  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.
