# Discovery of partial differential equations from highly noisy and sparse data with physics-informed information criterion

arXiv: https://arxiv.org/abs/2208.03322 · HTML: https://mlsanity.com/paper/2208.03322/

## Key facts

- Cited by 16 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 34 references that are in the map.
- First posted on arXiv 5 Aug 2022.

## Builds on (most cited first, 17 of 17)

- [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)
- [SINDy-PI: A Robust Algorithm for Parallel Implicit Sparse Identification of Nonlinear Dynamics](https://mlsanity.com/paper/2004.02322/) (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)
- [DLGA-PDE: Discovery of PDEs with incomplete candidate library via combination of deep learning and genetic algorithm](https://mlsanity.com/paper/2001.07305/) (2020)
- [Data-driven discovery of PDEs in complex datasets](https://mlsanity.com/paper/1808.10788/) (2018)
- [Any equation is a forest: Symbolic genetic algorithm for discovering open-form partial differential equations (SGA-PDE)](https://mlsanity.com/paper/2106.11927/) (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)
- [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)
- [Deep-learning of Parametric Partial Differential Equations from Sparse and Noisy Data](https://mlsanity.com/paper/2005.07916/) (2020)
- [Deep-learning based discovery of partial differential equations in integral form from sparse and noisy data](https://mlsanity.com/paper/2011.11981/) (2020)
- [Data-driven PDE discovery with evolutionary approach](https://mlsanity.com/paper/1903.08011/) (2019)
- [Deep-Learning Discovers Macroscopic Governing Equations for Viscous Gravity Currents from Microscopic Simulation Data](https://mlsanity.com/paper/2106.00009/) (2021)

## Built on by (most cited first, 16 of 16)

- [PDE-LEARN: Using Deep Learning to Discover Partial Differential Equations from Noisy, Limited Data](https://mlsanity.com/paper/2212.04971/) (2022)
- [Weak-PDE-LEARN: A Weak Form Based Approach to Discovering PDEs From Noisy, Limited Data](https://mlsanity.com/paper/2309.04699/) (2023)
- [Generative Discovery of Partial Differential Equations by Learning from Math Handbooks](https://mlsanity.com/paper/2505.05869/) (2025)
- [Physics-constrained robust learning of open-form partial differential equations from limited and noisy data](https://mlsanity.com/paper/2309.07672/) (2023)
- [Adaptive Uncertainty-Guided Model Selection for Data-Driven PDE Discovery](https://mlsanity.com/paper/2308.10283/) (2023)
- [Data-driven discovery of governing differential equations across physical systems](https://mlsanity.com/paper/2606.09638/) (2026)
- [Joint discovery of governing partial differential equations from multi-source datasets by competitive optimization](https://mlsanity.com/paper/2606.30699/) (2026)
- [Towards stable real-world equation discovery with assessing differentiating quality influence](https://mlsanity.com/paper/2311.05787/) (2023)
- [Equation identification for fluid flows via physics-informed neural networks](https://mlsanity.com/paper/2408.17271/) (2024)
- [Discovery and inversion of the viscoelastic wave equation in inhomogeneous media](https://mlsanity.com/paper/2409.18370/) (2024)
- [OmniFluids: Physics Pre-trained Modeling of Fluid Dynamics](https://mlsanity.com/paper/2506.10862/) (2025)
- [NewPINNs: Physics-Informing Neural Networks Using Conventional Solvers for Partial Differential Equations](https://mlsanity.com/paper/2601.17207/) (2026)
- [EqOD: Symmetry-Informed Stability Selection for PDE Identification](https://mlsanity.com/paper/2605.11524/) (2026)
- [On the post-hoc Evaluation of PDE Discovery: A Multifaceted Challenge of Scientific Advancement](https://mlsanity.com/paper/2607.23753/) (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)
- [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 discovery of PDEs has made tremendous progress recently, and many canonical PDEs have been discovered successfully for proof-of-concept. However, determining the most proper PDE without prior references remains challenging in terms of practical applications. In this work, a physics-informed information criterion (PIC) is proposed to measure the parsimony and precision of the discovered PDE synthetically. The proposed PIC achieves state-of-the-art robustness to highly noisy and sparse data on seven canonical PDEs from different physical scenes, which confirms its ability to handle difficult situations. The PIC is also employed to discover unrevealed macroscale governing equations from microscopic simulation data in an actual physical scene. The results show that the discovered macroscale PDE is precise and parsimonious, and satisfies underlying symmetries, which facilitates understanding and simulation of the physical process. The proposition of PIC enables practical applications of PDE discovery in discovering unrevealed governing equations in broader physical scenes.