mlsanityData to 14 Oct 2026

Weak Form Scientific Machine Learning: Test Function Construction for System Identification

arXiv 2507.03206 ↗First posted 3 Jul 2025math.NA cs.LG cs.NAcited by 5

Not on the map. Why?

Abstract (arXiv)

Weak form Scientific Machine Learning (WSciML) is a recently developed framework for data-driven modeling and scientific discovery. It leverages the weak form of equation error residuals to provide enhanced noise robustness in system identification via convolving model equations with test functions, reformulating the problem to avoid direct differentiation of data. The performance, however, relies on wisely choosing a set of compactly supported test functions. In this work, we mathematically motivate a novel data-driven method for constructing Single-scale-Local reference functions for creating the set of test functions. Our approach numerically approximates the integration error introduced by the quadrature and identifies the support size for which the error is minimal, without requiring access to the model parameter values. Through numerical experiments across various models, noise levels, and temporal resolutions, we demonstrate that the selected supports consistently align with regions of minimal parameter estimation error. We also compare the proposed method against the strategy for constructing Multi-scale-Global (and orthogonal) test functions introduced in our prior work, demonstrating the improved computational efficiency.

Key facts. Cited by 5 papers in the map (data through 14 Oct 2026). Builds on 9 of its 46 references that are in the map. First posted on arXiv 3 Jul 2025.

Builds on

9 of 46 references.

References · 9

  1. 2020Weak SINDy For Partial Differential Equationscited by 155
  2. 2021Learning Mean-Field Equations from Particle Data Using WSINDycited by 29
  3. 2023Direct Estimation of Parameters in ODE Models Using WENDy: Weak-form Estimation of Nonlinear Dynamicscited by 20
  4. 2023Weak-Form Latent Space Dynamics Identificationcited by 20
  5. 2024Physics-informed active learning with simultaneous weak-form latent space dynamics identificationcited by 13
  6. 2022Learning Anisotropic Interaction Rules from Individual Trajectories in a Heterogeneous Cellular Populationcited by 11
  7. 2023Coarse-Graining Hamiltonian Systems Using WSINDycited by 11
  8. 2024The Weak Form Is Stronger Than You Thinkcited by 11
  9. 2025A practical identifiability criterion leveraging weak-form parameter estimationcited by 5

Built on by

Cited by year (2026 to 14 Oct)

Citing papers per year: 2026: 5.

5 papers.

  1. 2026Learning effective models from network dynamics data with multiple initial conditions using weak form SINDycited by 2
  2. 2026WGFINNs: Weak formulation-based GENERIC formalism informed neural networkscited by 0
  3. 2026Time-Dependent PDE-Constrained Optimization via Weak-Form Latent Dynamicscited by 0
  4. 2026Weak Dominant Balance for Robust Identification of Dynamically Consistent Fluid Flow Structurecited by 0
  5. 2026Tensor-Train Weak SINDy: Identifying High-Dimensional Nonlinear Dynamicscited by 0