A Priori Denoising Strategies for Sparse Identification of Nonlinear Dynamical Systems: A Comparative Study

In recent years, identification of nonlinear dynamical systems from data has\nbecome increasingly popular. Sparse regression approaches, such as Sparse\nIdentification of Nonlinear Dynamics (SINDy), fostered the development of novel\ngoverning equation identification algorithms assuming the state variables are\nknown a priori and the governing equations lend themselves to sparse, linear\nexpansions in a (nonlinear) basis of the state variables. In the context of the\nidentification of governing equations of nonlinear dynamical systems, one faces\nthe problem of identifiability of model parameters when state measurements are\ncorrupted by noise. Measurement noise affects the stability of the recovery\nprocess yielding incorrect sparsity patterns and inaccurate estimation of\ncoefficients of the governing equations. In this work, we investigate and\ncompare the performance of several local and global smoothing techniques to a\npriori denoise the state measurements and numerically estimate the state\ntime-derivatives to improve the accuracy and robustness of two sparse\nregression methods to recover governing equations: Sequentially Thresholded\nLeast Squares (STLS) and Weighted Basis Pursuit Denoising (WBPDN) algorithms.\nWe empirically show that, in general, global methods, which use the entire\nmeasurement data set, outperform local methods, which employ a neighboring data\nsubset around a local point. We additionally compare Generalized Cross\nValidation (GCV) and Pareto curve criteria as model selection techniques to\nautomatically estimate near optimal tuning parameters, and conclude that Pareto\ncurves yield better results. The performance of the denoising strategies and\nsparse regression methods is empirically evaluated through well-known benchmark\nproblems of nonlinear dynamical systems.\n