Diffusion models improve sampling in complex lattice physics simulations
Diffusion models for the $O(3)$ nonlinear sigma model
Machine Learning
Summary
Sampling complex physical systems like the classical Heisenberg model can be difficult and slow. The authors train a type of AI called a diffusion model to help generate these samples more efficiently and accurately, using different scales of the system to speed things up. They also create new ways to check how well the AI is learning, without needing extra computation. Their work includes techniques to handle the mathematics of spherical spaces involved in these physics problems.
What this means in practice
- •For computational physicists: Generate accurate samples of spin lattice models more efficiently for studying phase transitions and critical phenomena.
- •For machine learning engineers: Implement multiscale diffusion models on spherical data to improve sampling quality and reduce runtime in generative modeling tasks.
Authors
Daniel Schubring
Abstract
A score-based diffusion model is trained on the two-dimensional classical Heisenberg model, the lattice representation of the $O(3)$ nonlinear sigma model. Neural networks are trained on three different coarse-grainings of the same training set to implement a multiscale sampling scheme that modestly improves both the accuracy and computational cost of direct sampling on the finest lattice. Stein (or Schwinger-Dyson) diagnostics for the two-point correlation function are introduced to assess the quality of the learned score without relying on reverse diffusion sampling. These diagnostics are also incorporated into an additional training loss. We discuss several technical considerations relevant to implementing diffusion models on $S^2$. These include a consideration of lifting the network input to $S^3$, as in the $CP^1$ representation, and developing a boundary-layer approximation to the short-time spherical heat kernel that extends its usefulness to larger diffusion times.