非高斯Levy噪声随机动力系统的数据驱动发现方法
A Data-Driven Approach for Discovering Stochastic Dynamical Systems with Non-Gaussian Levy Noise
摘要：
随着有价值的观测、实验和模拟数据的迅速增加，人们正致力于发现这些系统演化的规律。然而，现有的技术仅限于从数据中提取控制律，无论是确定性微分方程还是带有高斯噪声的随机微分方程。在目前的工作中，我们开发了一种新的数据驱动的方法来提取随机动力系统与非高斯对称Lévy噪声，以及高斯噪声。首先，我们建立了一个可行的理论框架，通过表达漂移系数，扩散系数和跳跃测度（即，异常扩散）的基本随机动力系统的样本路径数据。然后，我们设计了一个数值算法来计算漂移，扩散系数和跳跃措施，从而提取一个控制的随机微分方程的高斯和非高斯噪声。最后，我们证明了我们的方法的有效性和准确性，适用于几个原型的一，二，三维系统。这种新方法将成为一种工具，从嘈杂的数据集，从观察或模拟复杂的现象，如由随机波动触发的罕见事件与重以及轻尾统计特征中发现支配动力学规律。
英文摘要：
With the rapid increase of valuable observational, experimental and simulating data for complex systems, great efforts are 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.
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