With the rapid increase of valuable observational, experimental and simulating data for complex systems, much effort is being devoted to discovering governing laws underlying the evolution of these systems. However, the existing techniques are limited to extract governing laws from data as either deterministic differential equations or stochastic differential equations with Gaussian noise. In the present work, we develop a new data-driven approach to extract stochastic dynamical systems with non-Gaussian symmetric Lévy noise, as well as Gaussian noise. First, we establish a feasible theoretical framework, by expressing the drift coefficient, diffusion coefficient and jump measure (i.e., anomalous diffusion) for the underlying stochastic dynamical system in terms of sample paths data. We then design a numerical algorithm to compute the drift, diffusion coefficient and jump measure, and thus extract a governing stochastic differential equation with Gaussian and non-Gaussian noise. Finally, we demonstrate the efficacy and accuracy of our approach by applying to several prototypical one-, two- and three-dimensional systems. This new approach will become a tool in discovering governing dynamical laws from noisy data sets, from observing or simulating complex phenomena, such as rare events triggered by random fluctuations with heavy as well as light tail statistical features.

随着宝贵的观测，实验和模拟数据在复杂系统中的迅速增加，人们正致力于发现这些系统发展的基本规律。但是，现有技术仅限于从数据中提取控制律，例如确定性微分方程或具有高斯噪声的随机微分方程。在当前的工作中，我们开发了一种新的数据驱动方法来提取具有非高斯对称Lévy噪声和高斯噪声的随机动力学系统。首先，我们通过样本路径数据表示潜在的随机动力系统的漂移系数，扩散系数和跳跃度量（即异常扩散），建立了一个可行的理论框架。然后，我们设计一个数值算法来计算漂移，扩散系数和跳跃度量，从而提取具有高斯噪声和非高斯噪声的控制型随机微分方程。最后，我们通过将其应用于几种原型一维，二维和三维系统，证明了我们方法的有效性和准确性。这种新方法将成为从嘈杂的数据集中发现，观察或模拟复杂现象（例如，由随机波动引起的稀有事件以及重尾和轻尾统计特征）发现动态规律的工具。