Discovering explicit governing equations of stochastic dynamical systems with both (Gaussian) Brownian noise and (non-Gaussian) L\'evy noise from data is chanllenging due to possible intricate functional forms and the inherent complexity of L\'evy motion. This present research endeavors to develop an evolutionary symbol sparse regression (ESSR) approach to extract non-Gaussian stochastic dynamical systems from sample path data, based on nonlocal Kramers-Moyal formulas, genetic programming, and sparse regression. More specifically, the genetic programming is employed to generate a diverse array of candidate functions, the sparse regression technique aims at learning the coefficients associated with these candidates, and the nonlocal Kramers-Moyal formulas serve as the foundation for constructing the fitness measure in genetic programming and the loss function in sparse regression. The efficacy and capabilities of this approach are showcased through its application to several illustrative models. This approach stands out as a potent instrument for deciphering non-Gaussian stochastic dynamics from available datasets, indicating a wide range of applications across different fields.

翻译：从数据中发现同时含有（高斯）布朗噪声与（非高斯）Lévy噪声的随机动力系统的显式控制方程具有挑战性，这源于可能存在的复杂函数形式以及Lévy运动固有的复杂性。本研究致力于发展一种演化符号稀疏回归（ESSR）方法，基于非局部Kramers-Moyal公式、遗传编程与稀疏回归，从样本路径数据中提取非高斯随机动力系统。具体而言，遗传编程用于生成多样化的候选函数集合，稀疏回归技术旨在学习这些候选函数的对应系数，而非局部Kramers-Moyal公式则作为构建遗传编程中适应度度量与稀疏回归中损失函数的基础。通过对多个示例模型的应用，展示了该方法的有效性与能力。此方法作为从现有数据集中解析非高斯随机动力学的有力工具，显示出其在多个领域具有广泛的应用前景。