  In recent years, identification of nonlinear dynamical systems from data has
become increasingly popular. Sparse regression approaches, such as Sparse
Identification of Nonlinear Dynamics (SINDy), fostered the development of novel
governing equation identification algorithms assuming 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. In the context of the
identification of governing equations of nonlinear dynamical systems, one faces
the problem of identifiability of model parameters when state measurements are
corrupted by noise. Measurement noise affects the stability of the recovery
process yielding incorrect sparsity patterns and inaccurate estimation of
coefficients of the governing equations. In this work, we investigate and
compare the performance of several local and global smoothing techniques to a
priori denoise the state measurements and numerically estimate the state
time-derivatives to improve the accuracy and robustness of two sparse
regression methods to recover governing equations: Sequentially Thresholded
Least Squares (STLS) and Weighted Basis Pursuit Denoising (WBPDN) algorithms.
We empirically show that, in general, global methods, which use the entire
measurement data set, outperform local methods, which employ a neighboring data
subset around a local point. We additionally compare Generalized Cross
Validation (GCV) and Pareto curve criteria as model selection techniques to
automatically estimate near optimal tuning parameters, and conclude that Pareto
curves yield better results. The performance of the denoising strategies and
sparse regression methods is empirically evaluated through well-known benchmark
problems of nonlinear dynamical systems.
