Extracting governing physics from data is a key challenge in many areas of science and technology. The existing techniques for equation discovery are mostly applicable to deterministic systems and require both input and state measurements. We here propose a novel data-driven framework for discovering nonlinear stochastic dynamical systems with Gaussian white noise. The proposed framework blends concepts of stochastic calculus, sparse learning algorithms, and Bayesian statistics to learn the governing physics from data. In particular, we combine sparsity–promoting spike and slab prior, Bayes law, and the Kramers–Moyal formula to identify stochastic differential equations from data. The proposed framework is highly efficient and works with sparse, noisy, and incomplete output measurements. The efficacy and robustness of the proposed approach are illustrated in several numerical examples involving both complete and partial state measurements. The results obtained indicate the potential of the proposed approach in discovering nonlinear stochastic dynamical systems subjected to Gaussian white noise excitation.

从数据中提取控制物理学是许多科学和技术领域的关键挑战。现有的方程发现技术主要适用于确定性系统，需要输入和状态测量。我们在这里提出了一种新的数据驱动框架，用于发现具有高斯白噪声的非线性随机动力系统。所提出的框架融合了随机微积分、稀疏学习算法和贝叶斯统计的概念，以从数据中学习支配物理。特别是，我们结合了稀疏性促进尖峰和平板先验、贝叶斯定律和 Kramers-Moyal 公式来从数据中识别随机微分方程。所提出的框架非常高效，适用于稀疏、嘈杂和不完整的输出测量。在涉及完整和部分状态测量的几个数值示例中说明了所提出方法的有效性和稳健性。获得的结果表明所提出的方法在发现受高斯白噪声激励的非线性随机动力系统方面的潜力。