论文标题
通过重新加权$ \ ell_1 $登记最小二乘，稀疏识别非线性动力学系统
Sparse Identification of Nonlinear Dynamical Systems via Reweighted $\ell_1$-regularized Least Squares
论文作者
论文摘要
这项工作提出了一种迭代性稀疏调查的回归方法，以从嘈杂的状态测量中恢复非线性动力学系统的管理方程。该方法的灵感来自{\ it [\ it [Brunton等，PNAS，PNAS，113（15）（2016）3932-3937]}的非线性动力学方法（SINDY）方法的稀疏识别，这些方法依赖于两个主要假设，这些假设依赖于两个国家变量：先验变量是{\ \ n nontraniel sprine sprine sprine sprine sprine sprine sprine sprine inlinial inlinion in lin in lin in lin in lin in lin in lin in lin in lin in llin ins in lin in llin in line in line in line in（nlin）in（状态变量。这项工作的目的是在存在状态测量噪声的情况下提高信德的准确性和鲁棒性。为此，开发了一个重量级的$ \ ell_1 $调查最小二乘求解器，其中从帕累托曲线的角点选择了正则化参数。使用加权$ \ ell_1 $ - 正规化的想法 - 而不是标准$ \ ell_1 $ -norm-是为了更好地促进管理方程式恢复的稀疏性，进而减轻状态变量中噪声的效果。我们还提出了一种从状态测量中恢复单个物理约束的方法。通过众所周知的非线性动力学系统的几个示例，我们从经验上证明了经过重新持续的$ \ ell_1 $调查最小二乘策略在状态测量噪声方面的准确性和鲁棒性，从而说明了其对广泛潜在应用的生存能力。
This work proposes an iterative sparse-regularized regression method to recover governing equations of nonlinear dynamical systems from noisy state measurements. The method is inspired by the Sparse Identification of Nonlinear Dynamics (SINDy) approach of {\it [Brunton et al., PNAS, 113 (15) (2016) 3932-3937]}, which relies on two main assumptions: the state variables are known {\it a priori} and the governing equations lend themselves to sparse, linear expansions in a (nonlinear) basis of the state variables. The aim of this work is to improve the accuracy and robustness of SINDy in the presence of state measurement noise. To this end, a reweighted $\ell_1$-regularized least squares solver is developed, wherein the regularization parameter is selected from the corner point of a Pareto curve. The idea behind using weighted $\ell_1$-norm for regularization -- instead of the standard $\ell_1$-norm -- is to better promote sparsity in the recovery of the governing equations and, in turn, mitigate the effect of noise in the state variables. We also present a method to recover single physical constraints from state measurements. Through several examples of well-known nonlinear dynamical systems, we demonstrate empirically the accuracy and robustness of the reweighted $\ell_1$-regularized least squares strategy with respect to state measurement noise, thus illustrating its viability for a wide range of potential applications.
