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# Computational Physics

PHYS 7321, Fall 2025

Instructor: Jim Halverson

E-mail: ``j.halverson@northeastern.edu``

TA & Grader: Joydeep Naskar, ``naskar.j@northeastern.edu``

# Course Information

**Course Schedule:** Lectures TF 9:50-11:30.

**Course Location:** Shillman 210

**Office Hours:** TF 8:45-9:45 in-person, virtual on Slack.

**Lectures:** I'll lecture and type code from a script. We will debug in real time. Pedagogically, I think this makes sense. Let's talk about it.

**Hackathons:** We or you will solve a problem in real time in class. You'll get to suggest problems and vote on them. I'll make sure we hit a sufficiently diverse array of topics.

**Description:** Here's the official course description listed by the registrar's office.

> Covers basic numerical methods for differentiation, integration, and matrix operations used in linear algebra problems, discrete Fourier transforms, and standard and stochastic ordinary and partial differential equations. Specific applications of these methods may include classical chaos, computation of eigenstates of simple quantum systems, classical phase transitions, boundary value problems, pattern formation, and molecular dynamics and classical/quantum Monte Carlo methods to simulate the equilibrium and nonequilibrium properties of condensed phases.

To be sure, this is important material and I will teach you a good deal of it.

However, missing from this written-years-ago description is any nod toward computer science's ongoing revolution in machine learning (ML) and artificial intelligence (AI). This interface is of increasing importance to an increasing number of Ph.D. students in Physics. As your professor, it would be remiss of me if I didn't teach you the basics. Therefore, the entire course will be in Python and we will also cover:

- Introduction to Python.
- Supervised Learning.
- Reinforcement Learning.
- Genetic Algorithms.
- Generative Models.
- ML Theory: NNGP and NTK.

This list covers a broad swath of the recent progress in ML, and in each case we will focus not only on the basics of the techniques, but also applications within Physics.

**Course Goals:** Every subfield in physics develops niche techniques crucial to its success. However, most also rely crucially on broad computational techniques that are applicable to many subfields. Our goals in this course address the latter, and are:
- Proficiency in Python.
- Proficiency in the basics of ML.
- Assessing problems with respect to the potential use of computational techniques.
- Ability to rapidly develop and implement computational techniques to solve problems.

For most of you, maximizing your strengths in these areas will be extremely useful to your Ph.D. research. These skills also have much broader application, e.g. in many different types of industry, where you may one day work.

**Homework:** You'll be given homeworks from previous years for a small completion grade. We might work some of the problems in a hackathon.

**Quizzes:** There will quizzes aimed at testing basic proficiency in the topics of this course.

**Project:** There will also be a final project, done in coordination with a Northeastern faculty member. This is an excellent opportunity to learn about a new field, and to make a good impression on a potential Ph.D. advisor. The goal of the project is to implement a non-trivial algorithm in service of a physics problem, and obtain some non-trivial resuls. At the beginning of the semester you will 1) choose a faculty member to work with, 2) develop a plan with them, and 3) present the plan in a slide template, as one of the early homework assignments, which must be approved.

**What is cheating?:** You're allowed to discuss homeworks, but you have to write your own code, though you are allowed to use any code from class as a template. People were caught last year with identical code or nearly identical code with different variable naming. First offense is a zero on the homework. Second offense leads a failing grade and an academic integrity violation. Regarding language models: feel free to use them, but still think. The quizzes will be hard if you don't do the homework.

**Grading:** Project 50%, 3 Quizzes 40% total broken down as (10%, 15%, 15%), Homework 10%.

# Course Material and Calendar

**Part 1: Introduction to Python**
- 9/5: Introduction and Baby Hackathon (Pi and CLT)
- 9/9: Syllabus and Essentials
- 9/12: Neural Networks and Optimization
- 9/16: Hackathon 2: Python at Your Speed
- 9/19: Quiz 1: Introduction to Python, Lecture

**Part 2: Computational Physics**
- 9/23: Numerical Differentiation and Integration
- 9/26: Classical Mechanics
- 9/30: Electrostatics
- 10/3: Hackathon 2: Classical Physics
- 10/7: Schrodinger Equation
- 10/10: Hackathon 3: Quantum Mechanics
- 10/14: Statistics, Monte Carlo
- 10/17: 2d Ising Model
- 10/21: Hackathon 4: Statistical Mechanics
- 10/24: Quiz 2: Computational Physics, Lecture

