Identification of nonlinear dynamical systems has been popularized by sparse identification of the nonlinear dynamics (SINDy) via the sequentially thresholded least squares (STLS) algorithm. Many extensions SINDy have emerged in the literature to deal with experimental data which are finite in length and noisy. Recently, the computationally intensive method of ensembling bootstrapped SINDy models (E-SINDy) was proposed for model identification, handling finite, highly noisy data. While the extensions of SINDy are numerous, their sparsity-promoting estimators occasionally provide sparse approximations of the dynamics as opposed to exact recovery. Furthermore, these estimators suffer under multicollinearity, e.g. the irrepresentable condition for the Lasso. In this paper, we demonstrate that the Trimmed Lasso for robust identification of models (TRIM) can provide exact recovery under more severe noise, finite data, and multicollinearity as opposed to E-SINDy. Additionally, the computational cost of TRIM is asymptotically equal to STLS since the sparsity parameter of the TRIM can be solved efficiently by convex solvers. We compare these methodologies on challenging nonlinear systems, specifically the Lorenz 63 system, the Bouc Wen oscillator from the nonlinear dynamics benchmark of No\"el and Schoukens, 2016, and a time delay system describing tool cutting dynamics. This study emphasizes the comparisons between STLS, reweighted $\ell_1$ minimization, and Trimmed Lasso in identification with respect to problems faced by practitioners: the problem of finite and noisy data, the performance of the sparse regression of when the library grows in dimension (multicollinearity), and automatic methods for choice of regularization parameters.

翻译：非线性动力系统辨识因通过序列阈值最小二乘（STLS）算法实现非线性动力学稀疏辨识（SINDy）而广受关注。文献中涌现出大量SINDy的扩展方法，用于处理有限长度且含噪声的实验数据。近期，为应对有限高噪声数据，研究人员提出了集成自举SINDy模型（E-SINDy）这一计算密集型方法。尽管SINDy的扩展方法众多，但其稀疏促进估计器有时仅能提供动力学的稀疏近似，而非精确恢复。此外，这些估计器存在多重共线性问题（例如Lasso的不可表示条件）。本文证明，与E-SINDy相比，用于鲁棒模型辨识的修剪套索（TRIM）可在更严重噪声、有限数据和多重共线性条件下实现精确恢复。由于TRIM的稀疏参数可通过凸优化求解器高效求解，其计算复杂度渐近等价于STLS。我们通过挑战性非线性系统（具体包括Lorenz 63系统、Noël与Schoukens（2016）非线性动力学基准中的Bouc Wen振荡器，以及描述刀具切削动力学的时间延迟系统）对这些方法进行对比。本研究重点比较STLS、加权$\ell_1$最小化和修剪套索在辨识过程中面临的实践难题：有限噪声数据问题、函数库维度增长时的稀疏回归性能（多重共线性），以及正则化参数的自动选择方法。