# Weak-PDE-LEARN: A Weak Form Based Approach to Discovering PDEs From Noisy, Limited Data

arXiv: https://arxiv.org/abs/2309.04699 · HTML: https://mlsanity.com/paper/2309.04699/

## Key facts

- Cited by 16 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 48 references that are in the map.
- First posted on arXiv 9 Sep 2023.

## Builds on (most cited first, 21 of 21)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [Hamiltonian Neural Networks](https://mlsanity.com/paper/1906.01563/) (2019)
- [AI Feynman: a Physics-Inspired Method for Symbolic Regression](https://mlsanity.com/paper/1905.11481/) (2019)
- [Deep Hidden Physics Models: Deep Learning of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1801.06637/) (2018)
- [Physics-informed learning of governing equations from scarce data](https://mlsanity.com/paper/2005.03448/) (2020)
- [Weak SINDy For Partial Differential Equations](https://mlsanity.com/paper/2007.02848/) (2020)
- [Rational neural networks](https://mlsanity.com/paper/2004.01902/) (2020)
- [DeepMoD: Deep learning for Model Discovery in noisy data](https://mlsanity.com/paper/1904.09406/) (2019)
- [DeepGreen: Deep Learning of Green's Functions for Nonlinear Boundary Value Problems](https://mlsanity.com/paper/2101.07206/) (2020)
- [DLGA-PDE: Discovery of PDEs with incomplete candidate library via combination of deep learning and genetic algorithm](https://mlsanity.com/paper/2001.07305/) (2020)
- [Data-driven discovery of Green's functions with human-understandable deep learning](https://mlsanity.com/paper/2105.00266/) (2021)
- [Data-driven discovery of PDEs in complex datasets](https://mlsanity.com/paper/1808.10788/) (2018)
- [PDE-READ: Human-readable Partial Differential Equation Discovery using Deep Learning](https://mlsanity.com/paper/2111.00998/) (2021)
- [PDE-LEARN: Using Deep Learning to Discover Partial Differential Equations from Noisy, Limited Data](https://mlsanity.com/paper/2212.04971/) (2022)
- [Data-driven discovery of free-form governing differential equations](https://mlsanity.com/paper/1910.05117/) (2019)
- [Discovery of partial differential equations from highly noisy and sparse data with physics-informed information criterion](https://mlsanity.com/paper/2208.03322/) (2022)
- [Asymptotic consistency of the WSINDy algorithm in the limit of continuum data](https://mlsanity.com/paper/2211.16000/) (2022)
- [Bayesian Deep Learning for Partial Differential Equation Parameter Discovery with Sparse and Noisy Data](https://mlsanity.com/paper/2108.04085/) (2021)
- [Learning the Delay Using Neural Delay Differential Equations](https://mlsanity.com/paper/2304.01329/) (2023)

## Built on by (most cited first, 16 of 16)

- [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)
- [Self-test loss functions for learning weak-form operators and gradient flows](https://mlsanity.com/paper/2412.03506/) (2024)
- [Data-driven discovery of governing differential equations across physical systems](https://mlsanity.com/paper/2606.09638/) (2026)
- [chebgreen: Learning and Interpolating Continuous Empirical Green's Functions from Data](https://mlsanity.com/paper/2501.18715/) (2025)
- [A Weak Penalty Neural ODE for Learning Chaotic Dynamics from Noisy Time Series](https://mlsanity.com/paper/2511.06609/) (2025)
- [Weak-PDE-Net: Discovering Open-Form PDEs via Differentiable Symbolic Networks and Weak Formulation](https://mlsanity.com/paper/2603.22951/) (2026)
- [Fourier Weak SINDy: Spectral Test Function Selection for Robust Model Identification](https://mlsanity.com/paper/2604.20141/) (2026)
- [Joint discovery of governing partial differential equations from multi-source datasets by competitive optimization](https://mlsanity.com/paper/2606.30699/) (2026)
- [Symbolic Graph Networks for Robust PDE Discovery from Noisy Sparse Data](https://mlsanity.com/paper/2603.22380/) (2026)
- [Balance-Guided Sparse Identification of Multiscale Nonlinear PDEs with Small-coefficient Terms](https://mlsanity.com/paper/2604.18414/) (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)
- [Hypothesize, Evaluate, Refine: A Scientific Agent for PDE Discovery with Unknown Spatial Coefficient Fields](https://mlsanity.com/paper/2608.27475/) (2026)
- [Learning PDE Dynamics between Submanifolds Using Green's Observation Operators](https://mlsanity.com/paper/2610.01697/) (2026)

## Abstract (arXiv)

We introduce Weak-PDE-LEARN, a Partial Differential Equation (PDE) discovery algorithm that can identify non-linear PDEs from noisy, limited measurements of their solutions. Weak-PDE-LEARN uses an adaptive loss function based on weak forms to train a neural network, U, to approximate the PDE solution while simultaneously identifying the governing PDE. This approach yields an algorithm that is robust to noise and can discover a range of PDEs directly from noisy, limited measurements of their solutions. We demonstrate the efficacy of Weak-PDE-LEARN by learning several benchmark PDEs.