# Multilevel Generative Samplers for Investigating Critical Phenomena

arXiv: https://arxiv.org/abs/2503.08918 · HTML: https://mlsanity.com/paper/2503.08918/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 35 of its 125 references that are in the map.
- First posted on arXiv 11 Mar 2025.

## Builds on (most cited first, 30 of 35)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [WaveNet: A Generative Model for Raw Audio](https://mlsanity.com/paper/1609.03499/) (2016)
- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [Conditional Image Generation with PixelCNN Decoders](https://mlsanity.com/paper/1606.05328/) (2016)
- [Pixel Recurrent Neural Networks](https://mlsanity.com/paper/1601.06759/) (2016)
- [PixelCNN++: Improving the PixelCNN with Discretized Logistic Mixture Likelihood and Other Modifications](https://mlsanity.com/paper/1701.05517/) (2017)
- [MADE: Masked Autoencoder for Distribution Estimation](https://mlsanity.com/paper/1502.03509/) (2015)
- [Symmetry-adapted generation of 3d point sets for the targeted discovery of molecules](https://mlsanity.com/paper/1906.00957/) (2019)
- [Flow-based generative models for Markov chain Monte Carlo in lattice field theory](https://mlsanity.com/paper/1904.12072/) (2019)
- [Neural Variational Inference and Learning in Belief Networks](https://mlsanity.com/paper/1402.0030/) (2014)
- [Why does deep and cheap learning work so well?](https://mlsanity.com/paper/1608.08225/) (2016)
- [Solving Statistical Mechanics Using Variational Autoregressive Networks](https://mlsanity.com/paper/1809.10606/) (2018)
- [Inverse design of 3d molecular structures with conditional generative neural networks](https://mlsanity.com/paper/2109.04824/) (2021)
- [Neural Network Renormalization Group](https://mlsanity.com/paper/1802.02840/) (2018)
- [Learning Lattice Quantum Field Theories with Equivariant Continuous Flows](https://mlsanity.com/paper/2207.00283/) (2022)
- [Mutual Information, Neural Networks and the Renormalization Group](https://mlsanity.com/paper/1704.06279/) (2017)
- [Stochastic normalizing flows as non-equilibrium transformations](https://mlsanity.com/paper/2201.08862/) (2022)
- [Detecting and Mitigating Mode-Collapse for Flow-based Sampling of Lattice Field Theories](https://mlsanity.com/paper/2302.14082/) (2023)
- [Normalizing flows for lattice gauge theory in arbitrary space-time dimension](https://mlsanity.com/paper/2305.02402/) (2023)
- [Advances in machine-learning-based sampling motivated by lattice quantum chromodynamics](https://mlsanity.com/paper/2309.01156/) (2023)
- [AR-Net: A simple Auto-Regressive Neural Network for time-series](https://mlsanity.com/paper/1911.12436/) (2019)
- [Learning Thermodynamics with Boltzmann Machines](https://mlsanity.com/paper/1606.02718/) (2016)
- [Renormalizing Diffusion Models](https://mlsanity.com/paper/2308.12355/) (2023)
- [Flow-Based Sampling for Entanglement Entropy and the Machine Learning of Defects](https://mlsanity.com/paper/2410.14466/) (2024)
- [Hierarchical autoregressive neural networks for statistical systems](https://mlsanity.com/paper/2203.10989/) (2022)
- [Sampling the lattice Nambu-Goto string using Continuous Normalizing Flows](https://mlsanity.com/paper/2307.01107/) (2023)
- [RG-Flow: A hierarchical and explainable flow model based on renormalization group and sparse prior](https://mlsanity.com/paper/2010.00029/) (2020)
- [Analysis of autocorrelation times in Neural Markov Chain Monte Carlo simulations](https://mlsanity.com/paper/2111.10189/) (2021)

## Built on by (most cited first, 6 of 6)

- [Sampling two-dimensional spin systems with transformers](https://mlsanity.com/paper/2604.27738/) (2026)
- [A scalable flow-based approach to mitigate topological freezing](https://mlsanity.com/paper/2601.20708/) (2026)
- [The critical slowing down in training diffusion models](https://mlsanity.com/paper/2605.12597/) (2026)
- [Variational Autoregressive Networks with probability priors](https://mlsanity.com/paper/2605.16020/) (2026)
- [Renormalization Group Flow Matching for Scalable Local Generative Modeling](https://mlsanity.com/paper/2608.23696/) (2026)
- [Data Free Training of Diffusion Models for Lattice Field Theories](https://mlsanity.com/paper/2610.18863/) (2026)

## Abstract (arXiv)

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.