Self-study notes at the intersection of physics, mathematics, and machine learning.
Live site: https://carlm451.github.io
These are my personal study notes, written while working through research papers and textbooks in computational physics and scientific machine learning. Each "course" is a guided breakdown of one or more papers, with worked numerical examples, SVG diagrams, and enough background math to follow the arguments from scratch.
This is a work in progress. Courses are added and revised as I study. Some are complete; others are in early stages or still planned. Errors are mine.
Credit where it's due. The core mathematics, algorithms, and scientific ideas in these notes belong to the authors of the cited papers. My contribution is the pedagogical scaffolding: the ordering, the worked examples, the diagrams, and the connecting narrative. Each course page cites its primary source paper.
Background for reading Hajij et al., "Topological Deep Learning: Going Beyond Graph Data" (2022). Eight chapters in three parts, covering graphs through simplicial/cell/combinatorial complexes to higher-order message passing and the architecture zoo. Includes a running example (4-vertex graph with 2 triangular faces) carried through every chapter with fully verified matrices.
Background for reading Wu, Wang & Zhang, "Solving Statistical Mechanics Using Variational Autoregressive Networks" (2019). Two chapters covering variational free energy, naive mean field theory on the 1D and 2D Ising model, autoregressive decomposition, and the one-layer VAN architecture with training loop.
Background for reading Li et al., "Fourier Neural Operator for Parametric Partial Differential Equations" (2020). Seven chapters in four parts: operator learning and the neural operator architecture (Part I), the Fourier-space parameterization (Part II), benchmarks on Burgers/Darcy/Navier-Stokes (Part III), and an application to ultrasonic nondestructive testing with angle beam crack detection (Part IV). Darcy flow running example throughout Parts I-III; parametric crack geometry running example in Part IV.
- Physics-Informed Neural Networks (PINNs)
- DeepONet & Neural Operator Theory
- PDEs for Machine Learning
- Optimization for Deep Learning
- Autodiff & Backpropagation from Scratch
- Molecular Dynamics Simulations
- Computational Electromagnetics
Static HTML site hosted on GitHub Pages. No build tools, no frameworks. MathJax for equations, inline SVG for diagrams, one shared CSS stylesheet.