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SubTSBR to tackle high noise and outliers for data-driven discovery of differential equations
Journal of Computational Physics ( IF 4.1 ) Pub Date : 2020-11-30 , DOI: 10.1016/j.jcp.2020.109962
Sheng Zhang , Guang Lin

Data-driven discovery of differential equations has been an emerging research topic. We propose a novel algorithm subsampling-based threshold sparse Bayesian regression (SubTSBR) to tackle high noise and outliers. The subsampling technique is used for improving the accuracy of the Bayesian learning algorithm. It has two parameters: subsampling size and the number of subsamples. When the subsampling size increases with fixed total sample size, the accuracy of our algorithm goes up and then down. When the number of subsamples increases, the accuracy of our algorithm keeps going up. We demonstrate how to use our algorithm step by step and compare our algorithm with threshold sparse Bayesian regression (TSBR) for the discovery of differential equations. We show that our algorithm produces better results. We also discuss the merits of discovering differential equations from data and demonstrate how to discover models with random initial and boundary condition as well as models with bifurcations. The numerical examples are: (1) predator-prey model with noise, (2) shallow water equations with outliers, (3) heat diffusion with random initial and boundary condition, and (4) fish-harvesting problem with bifurcations.



中文翻译:

SubTSBR解决高噪声和离群值,用于数据驱动的微分方程式发现

数据驱动的微分方程的发现已成为新兴的研究主题。我们提出了一种基于子采样的阈值稀疏贝叶斯回归算法(SubTSBR),以解决高噪声和离群值问题。二次采样技术用于提高贝叶斯学习算法的准确性。它有两个参数:子采样大小和子采样数。当子采样大小随着固定的总采样大小而增加时,我们算法的准确性便会上升,然后下降。当子样本数量增加时,我们算法的准确性就会不断提高。我们演示了如何逐步使用我们的算法,并将我们的算法与阈值稀疏贝叶斯回归(TSBR)进行比较以发现微分方程。我们证明了我们的算法产生了更好的结果。我们还将讨论从数据中发现微分方程的优点,并演示如何发现具有随机初始条件和边界条件的模型以及具有分支的模型。数值示例包括:(1)具有噪声的捕食者-捕食模型;(2)具有离群值的浅水方程;(3)具有随机初始条件和边界条件的热扩散;以及(4)带有分叉的鱼类捕捞问题。

更新日期:2021-01-12
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