# Learning Partial Differential Equations from Data Using Neural Networks

arXiv: https://arxiv.org/abs/1910.10262 · HTML: https://mlsanity.com/paper/1910.10262/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 22 Oct 2019.

## Built on by (most cited first, 8 of 8)

- [Symbolic Recovery of Differential Equations: The Identifiability Problem](https://mlsanity.com/paper/2210.08342/) (2022)
- [When is a System Discoverable from Data? Discovery Requires Chaos](https://mlsanity.com/paper/2511.08860/) (2025)
- [Parsimony-Enhanced Sparse Bayesian Learning for Robust Discovery of Partial Differential Equations](https://mlsanity.com/paper/2107.07040/) (2021)
- [Robust identifiability for symbolic recovery of differential equations](https://mlsanity.com/paper/2410.09938/) (2024)
- [Neuro-Spectral Architectures for Causal Physics-Informed Networks](https://mlsanity.com/paper/2509.04966/) (2025)
- [ST-PCNN: Spatio-Temporal Physics-Coupled Neural Networks for Dynamics Forecasting](https://mlsanity.com/paper/2108.05940/) (2021)
- [A Neural Network Ensemble Approach to System Identification](https://mlsanity.com/paper/2110.08382/) (2021)
- [Beyond Field Accuracy: Two-Axis Diagnosis of Inverse-PINN Parameter Error](https://mlsanity.com/paper/2608.15373/) (2026)

## Abstract (arXiv)

We develop a framework for estimating unknown partial differential equations from noisy data, using a deep learning approach. Given noisy samples of a solution to an unknown PDE, our method interpolates the samples using a neural network, and extracts the PDE by equating derivatives of the neural network approximation. Our method applies to PDEs which are linear combinations of user-defined dictionary functions, and generalizes previous methods that only consider parabolic PDEs. We introduce a regularization scheme that prevents the function approximation from overfitting the data and forces it to be a solution of the underlying PDE. We validate the model on simulated data generated by the known PDEs and added Gaussian noise, and we study our method under different levels of noise. We also compare the error of our method with a Cramer-Rao lower bound for an ordinary differential equation. Our results indicate that our method outperforms other methods in estimating PDEs, especially in the low signal-to-noise regime.