# PDE-READ: Human-readable Partial Differential Equation Discovery using Deep Learning

arXiv: https://arxiv.org/abs/2111.00998 · HTML: https://mlsanity.com/paper/2111.00998/

## Key facts

- Cited by 24 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 36 references that are in the map.
- First posted on arXiv 1 Nov 2021.

## Builds on (most cited first, 15 of 15)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Implicit Neural Representations with Periodic Activation Functions](https://mlsanity.com/paper/2006.09661/) (2020)
- [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)
- [Sparse learning of stochastic dynamic equations](https://mlsanity.com/paper/1712.02432/) (2017)
- [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)
- [Data-driven discovery of Green's functions with human-understandable deep learning](https://mlsanity.com/paper/2105.00266/) (2021)
- [DL-PDE: Deep-learning based data-driven discovery of partial differential equations from discrete and noisy data](https://mlsanity.com/paper/1908.04463/) (2019)
- [Data-driven discovery of free-form governing differential equations](https://mlsanity.com/paper/1910.05117/) (2019)
- [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)
- [Evolutionary-Based Sparse Regression for the Experimental Identification of Duffing Oscillator](https://mlsanity.com/paper/2005.08631/) (2020)

## Built on by (most cited first, 24 of 24)

- [GPLaSDI: Gaussian Process-based Interpretable Latent Space Dynamics Identification through Deep Autoencoder](https://mlsanity.com/paper/2308.05882/) (2023)
- [A Comprehensive Review of Latent Space Dynamics Identification Algorithms for Intrusive and Non-Intrusive Reduced-Order-Modeling](https://mlsanity.com/paper/2403.10748/) (2024)
- [PDE-LEARN: Using Deep Learning to Discover Partial Differential Equations from Noisy, Limited Data](https://mlsanity.com/paper/2212.04971/) (2022)
- [Discovery of partial differential equations from highly noisy and sparse data with physics-informed information criterion](https://mlsanity.com/paper/2208.03322/) (2022)
- [Weak-PDE-LEARN: A Weak Form Based Approach to Discovering PDEs From Noisy, Limited Data](https://mlsanity.com/paper/2309.04699/) (2023)
- [Noise-aware Physics-informed Machine Learning for Robust PDE Discovery](https://mlsanity.com/paper/2206.12901/) (2022)
- [Generative Discovery of Partial Differential Equations by Learning from Math Handbooks](https://mlsanity.com/paper/2505.05869/) (2025)
- [ADAM-SINDy: An Efficient Optimization Framework for Parameterized Nonlinear Dynamical System Identification](https://mlsanity.com/paper/2410.16528/) (2024)
- [Physics-constrained robust learning of open-form partial differential equations from limited and noisy data](https://mlsanity.com/paper/2309.07672/) (2023)
- [Discovering stochastic partial differential equations from limited data using variational Bayes inference](https://mlsanity.com/paper/2306.15873/) (2023)
- [Physics-informed Temporal Alignment for Auto-regressive PDE Foundation Models](https://mlsanity.com/paper/2505.10930/) (2025)
- [Physics-Constrained Learning for PDE Systems with Uncertainty Quantified Port-Hamiltonian Models](https://mlsanity.com/paper/2406.11809/) (2024)
- [Data-Driven Autoencoder Numerical Solver with Uncertainty Quantification for Fast Physical Simulations](https://mlsanity.com/paper/2312.01021/) (2023)
- [Physics-informed AI and ML-based sparse system identification algorithm for discovery of PDE's representing nonlinear dynamic systems](https://mlsanity.com/paper/2410.10023/) (2024)
- [Data-driven discovery of governing differential equations across physical systems](https://mlsanity.com/paper/2606.09638/) (2026)
- [Higher-Order LaSDI: Reduced Order Modeling with Multiple Time Derivatives](https://mlsanity.com/paper/2512.15997/) (2025)
- [chebgreen: Learning and Interpolating Continuous Empirical Green's Functions from Data](https://mlsanity.com/paper/2501.18715/) (2025)
- [Plug-and-Play Physics-informed Learning using Uncertainty Quantified Port-Hamiltonian Models](https://mlsanity.com/paper/2504.17966/) (2025)
- [The Machine Learning Approach to Moment Closure Relations for Plasma: A Review](https://mlsanity.com/paper/2511.22486/) (2025)
- [Joint discovery of governing partial differential equations from multi-source datasets by competitive optimization](https://mlsanity.com/paper/2606.30699/) (2026)
- [WGFINNs: Weak formulation-based GENERIC formalism informed neural networks](https://mlsanity.com/paper/2604.02601/) (2026)
- [On the post-hoc Evaluation of PDE Discovery: A Multifaceted Challenge of Scientific Advancement](https://mlsanity.com/paper/2607.23753/) (2026)
- [Hypothesize, Evaluate, Refine: A Scientific Agent for PDE Discovery with Unknown Spatial Coefficient Fields](https://mlsanity.com/paper/2608.27475/) (2026)
- [See, Hypothesize, Validate: Multimodal Agentic Framework for Discovering Governing PDEs](https://mlsanity.com/paper/2608.27869/) (2026)

## Abstract (arXiv)

PDE discovery shows promise for uncovering predictive models of complex physical systems but has difficulty when measurements are sparse and noisy. We introduce a new approach for PDE discovery that uses two Rational Neural Networks and a principled sparse regression algorithm to identify the hidden dynamics that govern a system's response. The first network learns the system response function, while the second learns a hidden PDE describing the system's evolution. We then use a parameter-free sparse regression algorithm to extract a human-readable form of the hidden PDE from the second network. We implement our approach in an open-source library called PDE-READ. Our approach successfully identifies the governing PDE in six benchmark examples. We demonstrate that our approach is robust to both sparsity and noise and it, therefore, holds promise for application to real-world observational data.