Can you make a neural network satisfy a physical law? There are two main types of these laws: symmetries and ordinary/partial differential equations. I will focus on differential equations in this short presentation. The simplest way to bake information about a differential equation with neural networks is to create a regularization term for the loss function used in training. I will explain the mathematics of this idea. I will also talk about applying physics-informed neural networks to a plethora of applications spanning the range from solving differential equations for all possible parameters in one sweep (e.g., solve for all boundary conditions) to calibrating differential equations using data to design optimization. Then, we will work on a hands-on activity that shows you to implement the ideas in PyTorch. I am assuming some familiarity with how conventional neural networks are trained (stochastic gradient descent).

Also, you need to know the basics of PyTorch to follow along. Going over this tutorial should be sufficient: https://pytorch.org/tutorials/beginner/pytorch_with_examples.html.

This tutorial uses the Jupyter notebook A Hands-on Introduction to Physics-Informed Neural Networks found on nanoHUB.

Atharva finished his undergraduate from Purdue University in 2019 and started his Ph.D., wherein he is working towards developing models to estimate an individual’s state of knowledge about a specific engineering field.

Network for Computational Nanotechnology (NCN)

Researchers should cite this work as follows:

• Ilias Bilionis, Atharva Hans (2021), "A Hands-on Introduction to Physics-Informed Neural Networks," https://nanohub.org/resources/35060.

  BibTex | EndNote

1. data-driven materials discovery
2. data science
3. machine learning
4. materials science
5. NCN Group - Machine Learning Hands-on
6. neural networks
7. ODEs
8. PDEs
9. physics-based machine learning
10. Physics-Informed Neural Networks notebook
11. PyTorch