时变偏微分方程的贝叶斯稀疏识别方法

doi:10.19596/j.cnki.1001-246x.8189

计算物理 ›› 2021, Vol. 38 ›› Issue (1): 25-34.DOI: 10.19596/j.cnki.1001-246x.8189

时变偏微分方程的贝叶斯稀疏识别方法

胡军, 刘全, 倪国喜

北京应用物理与计算数学研究所计算物理实验室, 北京 100088

收稿日期:2019-12-19 修回日期:2020-03-03 出版日期:2021-01-25 发布日期:2021-01-25
作者简介:胡军(1978-),男,博士,副研究员,从事流体动力学不稳定性以及机器学习研究,E-mail:hu_jun@iapcm.ac.cn
基金资助:
国家自然科学基金（11672046）及计算物理实验室基金资助项目

Bayesian Sparse Identification of Time-varying Partial Differential Equations

HU Jun, LIU Quan, NI Guoxi

Laboratory of Computational Physics, Institute of Applied Physics and Computational Mathematics, P. O. Box 8009, Beijing 100088, China

Received:2019-12-19 Revised:2020-03-03 Online:2021-01-25 Published:2021-01-25

摘要/Abstract

摘要： 在数据驱动的建模中，通过测量或模拟得到时空数据，我们发现基于拉普拉斯先验的贝叶斯稀疏识别方法能有效地恢复时变偏微分方程的稀疏系数。本文将贝叶斯稀疏识别方法运用于各种时变偏微分方程模型（KdV方程、Burgers方程、Kuramoto-Sivashinsky方程、反应-扩散方程、非线性薛定谔方程和纳维-斯托克斯方程）的方程系数恢复，将贝叶斯稀疏恢复结果与PDE-FIND稀疏恢复算法进行比较，证实贝叶斯稀疏识别方法对偏微分方程具有非常强的稀疏恢复能力。同时，研究中发现贝叶斯稀疏方法对噪声更敏感，可以识别更多的附加项。此外，贝叶斯方法可以直接得到稀疏恢复解的误差方差，由此可以直接判定稀疏恢复的效果和可靠性。

关键词: 贝叶斯方法, 稀疏识别, 偏微分方程, 纳维-斯托克斯方程

Abstract: In data-driven modeling, Bayesian sparse identification method with Laplace priors was found and confirmed to recover sparse coefficients of governing partial differential equations(PDEs) by spatiotemporal data from measurement or simulation. Verification results of Bayesian sparse identification method for various canonical models (KdV equation, Burgers equation, Kuramoto-Sivashinsky equation, reaction-diffusion equations, nonlinear Schr dinger equation and Navier-Stokes equations) are compared with those of Rudy's PDE-FIND algorithm. Very well agreement between these two methods shows Bayesian sparse method has strong identification capability of PDE. However, it is also found that the Bayesian sparse method is much more sensitive to noise, which may identify more extra terms. In addition, relatively small error variances of Bayesian sparse solutions are obtained and exhibit clearly the successful identification of PDE.

Key words: Bayesian method, sparse identification, partial differential equation, Navier-Stokes equations

中图分类号:

胡军, 刘全, 倪国喜. 时变偏微分方程的贝叶斯稀疏识别方法[J]. 计算物理, 2021, 38(1): 25-34.

HU Jun, LIU Quan, NI Guoxi. Bayesian Sparse Identification of Time-varying Partial Differential Equations[J]. CHINESE JOURNAL OF COMPUTATIONAL PHYSICS, 2021, 38(1): 25-34.

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时变偏微分方程的贝叶斯稀疏识别方法

Bayesian Sparse Identification of Time-varying Partial Differential Equations

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