# WeakIdent: Weak formulation for Identifying Differential Equations using Narrow-fit and Trimming arXiv: https://arxiv.org/abs/2211.03134 ยท HTML: https://mlsanity.com/paper/2211.03134/ ## Key facts - Cited by 18 papers in the map (data through 14 Oct 2026). - Builds on 12 of its 46 references that are in the map. - First posted on arXiv 6 Nov 2022. ## Builds on (most cited first, 12 of 12) - [Deep learning for universal linear embeddings of nonlinear dynamics](https://mlsanity.com/paper/1712.09707/) (2017) - [Hidden Physics Models: Machine Learning of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1708.00588/) (2017) - [Weak SINDy For Partial Differential Equations](https://mlsanity.com/paper/2007.02848/) (2020) - [SINDy-PI: A Robust Algorithm for Parallel Implicit Sparse Identification of Nonlinear Dynamics](https://mlsanity.com/paper/2004.02322/) (2020) - [Data Driven Governing Equations Approximation Using Deep Neural Networks](https://mlsanity.com/paper/1811.05537/) (2018) - [On the Convergence of the SINDy Algorithm](https://mlsanity.com/paper/1805.06445/) (2018) - [Data-Driven Deep Learning of Partial Differential Equations in Modal Space](https://mlsanity.com/paper/1910.06948/) (2019) - [DLGA-PDE: Discovery of PDEs with incomplete candidate library via combination of deep learning and genetic algorithm](https://mlsanity.com/paper/2001.07305/) (2020) - [DL-PDE: Deep-learning based data-driven discovery of partial differential equations from discrete and noisy data](https://mlsanity.com/paper/1908.04463/) (2019) - [Deep Neural Network Modeling of Unknown Partial Differential Equations in Nodal Space](https://mlsanity.com/paper/2106.03603/) (2021) - [Numerical Aspects for Approximating Governing Equations Using Data](https://mlsanity.com/paper/1809.09170/) (2018) - [Methods to Recover Unknown Processes in Partial Differential Equations Using Data](https://mlsanity.com/paper/2003.02387/) (2020) ## Built on by (most cited first, 18 of 18) - [A Comprehensive Review of Latent Space Dynamics Identification Algorithms for Intrusive and Non-Intrusive Reduced-Order-Modeling](https://mlsanity.com/paper/2403.10748/) (2024) - [Asymptotic consistency of the WSINDy algorithm in the limit of continuum data](https://mlsanity.com/paper/2211.16000/) (2022) - [Generative Discovery of Partial Differential Equations by Learning from Math Handbooks](https://mlsanity.com/paper/2505.05869/) (2025) - [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) - [Physics-constrained robust learning of open-form partial differential equations from limited and noisy data](https://mlsanity.com/paper/2309.07672/) (2023) - [Weak-Form Inference for Hybrid Dynamical Systems in Ecology](https://mlsanity.com/paper/2405.20591/) (2024) - [Beyond empirical models: Discovering new constitutive laws in solids with graph-based equation discovery](https://mlsanity.com/paper/2511.09906/) (2025) - [Group Projected Subspace Pursuit for Block Sparse Signal Reconstruction: Convergence Analysis and Applications](https://mlsanity.com/paper/2407.07707/) (2024) - [Robust Moment Identification for Nonlinear PDEs via a Neural ODE Approach](https://mlsanity.com/paper/2506.05245/) (2025) - [Joint discovery of governing partial differential equations from multi-source datasets by competitive optimization](https://mlsanity.com/paper/2606.30699/) (2026) - [Bias Inheritance in Neural-Symbolic Discovery of Constitutive Closures Under Function-Class Mismatch](https://mlsanity.com/paper/2604.01335/) (2026) - [WGFINNs: Weak formulation-based GENERIC formalism informed neural networks](https://mlsanity.com/paper/2604.02601/) (2026) - [EqOD: Symmetry-Informed Stability Selection for PDE Identification](https://mlsanity.com/paper/2605.11524/) (2026) - [Data Enrichment for Symbolic Regression Using Diffusion Models](https://mlsanity.com/paper/2606.00988/) (2026) - [Freeze, Then Select: Structured Field Adapters and Stability-Validated Weak Selection for PDE Discovery from Sparse Observations](https://mlsanity.com/paper/2607.29665/) (2026) - [Robust data-driven discovery of fractional differential equations via weak formulations and Pareto-based subset selection](https://mlsanity.com/paper/2608.12879/) (2026) - [See, Hypothesize, Validate: Multimodal Agentic Framework for Discovering Governing PDEs](https://mlsanity.com/paper/2608.27869/) (2026) ## Abstract (arXiv) Data-driven identification of differential equations is an interesting but challenging problem, especially when the given data are corrupted by noise. When the governing differential equation is a linear combination of various differential terms, the identification problem can be formulated as solving a linear system, with the feature matrix consisting of linear and nonlinear terms multiplied by a coefficient vector. This product is equal to the time derivative term, and thus generates dynamical behaviors. The goal is to identify the correct terms that form the equation to capture the dynamics of the given data. We propose a general and robust framework to recover differential equations using a weak formulation, for both ordinary and partial differential equations (ODEs and PDEs). The weak formulation facilitates an efficient and robust way to handle noise. For a robust recovery against noise and the choice of hyper-parameters, we introduce two new mechanisms, narrow-fit and trimming, for the coefficient support and value recovery, respectively. For each sparsity level, Subspace Pursuit is utilized to find an initial set of support from the large dictionary. Then, we focus on highly dynamic regions (rows of the feature matrix), and error normalize the feature matrix in the narrow-fit step. The support is further updated via trimming of the terms that contribute the least. Finally, the support set of features with the smallest Cross-Validation error is chosen as the result. A comprehensive set of numerical experiments are presented for both systems of ODEs and PDEs with various noise levels. The proposed method gives a robust recovery of the coefficients, and a significant denoising effect which can handle up to 100% noise-to-signal ratio for some equations. We compare the proposed method with several state-of-the-art algorithms for the recovery of differential equations.