# An Analysis of Physics-Informed Neural Networks

arXiv: https://arxiv.org/abs/2303.02890 · HTML: https://mlsanity.com/paper/2303.02890/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 11 of its 68 references that are in the map.
- First posted on arXiv 6 Mar 2023.

## Builds on (most cited first, 11 of 11)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [An overview of gradient descent optimization algorithms](https://mlsanity.com/paper/1609.04747/) (2016)
- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [A Survey of Deep Learning Techniques for Autonomous Driving](https://mlsanity.com/paper/1910.07738/) (2019)
- [Physics Informed Deep Learning (Part II): Data-driven Discovery of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1711.10566/) (2017)
- [ElasticFace: Elastic Margin Loss for Deep Face Recognition](https://mlsanity.com/paper/2109.09416/) (2021)
- [On Loss Functions for Deep Neural Networks in Classification](https://mlsanity.com/paper/1702.05659/) (2017)
- [Wave Physics as an Analog Recurrent Neural Network](https://mlsanity.com/paper/1904.12831/) (2019)
- [Physics-informed neural networks for myocardial perfusion MRI quantification](https://mlsanity.com/paper/2011.12844/) (2020)
- [Wavefield Reconstruction Inversion: an example](https://mlsanity.com/paper/2003.14181/) (2020)
- [Analysis of the Gradient Method with an Armijo-Wolfe Line Search on a Class of Nonsmooth Convex Functions](https://mlsanity.com/paper/1711.08517/) (2017)

## Abstract (arXiv)

Whilst the partial differential equations that govern the dynamics of our world have been studied in great depth for centuries, solving them for complex, high-dimensional conditions and domains still presents an incredibly large mathematical and computational challenge. Analytical methods can be cumbersome to utilise, and numerical methods can lead to errors and inaccuracies. On top of this, sometimes we lack the information or knowledge to pose the problem well enough to apply these kinds of methods. Here, we present a new approach to approximating the solution to physical systems - physics-informed neural networks. The concept of artificial neural networks is introduced, the objective function is defined, and optimisation strategies are discussed. The partial differential equation is then included as a constraint in the loss function for the optimisation problem, giving the network access to knowledge of the dynamics of the physical system it is modelling. Some intuitive examples are displayed, and more complex applications are considered to showcase the power of physics informed neural networks, such as in seismic imaging. Solution error is analysed, and suggestions are made to improve convergence and/or solution precision. Problems and limitations are also touched upon in the conclusions, as well as some thoughts as to where physics informed neural networks are most useful, and where they could go next.