(Left to Right): Avalanche activity cascades in a sandpile automaton; a vortex street formed by flow past a cylinder; and Turing patterns in a reaction-diffusion model. All simulations from the course homeworks; a higher-resolution video may be viewed here

Computational Physics

Summary

Materials for UT Austin's graduate computational physics course, taught by William Gilpin.

This course aims to provide a very broad survey of computational methods that are particularly relevant to modern physics research. We will aim to cover efficient algorithm design and performance analysis, traditional numerical recipes such as integration and matrix manipulation, and emerging methods in data analysis and machine learning. Our goal by the end of the class will be to feel comfortable approaching diverse, open-ended computational problems that arise during research, and to be ready to design and share new algorithms with the broader research community.

The class website is located here. If you are enrolled in the course at UT, the syllabus and calendar are here

Contents

Many links below direct to Google Colaboratory, and can be run-in-browser without any installation as long as you are signed into a Google account. To download the raw source files, please refer to the GitHub repository

Homework Assignments

• HW1: The sandpile cellular automaton and directed percolation. Covers recursion, runtime scaling, vectorization
• HW2: Linear dynamical systems and decomposing a chaotic flow. Covers numerical linear algebra, optimization, and unsupervised learning
• HW3: Turing patterns and phase separation. Covers numerical integration; finite-differences and spectral methods
• HW4: Predicting turbulence with operator methods. Covers Supervised learning, time series forecasting, ridge, kernel, and logistic regression
• Homework solutions

Lecture Slides

• Lecture 1: Python syntax for Scientific Computing
  [Live Notebook]

• Lecture 1b: Object-oriented programming to find first-passage times of Brownian motion
  [Live Notebook] [vid1] [vid2]

• Lecture 1c: Vectorization, arrays, and the Mandelbrot set
  [Live Notebook]

• Lecture 2: Runtime complexity, convolutions, and the continuous Game of Life
  [Raw Notebook]

• Lecture 3: Finding the Feigenbaum constant with recursion and dynamic programming
  [Raw Notebook]

• Lecture 4: Detecting the onset of turbulence with the Fast Fourier Transform
  [Raw Notebook]

• Lecture 5: Condition Number and the irreversibility of chaos
  [Raw Notebook]

• Lecture 6a: Random walks on complex graphs
  [Live Notebook]

• Lecture 6b: Probing collaborator graphs with LU matrix inversion
  [Live Notebook]

• Lecture 7: Spectral graph theory and the QR eigenvalue algorithm
  [Live Notebook]

• Lecture 8: Singular Value Decomposition
  [Live Notebook]

• Lecture 9: Krylov subspace methods & Conjugate gradient methods
  [Live Notebook]

• Lecture 10: Optimization in low dimensions
  [Live Notebook]

• Lecture 11a: Multivariate Optimization and Potential Flows [Live Notebook]

• Lecture 11b: Monte Carlo methods and Hard Sphere Packing [Live Notebook]

• Lecture 12: Numerical Integration and predicting chaos
  [Live Notebook]

• Lecture 13: Variable step integration, symplectic and stochastic systems
  [Live Notebook]

• Lecture 14: A pragmatist's guide to numerical integration

• Lecture 15: Diffusion, relaxation, and instability [Live Notebook]

• Lecture 16: Shocks, solitons, and hyperbolic partial differential equations
  [Live Notebook]

• Lecture 17: Spectral solving using the Dedalus Python Package

• Lecture 18: Supervised Learning & The Ising Model

• Lecture 19: Classification, Logistic Regression, and phases of matter

• Lecture 20: Overfitting, bias-variance tradeoff, and double-descent
  [Live Notebook]

• Lecture 21: Unsupervised learning, embedding, clustering

• Lecture 22: Time series representation, featurizing chaos, kernel methods
  [Live Notebook]

• Lecture 23: Gaussian mixtures, expectation-maximization, and superresolution microscopy
  [Live Notebook]

• Lecture 24: Predicting the Reynolds number of turbulence with deep learning
  [Live Notebook]

• Lecture 25: Types of neural networks; symmetries in physical systems

• Lecture 26: Training neural networks with backpropagation
  [Live Notebook]

Notes

• How to use the high-performance computing cluster
• Matrix Derivatives notes by Yuezhang Liu

Laboratory Exercises

• Lab 1: Getting started with Python
• Lab 2: git, GitHub, and GitHub Pages
• Lab 3: Documentation and Formatting
• Lab 4: Automatically creating online documentation with Sphinx
• Lab 5: Unit Testing
• Lab 6: Structuring an Open-Source Repository

Example Final Projects

• Quantum Reinforcement Learning with the Grover method
• Modelling the contractile dynamics of muscle
• Neural System Identification by Training Recurrent Neural Networks
• Testing particle phenomenology beyond the Standard Model with Bayesian classification

Usage and improvements

If you are teaching a similar course, please feel free to use any or all of these materials. If you have any suggestions for improvements or find any errors, I would very much appreciate any feedback.

For errors or typos, please consider opening an issue or submit a correction as a pull request on GitHub.

For students, course-related questions are best posted on GitHub as Discussions or Issues on the course repository; for other issues, I can be reached via email

Requirements

We will primarily use Python 3 with the following packages

• numpy
• matplotlib
• scipy
• scikit-learn
• jupyter

For projects and other parts of the class, you might also need

• ipykernel
• scikit-image
• umap-learn
• statsmodels
• pytorch
• jax
• numba

Attributions

Portions of the material in this course are adapted or inspired by other open-source classes, including: Pankaj Mehta's Machine Learning for Physics Course, Chris Rycroft's Numerical Recipe's Course, Volodymyr Kuleshov's Applied Machine Learning course, Fei-Fei Li's Deep Learning for Computer Vision course, Lorena Barba's CFD course and Jim Crutchfield's Nonlinear Dynamics course

<script async src="https://www.googletagmanager.com/gtag/js?id=G-37RSFCXBQY"></script> <script> window.dataLayer = window.dataLayer || []; function gtag(){dataLayer.push(arguments);} gtag('js', new Date()); gtag('config', 'G-37RSFCXBQY'); </script>