**Part 3: Machine Learning and Physics**
- 10/28: Optimization and the Cosmological Constant
- 10/31: Supervised Learning and Particle Physics
- 11/4: CNNs and Galaxy Classification
- 11/7: Reinforcement Learning and Physical Control
- 11/14: Unsupervised Learning and Topology
- 11/18: Hackathon 5: Supervised Learning and Optimization
- 11/21: Generative Models
- 11/25: Neural Nets and QFT
- 12/2: Hackathon 6: Advanced ML
- 12/5: Quiz 3: ML and Physics

# Resources

Given the constantly evolving nature of the subject, my lectures will rely on a number of different resources, including some written by friends: I am indebted to Fabian Ruehle, Sven Krippendorf, and especially Adrian Feiguin for sharing their resources. Two excellent resources for the intersection of physics and ML are:
- [IAIFI](http://www.iaifi.org), officially the NSF AI Institute for Artificial Intelligence and Fundamental Interactions, an interdisciplinary institute with a focus on the intersection of AI and physics in collaboration between Northeastern, MIT, Harvard, and Tufts.
- [Physics $\cap$ ML](http://www.physicsmeetsml.org), pronounced "Physics Meets ML," a virtual seminar series organized by myself and some friends that builds on a meeting we ran at Microsoft Research in 2019.

A simple perusal of the website will give you a sense of what's out there. Details can be found in links to slides and videos. More broadly, some canonical ML references are:
- [Deep Learning](https://www.deeplearningbook.org), by Ian Goodfellow, Yoshua Bengio, and Aaron Courville.
- [Geometric Deep Learning](https://geometricdeeplearning.com), by Michael Bronstein, Joan Bruna, Taco Cohen, and Petar Veličković.
- [Reinforcement Learning](http://incompleteideas.net/book/the-book-2nd.html), by Richard Sutton and Andrew Barto.

Of course, I am also heavily influenced by and indebted to my collaborators on related topics. Some of our recent research works include
- [NN-FT](https://arxiv.org/abs/2307.03223). Neural network field theories are a new approach to field theory inspired by ML theory. The above link presents the most modern and thorough treatment, see also [here](https://arxiv.org/abs/2008.08601) for our original work and [here](https://arxiv.org/abs/2112.04527) for a brief article that developed a number of new concepts, including when an NN-FT is a *quantum* field theory.
- [Searching for Ribbons with Machine Learning](https://arxiv.org/abs/2304.09304) uses ML to build a state of the art algorithm for verifying ribbonness of a knot. Together with slice obstructions, this may be used to study the Smooth Poincaré Conjecture in four dimensions (SPC4), a major open problem in topology. Using it, we ruled out over 800 potential counterexamples to SPC4. See also the fantastic [Man and Machine](https://arxiv.org/abs/0906.5177) article about SPC4.
- [Machine Learning in the String Landscape](https://arxiv.org/abs/1707.00655) one of the original papers on ML for String Theory. One aspect that has stood the test of time is the introduction of conjecture generation, a technique for making ML rigorous.

You might also enjoy articles I've written that cover a numbre of Physics $\cap$ ML topics:
- [Rigor with Machine Learning from Field Theory to the Poincare Conjecture](https://www.nature.com/articles/s42254-024-00709-0) where some friends and I discuss various ways in which ML techniques can lead to rigorous results.
- [TASI Lectures on Physics for Machine Learning](https://arxiv.org/abs/2408.00082) which represent my perspective (as of 2024) on how physics ideas may inform ML theory through the expressivity, statistics, and dynamics and neural networks.

# Language Model Usage

Finally, it's 2025 and this is a course on computational physics. Therefore, unless for some unforeseen reason it becomes a problem, large language models (LLMs) may be used as coding assistants. My only rule is that you use *free* tools, fairness reasons, such as [ChatGPT](http://chat.openai.com), [GitHub Copilot](https://github.com/features/copilot), or [Claude](http://www.claude.ai). The latter is free with a student account; to learn about that, post in ``#ask-joydeep``. While this might be controversial to some, a goal of this course is to teach you to be a *productive* computational scientist, and that means using the best tools available. Crucially from an educational perspective, though these are useful, LLMs can produce code that is either wrong or doesn't run, so you still have to know what you're doing. You might find if you're just learning that you shouldn't use LLMs for awhile, test it out and see what works.

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