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

arXiv: https://arxiv.org/abs/2507.03206 · HTML: https://mlsanity.com/paper/2507.03206/

## 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 (most cited first, 9 of 9)

- [Weak SINDy For Partial Differential Equations](https://mlsanity.com/paper/2007.02848/) (2020)
- [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)
- [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)
- [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)
- [A practical identifiability criterion leveraging weak-form parameter estimation](https://mlsanity.com/paper/2506.17373/) (2025)

## Built on by (most cited first, 5 of 5)

- [Learning effective models from network dynamics data with multiple initial conditions using weak form SINDy](https://mlsanity.com/paper/2605.30432/) (2026)
- [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)
- [Weak Dominant Balance for Robust Identification of Dynamically Consistent Fluid Flow Structure](https://mlsanity.com/paper/2606.29047/) (2026)
- [Tensor-Train Weak SINDy: Identifying High-Dimensional Nonlinear Dynamics](https://mlsanity.com/paper/2609.09434/) (2026)

## 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.