# Advances in machine-learning-based sampling motivated by lattice quantum chromodynamics

arXiv: https://arxiv.org/abs/2309.01156 · HTML: https://mlsanity.com/paper/2309.01156/

## Key facts

- Cited by 23 papers in the map (data through 14 Oct 2026).
- Builds on 58 of its 107 references that are in the map.
- First posted on arXiv 3 Sep 2023.

## Builds on (most cited first, 30 of 58)

- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [Diffusion Models Beat GANs on Image Synthesis](https://mlsanity.com/paper/2105.05233/) (2021)
- [Scaling Laws for Neural Language Models](https://mlsanity.com/paper/2001.08361/) (2020)
- [Photorealistic Text-to-Image Diffusion Models with Deep Language Understanding](https://mlsanity.com/paper/2205.11487/) (2022)
- [Training Compute-Optimal Large Language Models](https://mlsanity.com/paper/2203.15556/) (2022)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [WaveNet: A Generative Model for Raw Audio](https://mlsanity.com/paper/1609.03499/) (2016)
- [Glow: Generative Flow with Invertible 1x1 Convolutions](https://mlsanity.com/paper/1807.03039/) (2018)
- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [LaMDA: Language Models for Dialog Applications](https://mlsanity.com/paper/2201.08239/) (2022)
- [Scaling Language Models: Methods, Analysis & Insights from Training Gopher](https://mlsanity.com/paper/2112.11446/) (2021)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [Pixel Recurrent Neural Networks](https://mlsanity.com/paper/1601.06759/) (2016)
- [Masked Autoregressive Flow for Density Estimation](https://mlsanity.com/paper/1705.07057/) (2017)
- [SE(3)-Transformers: 3D Roto-Translation Equivariant Attention Networks](https://mlsanity.com/paper/2006.10503/) (2020)
- [Using DeepSpeed and Megatron to Train Megatron-Turing NLG 530B, A Large-Scale Generative Language Model](https://mlsanity.com/paper/2201.11990/) (2022)
- [Very Deep VAEs Generalize Autoregressive Models and Can Outperform Them on Images](https://mlsanity.com/paper/2011.10650/) (2020)
- [Learning Local Equivariant Representations for Large-Scale Atomistic Dynamics](https://mlsanity.com/paper/2204.05249/) (2022)
- [Equivariant Flows: Exact Likelihood Generative Learning for Symmetric Densities](https://mlsanity.com/paper/2006.02425/) (2020)
- [Representation Learning via Invariant Causal Mechanisms](https://mlsanity.com/paper/2010.07922/) (2020)
- [Stochastic Normalizing Flows](https://mlsanity.com/paper/2002.06707/) (2020)
- [Neural Importance Sampling](https://mlsanity.com/paper/1808.03856/) (2018)
- [Flow-based generative models for Markov chain Monte Carlo in lattice field theory](https://mlsanity.com/paper/1904.12072/) (2019)
- [Equivariant flow-based sampling for lattice gauge theory](https://mlsanity.com/paper/2003.06413/) (2020)
- [Annealed Flow Transport Monte Carlo](https://mlsanity.com/paper/2102.07501/) (2021)
- [Sampling using SU(N) gauge equivariant flows](https://mlsanity.com/paper/2008.05456/) (2020)
- [SE(3) Equivariant Graph Neural Networks with Complete Local Frames](https://mlsanity.com/paper/2110.14811/) (2021)
- [NeuTra-lizing Bad Geometry in Hamiltonian Monte Carlo Using Neural Transport](https://mlsanity.com/paper/1903.03704/) (2019)
- [Augmented Normalizing Flows: Bridging the Gap Between Generative Flows and Latent Variable Models](https://mlsanity.com/paper/2002.07101/) (2020)

