Exact identification of nonlinear dynamical systems by Trimmed Lasso
Shawn L. Kiser Mikhail Guskov Marc Rébillat Nicolas Ranc
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.