the resulting sparse regression problem of inferring the active feature terms in the drift and diffusion terms and their corresponding parameters from data. Finally, several numerical experiments are provided to illustrate the effectiveness and feasibility of the proposed method for inferring slow–fast SDEs from data. As a consequence, it is observed that the data binning facilitates mitigating the imbalance between the fast component and the slow one.