Robust Identification of Differential Equations by Numerical Techniques from a Single Set of Noisy Observation
Robust Identification of Differential Equations by Numerical Techniques from a Single Set of Noisy Observation
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DOI：
10.1137/20m134513x
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发表时间：
2020-06
期刊：
影响因子：
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通讯作者：
Yuchen He;S. Kang;Wenjing Liao;Hao Liu;Yingjie Liu
中科院分区：
文献类型：
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作者：
Yuchen He;S. Kang;Wenjing Liao;Hao Liu;Yingjie Liu
我们提出了鲁棒的方法来识别潜在的偏微分方程（PDE）从一组给定的噪声时间相关的数据。我们假设控制方程是一个线性组合的几个线性和非线性微分项在一个指定的字典。噪声数据使得这种识别特别具有挑战性。我们的目标是开发的方法，这是强大的对高水平的噪声，以及近似的基本无噪声动态。首先，我们介绍了一个Successfully去噪微分（SDD）计划，以稳定的放大噪声在数值微分。SDD有效地去除了给定数据和相应导数的噪声。其次，提出了两种PDE辨识算法：子空间追踪时间演化误差法（ST）和子空间追踪交叉验证法（SC）。我们的一般策略是首先使用子空间追踪（SP）贪婪算法找到一个候选集，然后通过时间进化或交叉验证来选择最好的一个。ST使用多激发数值时间演化，并选择产生最小演化误差的PDE。SC评估最小二乘拟合中的交叉验证误差，并选择给出最小验证误差的PDE。我们提出了一个统一的概念偏微分方程识别误差比较相关方法的目标。我们提出了各种数值实验来验证我们的方法。这两种方法都是有效的，对噪声具有鲁棒性。
We propose robust methods to identify underlying Partial Differential Equation (PDE) from a given set of noisy time dependent data. We assume that the governing equation is a linear combination of a few linear and nonlinear differential terms in a prescribed dictionary. Noisy data make such identification particularly challenging. Our objective is to develop methods which are robust against a high level of noise, and to approximate the underlying noise-free dynamics well. We first introduce a Successively Denoised Differentiation (SDD) scheme to stabilize the amplified noise in numerical differentiation. SDD effectively denoises the given data and the corresponding derivatives. Secondly, we present two algorithms for PDE identification: Subspace pursuit Time evolution error (ST) and Subspace pursuit Cross-validation (SC). Our general strategy is to first find a candidate set using the Subspace Pursuit (SP) greedy algorithm, then choose the best one via time evolution or cross validation. ST uses multi-shooting numerical time evolution and selects the PDE which yields the least evolution error. SC evaluates the cross-validation error in the least squares fitting and picks the PDE that gives the smallest validation error. We present a unified notion of PDE identification error to compare the objectives of related approaches. We present various numerical experiments to validate our methods. Both methods are efficient and robust to noise.