Sparse Identification of Nonlinear Dynamical Systems via Reweighted ℓ_1-regularized Least Squares
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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 [Brunton et al., PNAS, 113 (15) (2016) 3932-3937], which relies on two main assumptions: 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. 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 ℓ_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 ℓ_1-norm for regularization – instead of the standard ℓ_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 ℓ_1-regularized least squares strategy with respect to state measurement noise, thus illustrating its viability for a wide range of potential applications.
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