# PID-GAN: A GAN Framework based on a Physics-informed Discriminator for Uncertainty Quantification with Physics

arXiv: https://arxiv.org/abs/2106.02993 · HTML: https://mlsanity.com/paper/2106.02993/

## Key facts

- Cited by 21 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 24 references that are in the map.
- First posted on arXiv 6 Jun 2021.

## Builds on (most cited first, 20 of 20)

- [Dropout as a Bayesian Approximation: Representing Model Uncertainty in Deep Learning](https://mlsanity.com/paper/1506.02142/) (2015)
- [Improved Techniques for Training GANs](https://mlsanity.com/paper/1606.03498/) (2016)
- [Improved Training of Wasserstein GANs](https://mlsanity.com/paper/1704.00028/) (2017)
- [Hands-on Bayesian Neural Networks -- a Tutorial for Deep Learning Users](https://mlsanity.com/paper/2007.06823/) (2020)
- [Theory-guided Data Science: A New Paradigm for Scientific Discovery from Data](https://mlsanity.com/paper/1612.08544/) (2016)
- [Quantifying total uncertainty in physics-informed neural networks for solving forward and inverse stochastic problems](https://mlsanity.com/paper/1809.08327/) (2018)
- [Understanding and mitigating gradient pathologies in physics-informed neural networks](https://mlsanity.com/paper/2001.04536/) (2020)
- [Physics-guided Neural Networks (PGNN): An Application in Lake Temperature Modeling](https://mlsanity.com/paper/1710.11431/) (2017)
- [Adversarial Uncertainty Quantification in Physics-Informed Neural Networks](https://mlsanity.com/paper/1811.04026/) (2018)
- [Uncertainty in Neural Networks: Approximately Bayesian Ensembling](https://mlsanity.com/paper/1810.05546/) (2018)
- [Label-Free Supervision of Neural Networks with Physics and Domain Knowledge](https://mlsanity.com/paper/1609.05566/) (2016)
- [Towards Bayesian Deep Learning: A Framework and Some Existing Methods](https://mlsanity.com/paper/1608.06884/) (2016)
- [Physics Guided RNNs for Modeling Dynamical Systems: A Case Study in Simulating Lake Temperature Profiles](https://mlsanity.com/paper/1810.13075/) (2018)
- [Adversarial Distillation of Bayesian Neural Network Posteriors](https://mlsanity.com/paper/1806.10317/) (2018)
- [Physics-Guided Architecture (PGA) of Neural Networks for Quantifying Uncertainty in Lake Temperature Modeling](https://mlsanity.com/paper/1911.02682/) (2019)
- [Uncertainty Quantification with Generative Models](https://mlsanity.com/paper/1910.10046/) (2019)
- [Deep Echo State Networks with Uncertainty Quantification for Spatio-Temporal Forecasting](https://mlsanity.com/paper/1806.10728/) (2018)
- [Blending Diverse Physical Priors with Neural Networks](https://mlsanity.com/paper/1910.00201/) (2019)
- [The MBPEP: a deep ensemble pruning algorithm providing high quality uncertainty prediction](https://mlsanity.com/paper/1902.09238/) (2019)
- [Learning Structural Weight Uncertainty for Sequential Decision-Making](https://mlsanity.com/paper/1801.00085/) (2017)

## Built on by (most cited first, 21 of 21)

- [Uncertainty Quantification in Scientific Machine Learning: Methods, Metrics, and Comparisons](https://mlsanity.com/paper/2201.07766/) (2022)
- [A Survey on Uncertainty Quantification Methods for Deep Learning](https://mlsanity.com/paper/2302.13425/) (2023)
- [From PINNs to PIKANs: Recent Advances in Physics-Informed Machine Learning](https://mlsanity.com/paper/2410.13228/) (2024)
- [PI-VAE: Physics-Informed Variational Auto-Encoder for stochastic differential equations](https://mlsanity.com/paper/2203.11363/) (2022)
- [Ten Years of Generative Adversarial Nets (GANs): A survey of the state-of-the-art](https://mlsanity.com/paper/2308.16316/) (2023)
- [Bayesian Physics-Informed Extreme Learning Machine for Forward and Inverse PDE Problems with Noisy Data](https://mlsanity.com/paper/2205.06948/) (2022)
- [TrafficFlowGAN: Physics-informed Flow based Generative Adversarial Network for Uncertainty Quantification](https://mlsanity.com/paper/2206.09319/) (2022)
- [Physics-Informed Deep Learning For Traffic State Estimation: A Survey and the Outlook](https://mlsanity.com/paper/2303.02063/) (2023)
- [Quantifying Uncertainty In Traffic State Estimation Using Generative Adversarial Networks](https://mlsanity.com/paper/2206.09349/) (2022)
- [Generative Adversarial Reduced Order Modelling](https://mlsanity.com/paper/2305.15881/) (2023)
- [Precipitation Nowcasting Using Physics Informed Discriminator Generative Models](https://mlsanity.com/paper/2406.10108/) (2024)
- [Learning PDE Solvers with Physics and Data: A Unifying View of Physics-Informed Neural Networks and Neural Operators](https://mlsanity.com/paper/2601.14517/) (2026)
- [PI-VEGAN: Physics Informed Variational Embedding Generative Adversarial Networks for Stochastic Differential Equations](https://mlsanity.com/paper/2307.11289/) (2023)
- [Physics-guided training of GAN to improve accuracy in airfoil design synthesis](https://mlsanity.com/paper/2308.10038/) (2023)
- [Physics-Informed Deep Learning: A Promising Technique for System Reliability Assessment](https://mlsanity.com/paper/2108.10828/) (2021)
- [SPIEDiff: robust learning of long-time macroscopic dynamics from short-time particle simulations with quantified epistemic uncertainty](https://mlsanity.com/paper/2505.13501/) (2025)
- [Probabilistic Inverse Modeling: An Application in Hydrology](https://mlsanity.com/paper/2210.06213/) (2022)
- [Generating artificial digital image correlation data using physics-guided adversarial networks](https://mlsanity.com/paper/2303.15939/) (2023)
- [Label-free timing analysis of SiPM-based modularized detectors with physics-constrained deep learning](https://mlsanity.com/paper/2304.11930/) (2023)
- [Latent Generative Modeling of Random Fields from Limited Training Data](https://mlsanity.com/paper/2505.13007/) (2025)
- [Uncertainty Quantification in Inverse Models in Hydrology](https://mlsanity.com/paper/2310.02193/) (2023)

## Abstract (arXiv)

As applications of deep learning (DL) continue to seep into critical scientific use-cases, the importance of performing uncertainty quantification (UQ) with DL has become more pressing than ever before. In scientific applications, it is also important to inform the learning of DL models with knowledge of physics of the problem to produce physically consistent and generalized solutions. This is referred to as the emerging field of physics-informed deep learning (PIDL). We consider the problem of developing PIDL formulations that can also perform UQ. To this end, we propose a novel physics-informed GAN architecture, termed PID-GAN, where the knowledge of physics is used to inform the learning of both the generator and discriminator models, making ample use of unlabeled data instances. We show that our proposed PID-GAN framework does not suffer from imbalance of generator gradients from multiple loss terms as compared to state-of-the-art. We also empirically demonstrate the efficacy of our proposed framework on a variety of case studies involving benchmark physics-based PDEs as well as imperfect physics. All the code and datasets used in this study have been made available on this link : https://github.com/arkadaw9/PID-GAN.