# Deep-learning based discovery of partial differential equations in integral form from sparse and noisy data

arXiv: https://arxiv.org/abs/2011.11981 · HTML: https://mlsanity.com/paper/2011.11981/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 24 Nov 2020.

## Built on by (most cited first, 14 of 14)

- [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)
- [Integration of knowledge and data in machine learning](https://mlsanity.com/paper/2202.10337/) (2022)
- [Bayesian Deep Learning for Partial Differential Equation Parameter Discovery with Sparse and Noisy Data](https://mlsanity.com/paper/2108.04085/) (2021)
- [Combining physics-based and data-driven techniques for reliable hybrid analysis and modeling using the corrective source term approach](https://mlsanity.com/paper/2206.03451/) (2022)
- [Physics-informed Temporal Alignment for Auto-regressive PDE Foundation Models](https://mlsanity.com/paper/2505.10930/) (2025)
- [A novel corrective-source term approach to modeling unknown physics in aluminum extraction process](https://mlsanity.com/paper/2209.10861/) (2022)
- [Deep-Learning Discovers Macroscopic Governing Equations for Viscous Gravity Currents from Microscopic Simulation Data](https://mlsanity.com/paper/2106.00009/) (2021)
- [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)
- [Joint discovery of governing partial differential equations from multi-source datasets by competitive optimization](https://mlsanity.com/paper/2606.30699/) (2026)
- [Impilict Runge-Kutta based sparse identification of governing equations in biologically motivated systems](https://mlsanity.com/paper/2502.20319/) (2025)
- [EqOD: Symmetry-Informed Stability Selection for PDE Identification](https://mlsanity.com/paper/2605.11524/) (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)

## Abstract (arXiv)

Data-driven discovery of partial differential equations (PDEs) has attracted increasing attention in recent years. Although significant progress has been made, certain unresolved issues remain. For example, for PDEs with high-order derivatives, the performance of existing methods is unsatisfactory, especially when the data are sparse and noisy. It is also difficult to discover heterogeneous parametric PDEs where heterogeneous parameters are embedded in the partial differential operators. In this work, a new framework combining deep-learning and integral form is proposed to handle the above-mentioned problems simultaneously, and improve the accuracy and stability of PDE discovery. In the framework, a deep neural network is firstly trained with observation data to generate meta-data and calculate derivatives. Then, a unified integral form is defined, and the genetic algorithm is employed to discover the best structure. Finally, the value of parameters is calculated, and whether the parameters are constants or variables is identified. Numerical experiments proved that our proposed algorithm is more robust to noise and more accurate compared with existing methods due to the utilization of integral form. Our proposed algorithm is also able to discover PDEs with high-order derivatives or heterogeneous parameters accurately with sparse and noisy data.