时变偏微分方程的贝叶斯稀疏识别方法

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。

Bayesian Sparse Identification of Time-varying Partial Differential Equations

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.