# Asymptotic consistency of the WSINDy algorithm in the limit of continuum data arXiv: https://arxiv.org/abs/2211.16000 · HTML: https://mlsanity.com/paper/2211.16000/ ## Key facts - Cited by 14 papers in the map (data through 14 Oct 2026). - Builds on 16 of its 51 references that are in the map. - First posted on arXiv 29 Nov 2022. ## Builds on (most cited first, 16 of 16) - [Weak SINDy For Partial Differential Equations](https://mlsanity.com/paper/2007.02848/) (2020) - [Ensemble-SINDy: Robust sparse model discovery in the low-data, high-noise limit, with active learning and control](https://mlsanity.com/paper/2111.10992/) (2021) - [On the Convergence of the SINDy Algorithm](https://mlsanity.com/paper/1805.06445/) (2018) - [Automatic Differentiation to Simultaneously Identify Nonlinear Dynamics and Extract Noise Probability Distributions from Data](https://mlsanity.com/paper/2009.08810/) (2020) - [Learning Mean-Field Equations from Particle Data Using WSINDy](https://mlsanity.com/paper/2110.07756/) (2021) - [Learning Sparse Nonlinear Dynamics via Mixed-Integer Optimization](https://mlsanity.com/paper/2206.00176/) (2022) - [Sparse Identification of Nonlinear Dynamical Systems via Reweighted ℓ1-regularized Least Squares](https://mlsanity.com/paper/2005.13232/) (2020) - [WeakIdent: Weak formulation for Identifying Differential Equations using Narrow-fit and Trimming](https://mlsanity.com/paper/2211.03134/) (2022) - [Sharp Concentration Results for Heavy-Tailed Distributions](https://mlsanity.com/paper/2003.13819/) (2020) - [Variational system identification of the partial differential equations governing microstructure evolution in materials: Inference over sparse and spatially unrelated data](https://mlsanity.com/paper/2001.04816/) (2020) - [Online Weak-form Sparse Identification of Partial Differential Equations](https://mlsanity.com/paper/2203.03979/) (2022) - [Learning Anisotropic Interaction Rules from Individual Trajectories in a Heterogeneous Cellular Population](https://mlsanity.com/paper/2204.14141/) (2022) - [Convergence of weak-SINDy Surrogate Models](https://mlsanity.com/paper/2209.15573/) (2022) - [Simultaneous Identification and Denoising of Dynamical Systems](https://mlsanity.com/paper/2203.13837/) (2022) - [Weak Collocation Regression method: fast reveal hidden stochastic dynamics from high-dimensional aggregate data](https://mlsanity.com/paper/2209.02628/) (2022) - [Methods to Recover Unknown Processes in Partial Differential Equations Using Data](https://mlsanity.com/paper/2003.02387/) (2020) ## Built on by (most cited first, 14 of 14) - [Direct Estimation of Parameters in ODE Models Using WENDy: Weak-form Estimation of Nonlinear Dynamics](https://mlsanity.com/paper/2302.13271/) (2023) - [Benchmarking sparse system identification with low-dimensional chaos](https://mlsanity.com/paper/2302.10787/) (2023) - [Weak-PDE-LEARN: A Weak Form Based Approach to Discovering PDEs From Noisy, Limited Data](https://mlsanity.com/paper/2309.04699/) (2023) - [Physics-informed active learning with simultaneous weak-form latent space dynamics identification](https://mlsanity.com/paper/2407.00337/) (2024) - [Coarse-Graining Hamiltonian Systems Using WSINDy](https://mlsanity.com/paper/2310.05879/) (2023) - [The Weak Form Is Stronger Than You Think](https://mlsanity.com/paper/2409.06751/) (2024) - [Weak-Form Inference for Hybrid Dynamical Systems in Ecology](https://mlsanity.com/paper/2405.20591/) (2024) - [Statistical Mechanics of Dynamical System Identification](https://mlsanity.com/paper/2403.01723/) (2024) - [Learning Physically Interpretable Atmospheric Models from Data with WSINDy](https://mlsanity.com/paper/2501.00738/) (2025) - [Learning effective models from network dynamics data with multiple initial conditions using weak form SINDy](https://mlsanity.com/paper/2605.30432/) (2026) - [SODAs: Sparse Optimization for the Discovery of Differential and Algebraic Equations](https://mlsanity.com/paper/2503.05993/) (2025) - [Learning dynamical systems from noisy data with Weak-form Kernel Ridge Regression](https://mlsanity.com/paper/2607.00257/) (2026) - [LLM-PDESR: Robust PDE Discovery via Subdomain Weighted Residuals and LLM-Guided Symbolic Hypothesis Generation](https://mlsanity.com/paper/2607.10546/) (2026) - [Time-Dependent PDE-Constrained Optimization via Weak-Form Latent Dynamics](https://mlsanity.com/paper/2605.20639/) (2026) ## Abstract (arXiv) In this work we study the asymptotic consistency of the weak-form sparse identification of nonlinear dynamics algorithm (WSINDy) in the identification of differential equations from noisy samples of solutions. We prove that the WSINDy estimator is unconditionally asymptotically consistent for a wide class of models which includes the Navier-Stokes equations and the Kuramoto-Sivashinsky equation. We thus provide a mathematically rigorous explanation for the observed robustness to noise of weak-form equation learning. Conversely, we also show that in general the WSINDy estimator is only conditionally asymptotically consistent, yielding discovery of spurious terms with probability one if the noise level is above some critical threshold and the nonlinearities exhibit sufficiently fast growth. We derive explicit bounds on the critical noise threshold in the case of Gaussian white noise and provide an explicit characterization of these spurious terms in the case of trigonometric and/or polynomial model nonlinearities. However, a silver lining to this negative result is that if the data is suitably denoised (a simple moving average filter is sufficient), then we recover unconditional asymptotic consistency on the class of models with locally-Lipschitz nonlinearities. Altogether, our results reveal several important aspects of weak-form equation learning which may be used to improve future algorithms. We demonstrate our results numerically using the Lorenz system, the cubic oscillator, a viscous Burgers growth model, and a Kuramoto-Sivashinsky-type higher-order PDE.