IDENT: Identifying Differential Equations with Numerical Time Evolution
IDENT: Identifying Differential Equations with Numerical Time Evolution
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DOI：
10.1007/s10915-020-01404-9
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发表时间：
2019-04
影响因子：
2.5
通讯作者：
S. Kang;Wenjing Liao;Yingjie Liu
中科院分区：
文献类型：
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作者：
S. Kang;Wenjing Liao;Yingjie Liu
从给定的离散时变数据中识别未知微分方程是一个具有挑战性的问题。少量的噪声会使恢复不稳定。非线性和变系数增加了问题的复杂性。我们假设偏微分方程（PDE）是一个线性组合的几个微分项在一个指定的字典，本文的目的是找到正确的系数。我们提出了一个新的方向的基础上的基本收敛性原则的数值偏微分格式。我们利用Lasso来提高效率，并基于不相干属性建立性能保证。主要贡献是通过时间演化误差（TEE）来验证和校正结果。针对非周期边界条件、噪声数据和变系数偏微分方程，提出了一种新的数值时间演化微分方程辨识算法（IDENT）。基于Lasso的恢复理论，本文提出了一种新的信噪比定义，它能更好地反映偏微分方程辨识中的噪声水平。系统地分析和测试了数据生成和下采样的效果。对于噪声数据，我们提出了一种保序去噪方法，称为最小二乘移动平均（LSMA），对给定的数据进行预处理。对于具有变化系数的偏微分方程的识别，我们建议添加基元展开（BEE）来辅助计算。各种数值实验，从基本的测试，噪声数据，下采样效果和不同的系数。
Identifying unknown differential equations from a given set of discrete time dependent data is a challenging problem. A small amount of noise can make the recovery unstable. Nonlinearity and varying coefficients add complexity to the problem. We assume that the governing partial differential equation (PDE) is a linear combination of few differential terms in a prescribed dictionary, and the objective of this paper is to find the correct coefficients. We propose a new direction based on the fundamental convergence principle of numerical PDE schemes. We utilize Lasso for efficiency, and a performance guarantee is established based on an incoherence property. The main contribution is to validate and correct the results by time evolution error (TEE). A new algorithm, called identifying differential equations with numerical time evolution (IDENT), is explored for data with non-periodic boundary conditions, noisy data and PDEs with varying coefficients. Based on the recovery theory of Lasso, we propose a new definition of Noise-to-Signal ratio, which better represents the level of noise in the case of PDE identification. The effects of data generations and downsampling are systematically analyzed and tested. For noisy data, we propose an order preserving denoising method called least-squares moving average (LSMA), to preprocess the given data. For the identification of PDEs with varying coefficients, we propose to add Base Element Expansion (BEE) to aid the computation. Various numerical experiments from basic tests to noisy data, downsampling effects and varying coefficients are presented.