Transformer networks sample up to 180x180 2D Ising systems and 64x64 Edwards-Anderson systems by generating spin groups with probability approximations, yielding ~20x higher effective sample size than prior neural samplers at criticality.
Multilevel Generative Samplers for Investigating Critical Phenomena
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abstract
Investigating critical phenomena or phase transitions is of high interest in physics and chemistry, for which Monte Carlo (MC) simulations, a crucial tool for numerically analyzing macroscopic properties of given systems, are often hindered by an emerging divergence of correlation length -- known as scale invariance at criticality (SIC) in the renormalization group theory. SIC causes the system to behave the same at any length scale, from which many existing sampling methods suffer: long-range correlations cause critical slowing down in Markov chain Monte Carlo (MCMC), and require intractably large receptive fields for generative samplers. In this paper, we propose a Renormalization-informed Generative Critical Sampler (RiGCS) -- a novel sampler specialized for near-critical systems, where SIC is leveraged as an advantage rather than a nuisance. Specifically, RiGCS builds on MultiLevel Monte Carlo (MLMC) with Heat Bath (HB) algorithms, which perform ancestral sampling from low-resolution to high-resolution lattice configurations with site-wise-independent conditional HB sampling. Although MLMC-HB is highly efficient under exact SIC, it suffers from a low acceptance rate under slight SIC violation. Notably, SIC violation always occurs in finite-size systems, and may induce long-range and higher-order interactions in the renormalized distributions, which are not considered by independent HB samplers. RiGCS enhances MLMC-HB by replacing a part of the conditional HB sampler with generative models that capture those residual interactions and improve the sampling efficiency. Our experiments show that the effective sample size of RiGCS is a few orders of magnitude higher than state-of-the-art generative model baselines in sampling configurations for 128x128 two-dimensional Ising systems.
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Fully convolutional diffusion models trained on small lattices transfer to unseen larger volumes for 2D/3D phi^4 sampling across phases, matching or beating same-size training on most observables.
Diffusion models suffer critical slowing down when sampling near criticality in the O(n) model but deeper local architectures reduce training-time scaling from quadratic to logarithmic in system size.
Incorporating probability priors into variational autoregressive networks reduces training burden and enables larger system sizes for sampling in the Ising and Edwards-Anderson models.
citing papers explorer
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Sampling two-dimensional spin systems with transformers
Transformer networks sample up to 180x180 2D Ising systems and 64x64 Edwards-Anderson systems by generating spin groups with probability approximations, yielding ~20x higher effective sample size than prior neural samplers at criticality.
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Diffusion Models for Sampling Near Criticality in Lattice Field Theories
Fully convolutional diffusion models trained on small lattices transfer to unseen larger volumes for 2D/3D phi^4 sampling across phases, matching or beating same-size training on most observables.
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The critical slowing down in diffusion models
Diffusion models suffer critical slowing down when sampling near criticality in the O(n) model but deeper local architectures reduce training-time scaling from quadratic to logarithmic in system size.
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Variational Autoregressive Networks with probability priors
Incorporating probability priors into variational autoregressive networks reduces training burden and enables larger system sizes for sampling in the Ising and Edwards-Anderson models.