On the Convergence of the SINDy Algorithm

Abstract

One way to understand time-series data is to identify the underlying dynamical system which generates it. This task can be done by selecting an appropriate model and a set of parameters which best fits the dynamics while providing the simplest representation (i.e. the smallest amount of terms). One such approach is the sparse identification of nonlinear dynamics framework [6] which uses a sparsity-promoting algorithm that iterates between a partial least-squares fit and a thresholding (sparsity-promoting) step. In this work, we provide some theoretical results on the behavior and convergence of the algorithm proposed in [6]. In particular, we prove that the algorithm approximates local minimizers of an unconstrained ${\ell }^0$-penalized least-squares problem. From this, we provide sufficient conditions for general convergence, rate of convergence, and conditions for one-step recovery. Examples illustrate that the rates of convergence are sharp. In addition, our results extend to other algorithms related to the algorithm in [6], and provide theoretical verification to several observed phenomena.

arXiv comment24 pages, 4 figures, 3 tables

References in corpus (5)

• Discovering governing equations from data: Sparse identification of nonlinear dynamical systems
• Hidden Physics Models: Machine Learning of Nonlinear Partial Differential Equations
• Sparse Identification for Nonlinear Optical Communication Systems: SINO Method
• Learning Dynamical Systems and Bifurcation via Group Sparsity
• Extracting structured dynamical systems using sparse optimization with very few samples

Cited by in corpus (4)

• Data-driven discovery of coordinates and governing equations
• Sparse dynamical system identification with simultaneous structural parameters and initial condition estimation
• Machine Discovery of Partial Differential Equations from Spatiotemporal Data
• Unsupervised learning of control signals and their encodings in $\text{C. elegans}$ whole-brain recordings