# Physics-informed learning of governing equations from scarce data

arXiv: https://arxiv.org/abs/2005.03448 · HTML: https://mlsanity.com/paper/2005.03448/

## Key facts

- Cited by 167 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 5 May 2020.

## Built on by (most cited first, 30 of 167)

- [Finite Basis Physics-Informed Neural Networks (FBPINNs): a scalable domain decomposition approach for solving differential equations](https://mlsanity.com/paper/2107.07871/) (2021)
- [PhyCRNet: Physics-informed Convolutional-Recurrent Network for Solving Spatiotemporal PDEs](https://mlsanity.com/paper/2106.14103/) (2021)
- [Physics-Informed Machine Learning: A Survey on Problems, Methods and Applications](https://mlsanity.com/paper/2211.08064/) (2022)
- [From PINNs to PIKANs: Recent Advances in Physics-Informed Machine Learning](https://mlsanity.com/paper/2410.13228/) (2024)
- [Symbolic Physics Learner: Discovering governing equations via Monte Carlo tree search](https://mlsanity.com/paper/2205.13134/) (2022)
- [PINNsFormer: A Transformer-Based Framework For Physics-Informed Neural Networks](https://mlsanity.com/paper/2307.11833/) (2023)
- [Encoding physics to learn reaction-diffusion processes](https://mlsanity.com/paper/2106.04781/) (2021)
- [Physics-Guided Deep Learning for Dynamical Systems: A Survey](https://mlsanity.com/paper/2107.01272/) (2021)
- [Any equation is a forest: Symbolic genetic algorithm for discovering open-form partial differential equations (SGA-PDE)](https://mlsanity.com/paper/2106.11927/) (2021)
- [Correcting model misspecification in physics-informed neural networks (PINNs)](https://mlsanity.com/paper/2310.10776/) (2023)
- [PINNACLE: PINN Adaptive ColLocation and Experimental points selection](https://mlsanity.com/paper/2404.07662/) (2024)
- [GPLaSDI: Gaussian Process-based Interpretable Latent Space Dynamics Identification through Deep Autoencoder](https://mlsanity.com/paper/2308.05882/) (2023)
- [Uncertainty quantification for noisy inputs-outputs in physics-informed neural networks and neural operators](https://mlsanity.com/paper/2311.11262/) (2023)
- [Discovering Sparse Interpretable Dynamics from Partial Observations](https://mlsanity.com/paper/2107.10879/) (2021)
- [PDE-READ: Human-readable Partial Differential Equation Discovery using Deep Learning](https://mlsanity.com/paper/2111.00998/) (2021)
- [Discovering Nonlinear PDEs from Scarce Data with Physics-encoded Learning](https://mlsanity.com/paper/2201.12354/) (2022)
- [Physics-informed Dyna-Style Model-Based Deep Reinforcement Learning for Dynamic Control](https://mlsanity.com/paper/2108.00128/) (2021)
- [A Comprehensive Review of Latent Space Dynamics Identification Algorithms for Intrusive and Non-Intrusive Reduced-Order-Modeling](https://mlsanity.com/paper/2403.10748/) (2024)
- [PDE-LEARN: Using Deep Learning to Discover Partial Differential Equations from Noisy, Limited Data](https://mlsanity.com/paper/2212.04971/) (2022)
- [Physics-informed Spline Learning for Nonlinear Dynamics Discovery](https://mlsanity.com/paper/2105.02368/) (2021)
- [PhyMPGN: Physics-encoded Message Passing Graph Network for spatiotemporal PDE systems](https://mlsanity.com/paper/2410.01337/) (2024)
- [Variational Physics Informed Neural Networks: the role of quadratures and test functions](https://mlsanity.com/paper/2109.02035/) (2021)
- [Discovery of partial differential equations from highly noisy and sparse data with physics-informed information criterion](https://mlsanity.com/paper/2208.03322/) (2022)
- [SeismicNet: Physics-informed neural networks for seismic wave modeling in semi-infinite domain](https://mlsanity.com/paper/2210.14044/) (2022)
- [Weak-PDE-LEARN: A Weak Form Based Approach to Discovering PDEs From Noisy, Limited Data](https://mlsanity.com/paper/2309.04699/) (2023)
- [PROSE: Predicting Operators and Symbolic Expressions using Multimodal Transformers](https://mlsanity.com/paper/2309.16816/) (2023)
- [Deciphering and integrating invariants for neural operator learning with various physical mechanisms](https://mlsanity.com/paper/2311.14361/) (2023)
- [Noise-aware Physics-informed Machine Learning for Robust PDE Discovery](https://mlsanity.com/paper/2206.12901/) (2022)
- [Integration of knowledge and data in machine learning](https://mlsanity.com/paper/2202.10337/) (2022)
- [Bayesian Spline Learning for Equation Discovery of Nonlinear Dynamics with Quantified Uncertainty](https://mlsanity.com/paper/2210.08095/) (2022)

## Abstract (arXiv)

Harnessing data to discover the underlying governing laws or equations that describe the behavior of complex physical systems can significantly advance our modeling, simulation and understanding of such systems in various science and engineering disciplines. This work introduces a novel physics-informed deep learning framework to discover governing partial differential equations (PDEs) from scarce and noisy data for nonlinear spatiotemporal systems. In particular, this approach seamlessly integrates the strengths of deep neural networks for rich representation learning, physics embedding, automatic differentiation and sparse regression to (1) approximate the solution of system variables, (2) compute essential derivatives, as well as (3) identify the key derivative terms and parameters that form the structure and explicit expression of the PDEs. The efficacy and robustness of this method are demonstrated, both numerically and experimentally, on discovering a variety of PDE systems with different levels of data scarcity and noise accounting for different initial/boundary conditions. The resulting computational framework shows the potential for closed-form model discovery in practical applications where large and accurate datasets are intractable to capture.