# PDE-LEARN: Using Deep Learning to Discover Partial Differential Equations from Noisy, Limited Data

arXiv: https://arxiv.org/abs/2212.04971 · HTML: https://mlsanity.com/paper/2212.04971/

## Key facts

- Cited by 20 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 39 references that are in the map.
- First posted on arXiv 9 Dec 2022.

## Builds on (most cited first, 14 of 14)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [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)
- [DLGA-PDE: Discovery of PDEs with incomplete candidate library via combination of deep learning and genetic algorithm](https://mlsanity.com/paper/2001.07305/) (2020)
- [Any equation is a forest: Symbolic genetic algorithm for discovering open-form partial differential equations (SGA-PDE)](https://mlsanity.com/paper/2106.11927/) (2021)
- [PDE-READ: Human-readable Partial Differential Equation Discovery using Deep Learning](https://mlsanity.com/paper/2111.00998/) (2021)
- [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)
- [Bayesian Deep Learning for Partial Differential Equation Parameter Discovery with Sparse and Noisy Data](https://mlsanity.com/paper/2108.04085/) (2021)
- [Neural network augmented inverse problems for PDEs](https://mlsanity.com/paper/1712.09685/) (2017)

## Built on by (most cited first, 20 of 20)

- [GPLaSDI: Gaussian Process-based Interpretable Latent Space Dynamics Identification through Deep Autoencoder](https://mlsanity.com/paper/2308.05882/) (2023)
- [Weak-PDE-LEARN: A Weak Form Based Approach to Discovering PDEs From Noisy, Limited Data](https://mlsanity.com/paper/2309.04699/) (2023)
- [Spatio-spectral graph neural operator for solving computational mechanics problems on irregular domain and unstructured grid](https://mlsanity.com/paper/2409.00604/) (2024)
- [When is a System Discoverable from Data? Discovery Requires Chaos](https://mlsanity.com/paper/2511.08860/) (2025)
- [Mechanistic PDE Networks for Discovery of Governing Equations](https://mlsanity.com/paper/2502.18377/) (2025)
- [Data-driven discovery of governing differential equations across physical systems](https://mlsanity.com/paper/2606.09638/) (2026)
- [Data-Driven Discovery of PDEs via the Adjoint Method](https://mlsanity.com/paper/2401.17177/) (2024)
- [Meta-learning Loss Functions of Parametric Partial Differential Equations Using Physics-Informed Neural Networks](https://mlsanity.com/paper/2412.00225/) (2024)
- [Higher-Order LaSDI: Reduced Order Modeling with Multiple Time Derivatives](https://mlsanity.com/paper/2512.15997/) (2025)
- [Learning Coupled System Dynamics under Incomplete Physical Constraints and Missing Data](https://mlsanity.com/paper/2512.23761/) (2025)
- [Physics-Informed Neural Networks for Biological 2D+t Reaction-Diffusion Systems](https://mlsanity.com/paper/2604.18548/) (2026)
- [chebgreen: Learning and Interpolating Continuous Empirical Green's Functions from Data](https://mlsanity.com/paper/2501.18715/) (2025)
- [CTSR: Cartesian tensor-based sparse regression for data-driven discovery of high-dimensional invariant governing equations](https://mlsanity.com/paper/2504.07618/) (2025)
- [Weak-PDE-Net: Discovering Open-Form PDEs via Differentiable Symbolic Networks and Weak Formulation](https://mlsanity.com/paper/2603.22951/) (2026)
- [A Spectral Approach for Learning Spatiotemporal Neural Differential Equations](https://mlsanity.com/paper/2309.16131/) (2023)
- [Taylor-Sensus Network: Embracing Noise to Enlighten Uncertainty for Scientific Data](https://mlsanity.com/paper/2409.07942/) (2024)
- [Data Enrichment for Symbolic Regression Using Diffusion Models](https://mlsanity.com/paper/2606.00988/) (2026)
- [On the post-hoc Evaluation of PDE Discovery: A Multifaceted Challenge of Scientific Advancement](https://mlsanity.com/paper/2607.23753/) (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)
- [Hypothesize, Evaluate, Refine: A Scientific Agent for PDE Discovery with Unknown Spatial Coefficient Fields](https://mlsanity.com/paper/2608.27475/) (2026)

## Abstract (arXiv)

In this paper, we introduce PDE-LEARN, a novel deep learning algorithm that can identify governing partial differential equations (PDEs) directly from noisy, limited measurements of a physical system of interest. PDE-LEARN uses a Rational Neural Network, U, to approximate the system response function and a sparse, trainable vector, xi, to characterize the hidden PDE that the system response function satisfies. Our approach couples the training of U and xi using a loss function that (1) makes U approximate the system response function, (2) encapsulates the fact that U satisfies a hidden PDE that xi characterizes, and (3) promotes sparsity in xi using ideas from iteratively reweighted least-squares. Further, PDE-LEARN can simultaneously learn from several data sets, allowing it to incorporate results from multiple experiments. This approach yields a robust algorithm to discover PDEs directly from realistic scientific data. We demonstrate the efficacy of PDE-LEARN by identifying several PDEs from noisy and limited measurements.