Diffusion models for the O(3) nonlinear sigma model
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 S2. These include a consideration of lifting the network input to S3, as in the CP1 representation, and developing a boundary-layer approximation to the short-time spherical heat kernel that extends its usefulness to larger diffusion times.
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BibTeX
@misc{schubring2026diffusion,
title = {{Diffusion models for the $O(3)$ nonlinear sigma model}},
author = {Daniel Schubring},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.39568},
eprint = {2610.39568},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.39568},
}
RIS
TY - UNPB
TI - Diffusion models for the O(3) nonlinear sigma model
AU - Daniel Schubring
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.39568
UR - https://arxiv.org/abs/2610.39568
AB - 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.
ER -