Sparse identification of nonlinear dynamical systems via reweighted [formula omitted]-regularized least squares

Abstract

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. (2016), 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 ℓ1-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 ℓ1-norm for regularization – instead of the standard ℓ1-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 ℓ1-regularized least squares strategy with respect to state measurement noise, thus illustrating its viability for a wide range of potential applications.