WeakIdent: Weak formulation for identifying differential equation using narrow-fit and trimming
WeakIdent: Weak formulation for identifying differential equation using narrow-fit and trimming
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WeakIdent：使用窄拟合和修剪识别微分方程的弱公式
DOI：
10.1016/j.jcp.2023.112069
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发表时间：
2023
影响因子：
4.1
通讯作者：
Kang, Sung Ha
中科院分区：
文献类型：
--
作者：
Tang, Mengyi;Liao, Wenjing;Kuske, Rachel;Kang, Sung Ha
数据驱动的微分方程辨识是一个有趣但具有挑战性的问题，特别是当给定的数据被噪声破坏时。当控制微分方程是各种微分项的线性组合时，识别问题可以被公式化为求解线性系统，其中特征矩阵由线性和非线性项乘以系数向量组成。该乘积等于时间导数项，从而产生动力学行为。目标是识别形成方程的正确项，以捕获给定数据的动态。我们提出了一个通用且鲁棒的框架，使用弱公式来恢复微分方程，其中包括两种新机制：窄拟合和修剪，用于普通和偏微分方程（ODE和PDE）。弱公式有利于一个有效的和强大的方式来处理噪声，和两个新的机制，窄适应和修剪，提高系数支持和值恢复分别。对于每个稀疏水平，子空间追踪被用来从大字典中找到一个初始的支持集。然后，我们专注于高度动态区域（特征矩阵的行），并在窄拟合步骤中对特征矩阵进行误差归一化。通过修剪贡献最小的项来进一步更新支持。最后，选择具有最小交叉验证误差的特征的支持集作为结果。一套全面的数值实验的常微分方程和偏微分方程的系统与各种噪声水平。所提出的方法给出了一个强大的恢复系数，和一个显着的去噪效果，可以处理高达100%的噪声信号比的一些方程。我们比较所提出的方法与几个国家的最先进的算法恢复微分方程。
Data-driven identification of differential equations is an interesting but challenging problem, especially when the given data are corrupted by noise. When the governing differential equation is a linear combination of various differential terms, the identification problem can be formulated as solving a linear system, with the feature matrix consisting of linear and nonlinear terms multiplied by a coefficient vector. This product is equal to the time derivative term, and thus generates dynamical behaviors. The goal is to identify the correct terms that form the equation to capture the dynamics of the given data. We propose a general and robust framework to recover differential equations using a weak formulation with two new mechanisms, narrow-fit and trimming, for both ordinary and partial differential equations (ODEs and PDEs). The weak formulation facilitates an efficient and robust way to handle noise, and two new mechanisms, narrow-fit and trimming, improve the coefficient support and value recoveries respectively. For each sparsity level, Subspace Pursuit is utilized to find an initial set of support from the large dictionary. Then, we focus on highly dynamic regions (rows of the feature matrix), and error normalize the feature matrix in the narrow-fit step. The support is further updated via trimming the terms that contribute the least. Finally, the support set of features with the smallest Cross-Validation error is chosen as the result. A comprehensive set of numerical experiments are presented for both systems of ODEs and PDEs with various noise levels. The proposed method gives a robust recovery of the coefficients, and a significant denoising effect which can handle up to 100% noise-to-signal ratio for some equations. We compare the proposed method with several state-of-the-art algorithms for the recovery of differential equations.