# Physics-Guided, Physics-Informed, and Physics-Encoded Neural Networks in Scientific Computing

**Authors:** Salah A Faroughi, Nikhil Pawar, Celio Fernandes, Subasish Das, Nima K. Kalantari, Seyed Kourosh Mahjour  
**arXiv:** [2211.07377](https://arxiv.org/abs/2211.07377)  
**Date:** 2022-11-14  
**Summary page:** https://www.summarizepaper.com/en/paper/2211.07377v1/

## Key points

- Recent breakthroughs in computing power have enabled the integration of machine learning and deep learning techniques into scientific computing across fields like fluid mechanics, solid mechanics, and materials science.
- Neural networks, particularly physics-guided neural networks (PgNN), physics-informed neural networks (PiNN), and physics-encoded neural networks (PeNN), play a crucial role in incorporating physical-driven or knowledge-based constraints during training.
- Conventional neural networks face challenges with sparse data common in scientific fields, leading to the emergence of specialized frameworks like PgNN, PiNN, and PeNN.
- Each framework offers unique advantages in modeling complex multiscale multi-physics phenomena but also presents limitations related to stability, convergence, and generalization that require further research.
- The combination of scientific computing and deep learning approaches enhances forward and inverse modeling for high-dimensional problems characterized by noisy data and complex meshes governed by non-linear differential equations.
- PgNN models extract latent interdependencies from sparse data, PiNN models deduce governing equations from interpolated data, and PeNN models explore extrapolation based on strong mathematical forms derived from combined datasets.
- Advancements are needed to enable the use of PgNNs, PiNNs, and PeNNs in large-scale three-dimensional or multi-dimensional problems through accelerated training methods using different parallelization paradigms.

## Summary

Recent breakthroughs in computing power have paved the way for the integration of machine learning and deep learning techniques into scientific computing across various fields such as fluid mechanics, solid mechanics, and materials science. Neural networks, in particular, play a crucial role in this hybridization, offering a foundation to incorporate physical-driven or knowledge-based constraints during training. However, conventional neural networks face challenges when dealing with sparse data common in scientific fields. To address these challenges, three distinct neural network frameworks have emerged: physics-guided neural networks (PgNN), physics-informed neural networks (PiNN), and physics-encoded neural networks (PeNN). Each framework offers unique advantages in accelerating the modeling of complex multiscale multi-physics phenomena but also presents limitations that require further research to overcome issues related to stability, convergence, and generalization. This study provides an in-depth review of these three neural network frameworks used in scientific computing research. It discusses state-of-the-art architectures, applications, limitations, and future research opportunities aimed at improving algorithms, considering causalities, expanding applications, and integrating scientific and deep learning solvers. The comparison table highlights the main characteristics of PgNNs,PiNNs,and PeNNs for modeling non-linear multiscale physiochemical phenomena with sparse datasets. The combination of scientific computing and deep learning approaches enhances forward and inverse modeling for high-dimensional problems characterized by noisy data and complex meshes governed by non-linear differential equations. This integration surpasses traditional computational mechanics solvers in scenarios where experimental data is sparse or hard-to-acquire. By utilizing PgNN models for extracting latent interdependencies from sparse data, PiNN models for deducing governing equations from interpolated data, and PeNN models for extrapolation exploration based on strong mathematical forms derived from the combined data sets. Looking ahead, further advancements are needed to enable the use of PgNNs,PiNNs,and PeNNs in large-scale three-dimensional or multi-dimensional problems. Accelerating the training of complex deep learning models through different parallelization paradigms will be essential to unlock the full potential of these frameworks in tackling challenging computational tasks across various scales. This critical review serves as a valuable resource for researchers and engineers seeking to integrate physics-based constraints into neural networks effectively.