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 Brunton et al. (2016), which relies on two main assumptions: the state variables are known 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.

这项工作提出了一种迭代的稀疏-正则化回归方法，以从噪声状态测量中恢复非线性动力学系统的控制方程。该方法受到Brunton等人的非线性动力学稀疏识别（SINDy）方法的启发。（2016），它基于两个主要假设：状态变量是先验的，控制方程式使状态变量在（非线性）基础上进行稀疏的线性扩展。这项工作的目的是在存在状态测量噪声的情况下提高SINDy的准确性和鲁棒性。为此，重新加权${\ell }_{\mathrm{1个}}$开发了正则化的最小二乘法求解器，其中，所述正则化参数是从帕累托曲线的角点中选择的。使用加权的想法${\ell }_{\mathrm{1个}}$-规范化-代替标准 ${\ell }_{\mathrm{1个}}$-norm –是为了更好地促进控制方程的恢复中的稀疏性，进而减轻状态变量中噪声的影响。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。通过一些著名的非线性动力学系统的例子，我们通过经验证明了重新加权的准确性和鲁棒性${\ell }_{\mathrm{1个}}$关于状态测量噪声的正则化最小二乘策略，从而说明了其在各种潜在应用中的可行性。