# Flow-based sampling for fermionic lattice field theories

arXiv: https://arxiv.org/abs/2106.05934 · HTML: https://mlsanity.com/paper/2106.05934/

## Key facts

- Cited by 27 papers in the map (data through 14 Oct 2026).
- Builds on 1 of its 55 references that are in the map.
- First posted on arXiv 10 Jun 2021.

## Builds on (most cited first, 1 of 1)

- [A high-bias, low-variance introduction to Machine Learning for physicists](https://mlsanity.com/paper/1803.08823/) (2018)

## Built on by (most cited first, 27 of 27)

- [Learning Interpolations between Boltzmann Densities](https://mlsanity.com/paper/2301.07388/) (2023)
- [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)
- [Advances in machine-learning-based sampling motivated by lattice quantum chromodynamics](https://mlsanity.com/paper/2309.01156/) (2023)
- [Applications of Machine Learning to Lattice Quantum Field Theory](https://mlsanity.com/paper/2202.05838/) (2022)
- [Sampling the lattice Nambu-Goto string using Continuous Normalizing Flows](https://mlsanity.com/paper/2307.01107/) (2023)
- [Markovian Flow Matching: Accelerating MCMC with Continuous Normalizing Flows](https://mlsanity.com/paper/2405.14392/) (2024)
- [Geometrical aspects of lattice gauge equivariant convolutional neural networks](https://mlsanity.com/paper/2303.11448/) (2023)
- [AdvNF: Reducing Mode Collapse in Conditional Normalising Flows using Adversarial Learning](https://mlsanity.com/paper/2401.15948/) (2024)
- [Practical applications of machine-learned flows on gauge fields](https://mlsanity.com/paper/2404.11674/) (2024)
- [Scaling flow-based approaches for topology sampling in SU(3) gauge theory](https://mlsanity.com/paper/2510.25704/) (2025)
- [Machine Learning Trivializing Maps: A First Step Towards Understanding How Flow-Based Samplers Scale Up](https://mlsanity.com/paper/2112.15532/) (2021)
- [Simulating first-order phase transition with hierarchical autoregressive networks](https://mlsanity.com/paper/2212.04955/) (2022)
- [Machine Learning in Nuclear Physics](https://mlsanity.com/paper/2112.02309/) (2021)
- [Estimating the Euclidean quantum propagator with deep generative modeling of Feynman paths](https://mlsanity.com/paper/2202.02750/) (2022)
- [Understanding Deep Generative Models with Generalized Empirical Likelihoods](https://mlsanity.com/paper/2306.09780/) (2023)
- [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)
- [Multi-Lattice Sampling of Quantum Field Theories via Neural Operator-based Flows](https://mlsanity.com/paper/2401.00828/) (2024)
- [NeuMC -- a package for neural sampling for lattice field theories](https://mlsanity.com/paper/2503.11482/) (2025)
- [A scalable flow-based approach to mitigate topological freezing](https://mlsanity.com/paper/2601.20708/) (2026)
- [Flow Perturbation to Accelerate Unbiased Sampling of Boltzmann distribution](https://mlsanity.com/paper/2407.10666/) (2024)
- [A Critical Review of Monte Carlo Algorithms Balancing Performance and Probabilistic Accuracy with AI Augmented Framework](https://mlsanity.com/paper/2512.17968/) (2025)
- [Sampling the Schwinger Model with Gauge-Equivariant Diffusion](https://mlsanity.com/paper/2606.27481/) (2026)

## Abstract (arXiv)

Algorithms based on normalizing flows are emerging as promising machine learning approaches to sampling complicated probability distributions in a way that can be made asymptotically exact. In the context of lattice field theory, proof-of-principle studies have demonstrated the effectiveness of this approach for scalar theories, gauge theories, and statistical systems. This work develops approaches that enable flow-based sampling of theories with dynamical fermions, which is necessary for the technique to be applied to lattice field theory studies of the Standard Model of particle physics and many condensed matter systems. As a practical demonstration, these methods are applied to the sampling of field configurations for a two-dimensional theory of massless staggered fermions coupled to a scalar field via a Yukawa interaction.