# The Weak Form Is Stronger Than You Think

arXiv: https://arxiv.org/abs/2409.06751 · HTML: https://mlsanity.com/paper/2409.06751/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 144 references that are in the map.
- First posted on arXiv 10 Sep 2024.

## Builds on (most cited first, 19 of 19)

- [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)
- [gLaSDI: Parametric Physics-informed Greedy Latent Space Dynamics Identification](https://mlsanity.com/paper/2204.12005/) (2022)
- [Learning Mean-Field Equations from Particle Data Using WSINDy](https://mlsanity.com/paper/2110.07756/) (2021)
- [Direct Estimation of Parameters in ODE Models Using WENDy: Weak-form Estimation of Nonlinear Dynamics](https://mlsanity.com/paper/2302.13271/) (2023)
- [Weak-Form Latent Space Dynamics Identification](https://mlsanity.com/paper/2311.12880/) (2023)
- [WeakIdent: Weak formulation for Identifying Differential Equations using Narrow-fit and Trimming](https://mlsanity.com/paper/2211.03134/) (2022)
- [Data-Driven Discovery of Coarse-Grained Equations](https://mlsanity.com/paper/2002.00790/) (2020)
- [Asymptotic consistency of the WSINDy algorithm in the limit of continuum data](https://mlsanity.com/paper/2211.16000/) (2022)
- [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)
- [Physics-informed active learning with simultaneous weak-form latent space dynamics identification](https://mlsanity.com/paper/2407.00337/) (2024)
- [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)
- [Coarse-Graining Hamiltonian Systems Using WSINDy](https://mlsanity.com/paper/2310.05879/) (2023)
- [Weak-Form Inference for Hybrid Dynamical Systems in Ecology](https://mlsanity.com/paper/2405.20591/) (2024)
- [Parametric Estimation of Ordinary Differential Equations with Orthogonality Conditions](https://mlsanity.com/paper/1410.7566/) (2014)
- [Scalable Sparse Regression for Model Discovery: The Fast Lane to Insight](https://mlsanity.com/paper/2405.09579/) (2024)
- [A Tracking Approach to Parameter Estimation in Linear Ordinary Differential Equations](https://mlsanity.com/paper/1410.7554/) (2014)
- [Reference Tracking AND Observer Design for Space-Fractional Partial Differential Equation Modeling Gas Pressures in Fractured Media](https://mlsanity.com/paper/2107.02042/) (2021)

## Built on by (most cited first, 11 of 11)

- [Weak Form Scientific Machine Learning: Test Function Construction for System Identification](https://mlsanity.com/paper/2507.03206/) (2025)
- [Self-test loss functions for learning weak-form operators and gradient flows](https://mlsanity.com/paper/2412.03506/) (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)
- [Fourier Weak SINDy: Spectral Test Function Selection for Robust Model Identification](https://mlsanity.com/paper/2604.20141/) (2026)
- [Learning dynamical systems from noisy data with Weak-form Kernel Ridge Regression](https://mlsanity.com/paper/2607.00257/) (2026)
- [Impilict Runge-Kutta based sparse identification of governing equations in biologically motivated systems](https://mlsanity.com/paper/2502.20319/) (2025)
- [From STLS to Projection-based Dictionary Selection in Sparse Regression for System Identification](https://mlsanity.com/paper/2512.14404/) (2025)
- [WGFINNs: Weak formulation-based GENERIC formalism informed neural networks](https://mlsanity.com/paper/2604.02601/) (2026)
- [Time-Dependent PDE-Constrained Optimization via Weak-Form Latent Dynamics](https://mlsanity.com/paper/2605.20639/) (2026)
- [Tensor-Train Weak SINDy: Identifying High-Dimensional Nonlinear Dynamics](https://mlsanity.com/paper/2609.09434/) (2026)

## Abstract (arXiv)

The weak form is a ubiquitous, well-studied, and widely-utilized mathematical tool in modern computational and applied mathematics. In this work we provide a survey of both the history and recent developments for several fields in which the weak form can play a critical role. In particular, we highlight several recent advances in weak form versions of equation learning, parameter estimation, and coarse graining, which offer surprising noise robustness, accuracy, and computational efficiency. We note that this manuscript is a companion piece to our October 2024 SIAM News article of the same name. Here we provide more detailed explanations of mathematical developments as well as a more complete list of references. Lastly, we note that the software with which to reproduce the results in this manuscript is also available on our group's GitHub website https://github.com/MathBioCU .