With the rapid increase of observational, experimental and simulated data for stochastic systems, tremendous efforts have been devoted to identifying governing laws underlying the evolution of these systems. Despite the broad applications of non-Gaussian fluctuations in numerous physical phenomena, the data-driven approaches to extracting stochastic dynamics with L\'{e}vy noise are relatively few. In this work, we propose a Weak Collocation Regression (WCR) to explicitly reveal unknown stochastic dynamical systems, i.e., the Stochastic Differential Equation (SDE) with both $\alpha$-stable L\'{e}vy noise and Gaussian noise, from discrete aggregate data. This method utilizes the evolution equation of the probability distribution function, i.e., the Fokker-Planck (FP) equation. With the weak form of the FP equation, the WCR constructs a linear system of unknown parameters where all integrals are evaluated by Monte Carlo method with the observations. Then, the unknown parameters are obtained by a sparse linear regression. For a SDE with L\'{e}vy noise, the corresponding FP equation is a partial integro-differential equation (PIDE), which contains nonlocal terms, and is difficult to deal with. The weak form can avoid complicated multiple integrals. Our approach can simultaneously distinguish mixed noise types, even in multi-dimensional problems. Numerical experiments demonstrate that our method is accurate and computationally efficient.

翻译：随着随机系统观测、实验与模拟数据的快速增长，大量研究致力于揭示这些系统演化背后的支配规律。尽管非高斯涨落在众多物理现象中具有广泛应用，但针对含莱维噪声随机动力学提取的数据驱动方法相对较少。本文提出一种弱配点回归法，用于从离散聚合数据中显式揭示未知随机动力系统——即同时包含α稳定莱维噪声与高斯噪声的随机微分方程。该方法利用概率分布函数的演化方程——福克-普朗克方程。通过福克-普朗克方程的弱形式，弱配点回归法构建了一个未知参数的线性系统，其中所有积分均采用蒙特卡洛方法结合观测数据进行求解，进而通过稀疏线性回归获得未知参数。对于含莱维噪声的随机微分方程，其对应的福克-普朗克方程为包含非局部项的部分积分微分方程，处理难度较大，而弱形式可避免复杂的多重积分问题。该方法能够同时区分混合噪声类型，甚至适用于多维问题。数值实验表明，我们的方法具有高精度与计算高效性。