# On the Convergence of the SINDy Algorithm

arXiv: https://arxiv.org/abs/1805.06445 · HTML: https://mlsanity.com/paper/1805.06445/

## Key facts

- Cited by 49 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 16 May 2018.

## Built on by (most cited first, 30 of 49)

- [Deeptime: a Python library for machine learning dynamical models from time series data](https://mlsanity.com/paper/2110.15013/) (2021)
- [Discovery of Nonlinear Dynamical Systems using a Runge-Kutta Inspired Dictionary-based Sparse Regression Approach](https://mlsanity.com/paper/2105.04869/) (2021)
- [Learning Mean-Field Equations from Particle Data Using WSINDy](https://mlsanity.com/paper/2110.07756/) (2021)
- [Bayesian autoencoders for data-driven discovery of coordinates, governing equations and fundamental constants](https://mlsanity.com/paper/2211.10575/) (2022)
- [DySMHO: Data-Driven Discovery of Governing Equations for Dynamical Systems via Moving Horizon Optimization](https://mlsanity.com/paper/2108.00069/) (2021)
- [WeakIdent: Weak formulation for Identifying Differential Equations using Narrow-fit and Trimming](https://mlsanity.com/paper/2211.03134/) (2022)
- [A toolkit for data-driven discovery of governing equations in high-noise regimes](https://mlsanity.com/paper/2111.04870/) (2021)
- [Convergence of uncertainty estimates in Ensemble and Bayesian sparse model discovery](https://mlsanity.com/paper/2301.12649/) (2023)
- [On sparse regression, Lp-regularization, and automated model discovery](https://mlsanity.com/paper/2310.06872/) (2023)
- [Rapid Bayesian identification of sparse nonlinear dynamics from scarce and noisy data](https://mlsanity.com/paper/2402.15357/) (2024)
- [Asymptotic consistency of the WSINDy algorithm in the limit of continuum data](https://mlsanity.com/paper/2211.16000/) (2022)
- [MODNO: Multi Operator Learning With Distributed Neural Operators](https://mlsanity.com/paper/2404.02892/) (2024)
- [Online Weak-form Sparse Identification of Partial Differential Equations](https://mlsanity.com/paper/2203.03979/) (2022)
- [Bayesian Spline Learning for Equation Discovery of Nonlinear Dynamics with Quantified Uncertainty](https://mlsanity.com/paper/2210.08095/) (2022)
- [Convergence of weak-SINDy Surrogate Models](https://mlsanity.com/paper/2209.15573/) (2022)
- [Coarse-Graining Hamiltonian Systems Using WSINDy](https://mlsanity.com/paper/2310.05879/) (2023)
- [Random Feature Models for Learning Interacting Dynamical Systems](https://mlsanity.com/paper/2212.05591/) (2022)
- [SHRIMP: Sparser Random Feature Models via Iterative Magnitude Pruning](https://mlsanity.com/paper/2112.04002/) (2021)
- [Dynamical System Identification, Model Selection and Model Uncertainty Quantification by Bayesian Inference](https://mlsanity.com/paper/2401.16943/) (2024)
- [Weak-Form Inference for Hybrid Dynamical Systems in Ecology](https://mlsanity.com/paper/2405.20591/) (2024)
- [Time-Series Forecasting, Knowledge Distillation, and Refinement within a Multimodal PDE Foundation Model](https://mlsanity.com/paper/2409.11609/) (2024)
- [Fourier PINNs: From Strong Boundary Conditions to Adaptive Fourier Bases](https://mlsanity.com/paper/2410.03496/) (2024)
- [Data-Efficient Kernel Methods for Learning Differential Equations and Their Solution Operators: Algorithms and Error Analysis](https://mlsanity.com/paper/2503.01036/) (2025)
- [A Robust SINDy Approach by Combining Neural Networks and an Integral Form](https://mlsanity.com/paper/2309.07193/) (2023)
- [A Priori Denoising Strategies for Sparse Identification of Nonlinear Dynamical Systems: A Comparative Study](https://mlsanity.com/paper/2201.12683/) (2022)
- [Statistical Mechanics of Dynamical System Identification](https://mlsanity.com/paper/2403.01723/) (2024)
- [Dynamical system prediction from sparse observations using deep neural networks with Voronoi tessellation and physics constraint](https://mlsanity.com/paper/2409.00458/) (2024)
- [Scalable Sparse Regression for Model Discovery: The Fast Lane to Insight](https://mlsanity.com/paper/2405.09579/) (2024)
- [Multiphysics discovery with moving boundaries using Ensemble SINDy and Peridynamic Differential Operator](https://mlsanity.com/paper/2303.15631/) (2023)
- [GN-SINDy: Greedy Sampling Neural Network in Sparse Identification of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/2405.08613/) (2024)

## Abstract (arXiv)

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 ℓ0-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.