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.
Generative samplers for lattice field theory are commonly evaluated by their effective sample size (ESS). We show that a small global ESS need not imply inaccurate long-distance observables: errors spread over many short-distance modes can dominate this diagnostic. We introduce a scale-resolved diagnostic based on the overlap between coarse-grained model and target distributions. At the lattice cutoff, this overlap determines the asymptotic ESS fraction; lower cutoffs isolate long-wavelength errors. In the exactly solvable free theory, two samplers with comparable ESS yield a $32\%$ bias and no bias in the susceptibility, respectively. In two-dimensional $\phi^4$ theory near criticality, trained diffusion samplers show the same separation of scales: their path-weight ESS fraction falls rapidly over the volumes studied, while low moments of the magnetization remain close to their target values. Varying the reference parameters substantially changes infrared observables, with little effect on the log-weight variance. Given the target magnetization distribution, a marginal Metropolis test maintains an acceptance rate of order one in our tests. Combined with exact local updates, it reduces the remaining mismatch, although residual bias remains at finite relaxation length. In the free field, uniformly bounded relative errors in the proposal mode variances give a volume-independent bound on the number of updates needed for a prescribed accuracy of the chain's stationary mode variances at fixed correlation length. The target marginal must be supplied or computed separately; estimating it from the samplers' own weights can inherit the poor global overlap.
Machine learning (ML) methods have been explored as a means of optimizing lattice field theory calculations, with increasing interest in recent years. This chapter gives an introduction and summary to the use of ML methods to optimize lattice field theory without introducing any uncontrolled approximations. We focus on those directions which are clearly established as of late 2025, marking the 50th anniversary of the beginning of the study of lattice gauge theory. This chapter is a contribution to the online book ``Lattice QCD at 50 years,'' (LQCD@50), edited by Tanmoy Bhattacharya, Maarten Golterman, Rajan Gupta, Laurent Lellouch, and Steve Sharpe.
The Hubbard model at finite chemical potential is a cornerstone for understanding doped correlated systems, but simulations are severely limited by the sign problem. In the auxiliary-field formulation, the spin basis mitigates the sign problem, yet severe ergodicity issues have limited its use. We extend recent advances with normalizing flows at half-filling to finite chemical potential by introducing an annealing scheme enabling ergodic sampling. Compared to state-of-the-art hybrid Monte Carlo in the charge basis, our approach accurately reproduces exact diagonalization results while reducing statistical uncertainties by an order of magnitude, opening a new path for simulations of doped correlated systems.