# DL-PDE: Deep-learning based data-driven discovery of partial differential equations from discrete and noisy data

arXiv: https://arxiv.org/abs/1908.04463 · HTML: https://mlsanity.com/paper/1908.04463/

## Key facts

- Cited by 38 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 13 Aug 2019.

## Built on by (most cited first, 30 of 38)

- [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)
- [Robust discovery of partial differential equations in complex situations](https://mlsanity.com/paper/2106.00008/) (2021)
- [WeakIdent: Weak formulation for Identifying Differential Equations using Narrow-fit and Trimming](https://mlsanity.com/paper/2211.03134/) (2022)
- [Discovery of partial differential equations from highly noisy and sparse data with physics-informed information criterion](https://mlsanity.com/paper/2208.03322/) (2022)
- [Real-time Inference and Extrapolation via a Diffusion-inspired Temporal Transformer Operator (DiTTO)](https://mlsanity.com/paper/2307.09072/) (2023)
- [Bayesian Deep Learning for Partial Differential Equation Parameter Discovery with Sparse and Noisy Data](https://mlsanity.com/paper/2108.04085/) (2021)
- [Generative Discovery of Partial Differential Equations by Learning from Math Handbooks](https://mlsanity.com/paper/2505.05869/) (2025)
- [Physics-aware Machine Learning Revolutionizes Scientific Paradigm for Machine Learning and Process-based Hydrology](https://mlsanity.com/paper/2310.05227/) (2023)
- [Physics-constrained robust learning of open-form partial differential equations from limited and noisy data](https://mlsanity.com/paper/2309.07672/) (2023)
- [D-CIPHER: Discovery of Closed-form Partial Differential Equations](https://mlsanity.com/paper/2206.10586/) (2022)
- [Data-Efficient Kernel Methods for Learning Differential Equations and Their Solution Operators: Algorithms and Error Analysis](https://mlsanity.com/paper/2503.01036/) (2025)
- [Physics-Guided Discovery of Highly Nonlinear Parametric Partial Differential Equations](https://mlsanity.com/paper/2106.01078/) (2021)
- [Physics-informed Temporal Alignment for Auto-regressive PDE Foundation Models](https://mlsanity.com/paper/2505.10930/) (2025)
- [Deep-Learning Discovers Macroscopic Governing Equations for Viscous Gravity Currents from Microscopic Simulation Data](https://mlsanity.com/paper/2106.00009/) (2021)
- [Parsimony-Enhanced Sparse Bayesian Learning for Robust Discovery of Partial Differential Equations](https://mlsanity.com/paper/2107.07040/) (2021)
- [Predicting Shallow Water Dynamics using Echo-State Networks with Transfer Learning](https://mlsanity.com/paper/2112.09182/) (2021)
- [DISCOVER: Deep identification of symbolically concise open-form PDEs via enhanced reinforcement-learning](https://mlsanity.com/paper/2210.02181/) (2022)
- [An invariance constrained deep learning network for PDE discovery](https://mlsanity.com/paper/2402.03747/) (2024)
- [A Data-Driven Framework for Discovering Fractional Differential Equations in Complex Systems](https://mlsanity.com/paper/2412.03970/) (2024)
- [Physics-Coupled Spatio-Temporal Active Learning for Dynamical Systems](https://mlsanity.com/paper/2108.05385/) (2021)
- [Multigoal-oriented dual-weighted-residual error estimation using deep neural networks](https://mlsanity.com/paper/2112.11360/) (2021)
- [Predictions Based on Pixel Data: Insights from PDEs and Finite Differences](https://mlsanity.com/paper/2305.00723/) (2023)
- [Discovering an interpretable mathematical expression for a full wind-turbine wake with artificial intelligence enhanced symbolic regression](https://mlsanity.com/paper/2406.00695/) (2024)
- [ST-PCNN: Spatio-Temporal Physics-Coupled Neural Networks for Dynamics Forecasting](https://mlsanity.com/paper/2108.05940/) (2021)
- [Filtered Partial Differential Equations: a robust surrogate constraint in physics-informed deep learning framework](https://mlsanity.com/paper/2311.03776/) (2023)
- [Weak-PDE-Net: Discovering Open-Form PDEs via Differentiable Symbolic Networks and Weak Formulation](https://mlsanity.com/paper/2603.22951/) (2026)
- [Joint discovery of governing partial differential equations from multi-source datasets by competitive optimization](https://mlsanity.com/paper/2606.30699/) (2026)
- [Discovering Governing Equations by Machine Learning implemented with Invariance](https://mlsanity.com/paper/2203.15586/) (2022)
- [PDE Discovery for Soft Sensors Using Coupled Physics-Informed Neural Network with Akaike's Information Criterion](https://mlsanity.com/paper/2308.06132/) (2023)

## Abstract (arXiv)

In recent years, data-driven methods have been developed to learn dynamical systems and partial differential equations (PDE). The goal of such work is discovering unknown physics and the corresponding equations. However, prior to achieving this goal, major challenges remain to be resolved, including learning PDE under noisy data and limited discrete data. To overcome these challenges, in this work, a deep-learning based data-driven method, called DL-PDE, is developed to discover the governing PDEs of underlying physical processes. The DL-PDE method combines deep learning via neural networks and data-driven discovery of PDE via sparse regressions. In the DL-PDE, a neural network is first trained, and then a large amount of meta-data is generated, and the required derivatives are calculated by automatic differentiation. Finally, the form of PDE is discovered by sparse regression. The proposed method is tested with physical processes, governed by groundwater flow equation, convection-diffusion equation, Burgers equation and Korteweg-de Vries (KdV) equation, for proof-of-concept and applications in real-world engineering settings. The proposed method achieves satisfactory results when data are noisy and limited.