Exact identification of nonlinear dynamical systems by Trimmed Lasso

TL;DR

This paper introduces the Trimmed Lasso method for exact nonlinear dynamical system identification, outperforming existing sparse techniques like E-SINDy especially under noisy, finite, and multicollinear data conditions.

Contribution

The paper demonstrates that Trimmed Lasso achieves exact recovery in nonlinear system identification with noisy, limited, and multicollinear data, with computational efficiency comparable to standard methods.

Findings

Trimmed Lasso outperforms E-SINDy in noisy, finite data scenarios.

Exact recovery is possible with Trimmed Lasso under severe noise and multicollinearity.

The method is computationally efficient, comparable to STLS.

Abstract

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.