# Introduction to Normalizing Flows for Lattice Field Theory

arXiv: https://arxiv.org/abs/2101.08176 · HTML: https://mlsanity.com/paper/2101.08176/

## Key facts

- Cited by 35 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 27 references that are in the map.
- First posted on arXiv 20 Jan 2021.

## Builds on (most cited first, 8 of 8)

- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [Gauge Equivariant Convolutional Networks and the Icosahedral CNN](https://mlsanity.com/paper/1902.04615/) (2019)
- [Stochastic Normalizing Flows](https://mlsanity.com/paper/2002.06707/) (2020)
- [Riemannian Continuous Normalizing Flows](https://mlsanity.com/paper/2006.10605/) (2020)
- [How (not) to Train your Generative Model: Scheduled Sampling, Likelihood, Adversary?](https://mlsanity.com/paper/1511.05101/) (2015)
- [Lattice gauge equivariant convolutional neural networks](https://mlsanity.com/paper/2012.12901/) (2020)
- [Neural Ordinary Differential Equations on Manifolds](https://mlsanity.com/paper/2006.06663/) (2020)
- [Gauge equivariant neural networks for quantum lattice gauge theories](https://mlsanity.com/paper/2012.05232/) (2020)

## Built on by (most cited first, 30 of 35)

- [SE(3) diffusion model with application to protein backbone generation](https://mlsanity.com/paper/2302.02277/) (2023)
- [Continual Repeated Annealed Flow Transport Monte Carlo](https://mlsanity.com/paper/2201.13117/) (2022)
- [Learning Interpolations between Boltzmann Densities](https://mlsanity.com/paper/2301.07388/) (2023)
- [Learning Lattice Quantum Field Theories with Equivariant Continuous Flows](https://mlsanity.com/paper/2207.00283/) (2022)
- [Aspects of scaling and scalability for flow-based sampling of lattice QCD](https://mlsanity.com/paper/2211.07541/) (2022)
- [Stochastic normalizing flows as non-equilibrium transformations](https://mlsanity.com/paper/2201.08862/) (2022)
- [Scaling Up Machine Learning For Quantum Field Theory with Equivariant Continuous Flows](https://mlsanity.com/paper/2110.02673/) (2021)
- [Diffusion Models as Stochastic Quantization in Lattice Field Theory](https://mlsanity.com/paper/2309.17082/) (2023)
- [A deep generative model for probabilistic energy forecasting in power systems: normalizing flows](https://mlsanity.com/paper/2106.09370/) (2021)
- [Sampling the lattice Nambu-Goto string using Continuous Normalizing Flows](https://mlsanity.com/paper/2307.01107/) (2023)
- [Path-Gradient Estimators for Continuous Normalizing Flows](https://mlsanity.com/paper/2206.09016/) (2022)
- [Analysis of autocorrelation times in Neural Markov Chain Monte Carlo simulations](https://mlsanity.com/paper/2111.10189/) (2021)
- [Accelerating Markov Chain Monte Carlo sampling with diffusion models](https://mlsanity.com/paper/2309.01454/) (2023)
- [Sampling Multimodal Distributions with the Vanilla Score: Benefits of Data-Based Initialization](https://mlsanity.com/paper/2310.01762/) (2023)
- [Deep Learning Hamiltonian Monte Carlo](https://mlsanity.com/paper/2105.03418/) (2021)
- [Fluctuation without dissipation: Microcanonical Langevin Monte Carlo](https://mlsanity.com/paper/2303.18221/) (2023)
- [Machine learning spectral functions in lattice QCD](https://mlsanity.com/paper/2110.13521/) (2021)
- [Geometrical aspects of lattice gauge equivariant convolutional neural networks](https://mlsanity.com/paper/2303.11448/) (2023)
- [Simulating first-order phase transition with hierarchical autoregressive networks](https://mlsanity.com/paper/2212.04955/) (2022)
- [Fast and Unified Path Gradient Estimators for Normalizing Flows](https://mlsanity.com/paper/2403.15881/) (2024)
- [Mutual information of spin systems from autoregressive neural networks](https://mlsanity.com/paper/2304.13412/) (2023)
- [Embedded-model flows: Combining the inductive biases of model-free deep learning and explicit probabilistic modeling](https://mlsanity.com/paper/2110.06021/) (2021)
- [LeapfrogLayers: A Trainable Framework for Effective Topological Sampling](https://mlsanity.com/paper/2112.01582/) (2021)
- [Non-Perturbative Trivializing Flows for Lattice Gauge Theories](https://mlsanity.com/paper/2410.13161/) (2024)
- [Group-Equivariant Diffusion Models for Lattice Field Theory](https://mlsanity.com/paper/2510.26081/) (2025)
- [A Probabilistic Forecast-Driven Strategy for a Risk-Aware Participation in the Capacity Firming Market: extended version](https://mlsanity.com/paper/2105.13801/) (2021)
- [Gradient estimators for normalising flows](https://mlsanity.com/paper/2202.01314/) (2022)
- [Training normalizing flows with computationally intensive target probability distributions](https://mlsanity.com/paper/2308.13294/) (2023)
- [Neural Triangular Transport Maps: A New Approach Towards Sampling in Lattice QCD](https://mlsanity.com/paper/2510.13112/) (2025)
- [Deformations of Boltzmann Distributions](https://mlsanity.com/paper/2210.13772/) (2022)

## Abstract (arXiv)

This notebook tutorial demonstrates a method for sampling Boltzmann distributions of lattice field theories using a class of machine learning models known as normalizing flows. The ideas and approaches proposed in arXiv:1904.12072, arXiv:2002.02428, and arXiv:2003.06413 are reviewed and a concrete implementation of the framework is presented. We apply this framework to a lattice scalar field theory and to U(1) gauge theory, explicitly encoding gauge symmetries in the flow-based approach to the latter. This presentation is intended to be interactive and working with the attached Jupyter notebook is recommended.