Lecture 1PDEAutogradCollocationInverse Problems

Physics-Informed Neural Networks

Embedding physical laws directly into the loss function

Overview

Classical numerical solvers discretize space and time on a mesh. PINNs take a fundamentally different approach: they parameterize the solution as a neural network and enforce the governing PDE, initial conditions, and boundary conditions as soft constraints in the training loss. This lecture derives the PINN loss from first principles, explains why automatic differentiation enables exact PDE residuals without a mesh, and demonstrates the method on the viscous Burgers equation.

Benchmark & Results

Setup

Burgers equation (ν = 0.01/π), 5,081 parameters

Result

~5% relative L² error, zero labeled solution data

Physics-Informed Neural Networks benchmark results
Figure: Physics-Informed Neural Networks — benchmark results

Lecture Slides

The full slide deck for this lecture is available as a PDF. Each slide includes speaker notes for the presenter.

Open Slides (PDF) →Download PDF ↓

Code

The annotated implementation for this lecture is in pinn_burgers.py. All code is written in PyTorch and prioritizes clarity over cleverness.

# pinn_burgers.py # See the attached file for the full annotated implementation. # Key classes and functions are documented with docstrings.

References

  1. [1]Raissi, M., Perdikaris, P., & Karniadakis, G. E. (2019). Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics, 378, 686–707. DOI: 10.1016/j.jcp.2018.10.045
  2. [2]Lagaris, I. E., Likas, A., & Fotiadis, D. I. (1998). Artificial neural networks for solving ordinary and partial differential equations. IEEE Transactions on Neural Networks, 9(5), 987–1000. DOI: 10.1109/72.712178
  3. [3]Karniadakis, G. E., et al. (2021). Physics-informed machine learning. Nature Reviews Physics, 3(6), 422–440. DOI: 10.1038/s42254-021-00314-5

Cite As

If you use this lecture material in your research or teaching, please cite the primary reference:

@misc{jing2025sciml1,
  title  = {Lecture 1: Physics-Informed Neural Networks},
  author = {Jing, Cheng},
  year   = {2025},
  note   = {An Intro Course to Scientific Machine Learning, Arizona State University},
  url    = {https://jessecj.me/course/lecture-1-pinns},
  howpublished = {\url{https://jessecj.me/course/lecture-1-pinns}}
}