Multilevel Generative Samplers for Investigating Critical Phenomena

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.

Multilevel Generative Samplers for Investigating Critical Phenomena

The paper introduces a novel approach in sampling techniques within the context of critical phenomena, where phase transitions present significant challenges due to diverging correlation lengths. It emphasizes the shortcomings of traditional Monte Carlo methods, especially the prevalent issue of critical slowing down in Markov Chain Monte Carlo (MCMC) simulations. The researchers propose the Renormalization-informed Generative Critical Sampler (RiGCS), which innovatively combines multi-level Monte Carlo approaches with generative models to overcome these simulation bottlenecks.

Conceptual Foundation

The study is grounded in the difficulties Monte Carlo simulations encounter due to scale invariance at criticality (SIC), leading to inefficiency in traditional sampling methods due to long-range correlations. Standard cluster algorithms focus on non-local moves, but still struggle with critical slowing down as system sizes increase. The authors address this by leveraging Renormalization Group Theory, focusing on the MultiLevel Monte Carlo with Heat Bath (MLMC-HB) algorithms, which traditionally improve sampling efficiency by employing block-spin transformations.

Methodology

The proposed RiGCS enhances MLMC-HB by incorporating generative models that can accommodate long-range and high-order interactions that MLMC-HB might miss due to finite system sizes leading to SIC violations. The authors suggest a structured, multi-level scheme where sampling is conducted from coarsest to finest resolutions using conditional models. This method acknowledges the inefficiency of straightforward heat bath samplers at fine resolutions when the SIC doesn’t hold precisely. Instead, RiGCS uses generative models to adaptively capture residual interactions, achieving higher sampling efficiency.

Experimental Results and Performance

The method's efficacy is highlighted through experiments on two-dimensional Ising systems. The results indicate that RiGCS significantly enhances effective sample sizes, showing improvements by orders of magnitude over existing generative models and baseline sampling methods. Furthermore, the authors present a sequential training protocol that exploits SIC to effectively transfer model parameters across resolutions, accelerating the training process considerably.

Implications and Future Directions

Practically, the advancement of RiGCS suggests enhanced capabilities in simulations where traditional methods falter, particularly in biology, chemistry, and materials science, where understanding phase transitions is crucial. Theoretically, this work underscores the importance of integrating renormalization principles with machine learning methodologies to combat the pervasive issue of critical slowing down, thus broadening the scope for more efficient computational models in complex systems.

Future developments could explore adapting the RiGCS framework to other physical models and dimensions, refining the interaction of neural networks with renormalization concepts. It opens an avenue for hybrid approaches combining RiGCS with existing techniques to further tackle the inherent complexities in high-dimensional lattice simulations. As generative modeling continues to evolve, integrating these advanced samplers will undoubtedly progress the capabilities of computational physics, potentially revolutionizing the understanding and technological applications of critical phenomena.