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数据驱动的随机微分方程辨识
Yasen Wang, Huazhen Fang, Junyang Jin, Guijun Ma, Xin He, Xing Dai, Zuogong Yue, Cheng Cheng, Hai-Tao Zhang, Donglin Pu, Dongrui Wu, Ye Yuan, Jorge Gonçalves, Jürgen Kurths, Han Ding
工程（英文） ›› 2022, Vol. 17 ›› Issue (10) : 244-252.
PDF(2538 KB)
PDF(2538 KB)
数据驱动的随机微分方程辨识
Data-Driven Discovery of Stochastic Differential Equations
随机微分方程（SDE）是一种广泛用于描述受不同来源噪声干扰的复杂过程或现象的数学模型。由于数据固有的强随机性和系统动力学的复杂性，控制系统的SDE的辨识通常是一个挑战。辨识SDE的现有参数化方法的实用性通常受到数据资源不足的限制。本研究提出了一种通过稀疏贝叶斯学习（SBL）技术从候选基函数空间中搜索简洁但物理必需的表示形式来辨识SDE的新框架。更重要的是，利用SBL的解析可处理性开发了一种可以通过少量数据来高效地构建辨识SDE的线性回归模型的方法。本文利用股票和石油价格、轴承变化和风速的真实数据，以及包括广义维纳（Wiener）过程和朗之万方程在内的著名随机动力系统的仿真数据，验证了所提出框架的有效性。该框架旨在帮助专家从自然科学、经济学和工程领域的随机现象中提取随机数学模型，用于分析、预测和决策。
Stochastic differential equations (SDEs) are mathematical models that are widely used to describe complex processes or phenomena perturbed by random noise from different sources. The identification of SDEs governing a system is often a challenge because of the inherent strong stochasticity of data and the complexity of the system's dynamics. The practical utility of existing parametric approaches for identifying SDEs is usually limited by insufficient data resources. This study presents a novel framework for identifying SDEs by leveraging the sparse Bayesian learning (SBL) technique to search for a parsimonious, yet physically necessary representation from the space of candidate basis functions. More importantly, we use the analytical tractability of SBL to develop an efficient way to formulate the linear regression problem for the discovery of SDEs that requires considerably less time-series data. The effectiveness of the proposed framework is demonstrated using real data on stock and oil prices, bearing variation, and wind speed, as well as simulated data on well-known stochastic dynamical systems, including the generalized Wiener process and Langevin equation. This framework aims to assist specialists in extracting
stochastic mathematical models from random phenomena in the natural sciences, economics, and engineering fields for analysis, prediction, and decision making.
数据驱动方法 / 系统辨识 / 稀疏贝叶斯学习 / 随机微分方程 / 随机现象
Data-driven method / System identification / Sparse Bayesian learning / Stochastic differential equations / Random phenomena
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