## Built on by (most cited first, 23 of 23)

- [Generative Diffusion Models for Lattice Field Theory](https://mlsanity.com/paper/2311.03578/) (2023)
- [Flow-Based Sampling for Entanglement Entropy and the Machine Learning of Defects](https://mlsanity.com/paper/2410.14466/) (2024)
- [Applications of flow models to the generation of correlated lattice QCD ensembles](https://mlsanity.com/paper/2401.10874/) (2024)
- [On learning higher-order cumulants in diffusion models](https://mlsanity.com/paper/2410.21212/) (2024)
- [Scaling of Stochastic Normalizing Flows in SU(3) lattice gauge theory](https://mlsanity.com/paper/2412.00200/) (2024)
- [Bayesian RG Flow in Neural Network Field Theories](https://mlsanity.com/paper/2405.17538/) (2024)
- [Physics-Conditioned Diffusion Models for Lattice Gauge Theory](https://mlsanity.com/paper/2502.05504/) (2025)
- [Physics-Driven Learning for Inverse Problems in Quantum Chromodynamics](https://mlsanity.com/paper/2501.05580/) (2025)
- [Practical applications of machine-learned flows on gauge fields](https://mlsanity.com/paper/2404.11674/) (2024)
- [Numerical determination of the width and shape of the effective string using Stochastic Normalizing Flows](https://mlsanity.com/paper/2409.15937/) (2024)
- [Multilevel Generative Samplers for Investigating Critical Phenomena](https://mlsanity.com/paper/2503.08918/) (2025)
- [Scaling flow-based approaches for topology sampling in SU(3) gauge theory](https://mlsanity.com/paper/2510.25704/) (2025)
- [Fast and Unified Path Gradient Estimators for Normalizing Flows](https://mlsanity.com/paper/2403.15881/) (2024)
- [Equivariant Transformer is all you need](https://mlsanity.com/paper/2310.13222/) (2023)
- [Group-Equivariant Diffusion Models for Lattice Field Theory](https://mlsanity.com/paper/2510.26081/) (2025)
- [The NFLikelihood: an unsupervised DNNLikelihood from Normalizing Flows](https://mlsanity.com/paper/2309.09743/) (2023)
- [Toward Scalable Normalizing Flows for the Hubbard Model](https://mlsanity.com/paper/2601.18273/) (2026)
- [Simulating the Hubbard Model with Equivariant Normalizing Flows](https://mlsanity.com/paper/2501.07371/) (2025)
- [Phase diagram and eigenvalue dynamics of stochastic gradient descent in multilayer neural networks](https://mlsanity.com/paper/2509.01349/) (2025)
- [Stochastic normalizing flows for Effective String Theory](https://mlsanity.com/paper/2412.19109/) (2024)
- [Studying Effective String Theory using deep generative models](https://mlsanity.com/paper/2508.20610/) (2025)
- [What Neural Network Field Theory Can and Cannot Realise on a Computer](https://mlsanity.com/paper/2608.21523/) (2026)
- [Accurate Sampling from Diffusion Models](https://mlsanity.com/paper/2609.29902/) (2026)

## Abstract (arXiv)

Sampling from known probability distributions is a ubiquitous task in computational science, underlying calculations in domains from linguistics to biology and physics. Generative machine-learning (ML) models have emerged as a promising tool in this space, building on the success of this approach in applications such as image, text, and audio generation. Often, however, generative tasks in scientific domains have unique structures and features -- such as complex symmetries and the requirement of exactness guarantees -- that present both challenges and opportunities for ML. This Perspective outlines the advances in ML-based sampling motivated by lattice quantum field theory, in particular for the theory of quantum chromodynamics. Enabling calculations of the structure and interactions of matter from our most fundamental understanding of particle physics, lattice quantum chromodynamics is one of the main consumers of open-science supercomputing worldwide. The design of ML algorithms for this application faces profound challenges, including the necessity of scaling custom ML architectures to the largest supercomputers, but also promises immense benefits, and is spurring a wave of development in ML-based sampling more broadly. In lattice field theory, if this approach can realize its early promise it will be a transformative step towards first-principles physics calculations in particle, nuclear and condensed matter physics that are intractable with traditional approaches.