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

arXiv: https://arxiv.org/abs/2201.12683 · HTML: https://mlsanity.com/paper/2201.12683/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 120 references that are in the map.
- First posted on arXiv 29 Jan 2022.

## Builds on (most cited first, 15 of 15)

- [CVXPY: A Python-Embedded Modeling Language for Convex Optimization](https://mlsanity.com/paper/1603.00943/) (2016)
- [Physics-informed learning of governing equations from scarce data](https://mlsanity.com/paper/2005.03448/) (2020)
- [Weak SINDy For Partial Differential Equations](https://mlsanity.com/paper/2007.02848/) (2020)
- [On the Convergence of the SINDy Algorithm](https://mlsanity.com/paper/1805.06445/) (2018)
- [Adaptive piecewise polynomial estimation via trend filtering](https://mlsanity.com/paper/1304.2986/) (2013)
- [Automatic Differentiation to Simultaneously Identify Nonlinear Dynamics and Extract Noise Probability Distributions from Data](https://mlsanity.com/paper/2009.08810/) (2020)
- [Model Selection Techniques -- An Overview](https://mlsanity.com/paper/1810.09583/) (2018)
- [Sparse Regression: Scalable algorithms and empirical performance](https://mlsanity.com/paper/1902.06547/) (2019)
- [Seismic Signal Denoising and Decomposition Using Deep Neural Networks](https://mlsanity.com/paper/1811.02695/) (2018)
- [Sparse Identification of Nonlinear Dynamical Systems via Reweighted ℓ1-regularized Least Squares](https://mlsanity.com/paper/2005.13232/) (2020)
- [Physics-informed Spline Learning for Nonlinear Dynamics Discovery](https://mlsanity.com/paper/2105.02368/) (2021)
- [Deep Recurrent Neural Networks for ECG Signal Denoising](https://mlsanity.com/paper/1807.11551/) (2018)
- [Manifold Approximation by Moving Least-Squares Projection (MMLS)](https://mlsanity.com/paper/1606.07104/) (2016)
- [Fast and Flexible ADMM Algorithms for Trend Filtering](https://mlsanity.com/paper/1406.2082/) (2014)
- [Approximation of Functions over Manifolds: A Moving Least-Squares Approach](https://mlsanity.com/paper/1711.00765/) (2017)

## Built on by (most cited first, 5 of 5)

- [Sparse discovery of differential equations based on multi-fidelity Gaussian process](https://mlsanity.com/paper/2401.11825/) (2024)
- [Constrained or Unconstrained? Neural-Network-Based Equation Discovery from Data](https://mlsanity.com/paper/2406.02581/) (2024)
- [Ensemble WSINDy for Data Driven Discovery of Governing Equations from Laser-based Full-field Measurements](https://mlsanity.com/paper/2409.20510/) (2024)
- [Exact identification of nonlinear dynamical systems by Trimmed Lasso](https://mlsanity.com/paper/2308.01891/) (2023)
- [Machine learning for cerebral blood vessels' malformations](https://mlsanity.com/paper/2411.16349/) (2024)

## Abstract (arXiv)

In recent years, identification of nonlinear dynamical systems from data has become increasingly popular. Sparse regression approaches, such as Sparse Identification of Nonlinear Dynamics (SINDy), fostered the development of novel governing equation identification algorithms assuming 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. In the context of the identification of governing equations of nonlinear dynamical systems, one faces the problem of identifiability of model parameters when state measurements are corrupted by noise. Measurement noise affects the stability of the recovery process yielding incorrect sparsity patterns and inaccurate estimation of coefficients of the governing equations. In this work, we investigate and compare the performance of several local and global smoothing techniques to a priori denoise the state measurements and numerically estimate the state time-derivatives to improve the accuracy and robustness of two sparse regression methods to recover governing equations: Sequentially Thresholded Least Squares (STLS) and Weighted Basis Pursuit Denoising (WBPDN) algorithms. We empirically show that, in general, global methods, which use the entire measurement data set, outperform local methods, which employ a neighboring data subset around a local point. We additionally compare Generalized Cross Validation (GCV) and Pareto curve criteria as model selection techniques to automatically estimate near optimal tuning parameters, and conclude that Pareto curves yield better results. The performance of the denoising strategies and sparse regression methods is empirically evaluated through well-known benchmark problems of nonlinear dynamical systems.