mlsanityData to 14 Oct 2026

The Weak Form Is Stronger Than You Think

arXiv 2409.06751 ↗First posted 10 Sep 2024cs.LG cs.CE cs.NAcited by 11

Not on the map. Why?

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 .

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

19 of 144 references.

References · 19

  1. 2020Weak SINDy For Partial Differential Equationscited by 155
  2. 2021Ensemble-SINDy: Robust sparse model discovery in the low-data, high-noise limit, with active learning and controlcited by 137
  3. 2022gLaSDI: Parametric Physics-informed Greedy Latent Space Dynamics Identificationcited by 30
  4. 2021Learning Mean-Field Equations from Particle Data Using WSINDycited by 29
  5. 2023Direct Estimation of Parameters in ODE Models Using WENDy: Weak-form Estimation of Nonlinear Dynamicscited by 20
  6. 2023Weak-Form Latent Space Dynamics Identificationcited by 20
  7. 2022WeakIdent: Weak formulation for Identifying Differential Equations using Narrow-fit and Trimmingcited by 18
  8. 2020Data-Driven Discovery of Coarse-Grained Equationscited by 14
  9. 2022Asymptotic consistency of the WSINDy algorithm in the limit of continuum datacited by 14
  10. 2020Variational system identification of the partial differential equations governing microstructure evolution in materials: Inference over sparse and spatially unrelated datacited by 13
  11. 2024Physics-informed active learning with simultaneous weak-form latent space dynamics identificationcited by 13
  12. 2022Learning Anisotropic Interaction Rules from Individual Trajectories in a Heterogeneous Cellular Populationcited by 11
7 more
  1. 2022Convergence of weak-SINDy Surrogate Modelscited by 11
  2. 2023Coarse-Graining Hamiltonian Systems Using WSINDycited by 11
  3. 2024Weak-Form Inference for Hybrid Dynamical Systems in Ecologycited by 7
  4. 2014Parametric Estimation of Ordinary Differential Equations with Orthogonality Conditionscited by 5
  5. 2024Scalable Sparse Regression for Model Discovery: The Fast Lane to Insightcited by 4
  6. 2014A Tracking Approach to Parameter Estimation in Linear Ordinary Differential Equationscited by 1
  7. 2021Reference Tracking AND Observer Design for Space-Fractional Partial Differential Equation Modeling Gas Pressures in Fractured Mediacited by 1

Built on by

Cited by year (2026 to 14 Oct)

Citing papers per year: 2024: 1; 2025: 4; 2026: 6.

11 papers.

  1. 2025Weak Form Scientific Machine Learning: Test Function Construction for System Identificationcited by 5
  2. 2024Self-test loss functions for learning weak-form operators and gradient flowscited by 4
  3. 2025Learning Physically Interpretable Atmospheric Models from Data with WSINDycited by 4
  4. 2026Learning effective models from network dynamics data with multiple initial conditions using weak form SINDycited by 2
  5. 2026Fourier Weak SINDy: Spectral Test Function Selection for Robust Model Identificationcited by 1
  6. 2026Learning dynamical systems from noisy data with Weak-form Kernel Ridge Regressioncited by 1
  7. 2025Impilict Runge-Kutta based sparse identification of governing equations in biologically motivated systemscited by 0
  8. 2025From STLS to Projection-based Dictionary Selection in Sparse Regression for System Identificationcited by 0
  9. 2026WGFINNs: Weak formulation-based GENERIC formalism informed neural networkscited by 0
  10. 2026Time-Dependent PDE-Constrained Optimization via Weak-Form Latent Dynamicscited by 0
  11. 2026Tensor-Train Weak SINDy: Identifying High-Dimensional Nonlinear Dynamicscited by 0