# Deep-learning of Parametric Partial Differential Equations from Sparse and Noisy Data

arXiv: https://arxiv.org/abs/2005.07916 · HTML: https://mlsanity.com/paper/2005.07916/

## Key facts

- Cited by 19 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 16 May 2020.

## Built on by (most cited first, 19 of 19)

- [Discovering Nonlinear PDEs from Scarce Data with Physics-encoded Learning](https://mlsanity.com/paper/2201.12354/) (2022)
- [Deep Learning and Symbolic Regression for Discovering Parametric Equations](https://mlsanity.com/paper/2207.00529/) (2022)
- [Robust discovery of partial differential equations in complex situations](https://mlsanity.com/paper/2106.00008/) (2021)
- [Discovery of partial differential equations from highly noisy and sparse data with physics-informed information criterion](https://mlsanity.com/paper/2208.03322/) (2022)
- [A Method for Computing Inverse Parametric PDE Problems with Random-Weight Neural Networks](https://mlsanity.com/paper/2210.04338/) (2022)
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- [Physics-aware Machine Learning Revolutionizes Scientific Paradigm for Machine Learning and Process-based Hydrology](https://mlsanity.com/paper/2310.05227/) (2023)
- [Learning Differential Operators for Interpretable Time Series Modeling](https://mlsanity.com/paper/2209.01491/) (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)
- [Image features of a splashing drop on a solid surface extracted using a feedforward neural network](https://mlsanity.com/paper/2201.09541/) (2022)
- [An invariance constrained deep learning network for PDE discovery](https://mlsanity.com/paper/2402.03747/) (2024)
- [Data-driven discovery of governing differential equations across physical systems](https://mlsanity.com/paper/2606.09638/) (2026)
- [Discovering Governing Equations by Machine Learning implemented with Invariance](https://mlsanity.com/paper/2203.15586/) (2022)
- [Critical Sampling for Robust Evolution Operator Learning of Unknown Dynamical Systems](https://mlsanity.com/paper/2304.07485/) (2023)
- [Dynamic Gaussian Graph Operator: Learning parametric partial differential equations in arbitrary discrete mechanics problems](https://mlsanity.com/paper/2403.02810/) (2024)
- [Hypothesize, Evaluate, Refine: A Scientific Agent for PDE Discovery with Unknown Spatial Coefficient Fields](https://mlsanity.com/paper/2608.27475/) (2026)

## Abstract (arXiv)

Data-driven methods have recently made great progress in the discovery of partial differential equations (PDEs) from spatial-temporal data. However, several challenges remain to be solved, including sparse noisy data, incomplete candidate library, and spatially- or temporally-varying coefficients. In this work, a new framework, which combines neural network, genetic algorithm and adaptive methods, is put forward to address all of these challenges simultaneously. In the framework, a trained neural network is utilized to calculate derivatives and generate a large amount of meta-data, which solves the problem of sparse noisy data. Next, genetic algorithm is utilized to discover the form of PDEs and corresponding coefficients with an incomplete candidate library. Finally, a two-step adaptive method is introduced to discover parametric PDEs with spatially- or temporally-varying coefficients. In this method, the structure of a parametric PDE is first discovered, and then the general form of varying coefficients is identified. The proposed algorithm is tested on the Burgers equation, the convection-diffusion equation, the wave equation, and the KdV equation. The results demonstrate that this method is robust to sparse and noisy data, and is able to discover parametric PDEs with an incomplete candidate library.