# Flow-based sampling for multimodal and extended-mode distributions in lattice field theory

arXiv: https://arxiv.org/abs/2107.00734 · HTML: https://mlsanity.com/paper/2107.00734/

## Key facts

- Cited by 25 papers in the map (data through 14 Oct 2026).
- Builds on 0 of its 15 references that are in the map.
- First posted on arXiv 1 Jul 2021.

## Built on by (most cited first, 25 of 25)

- [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)
- [Gradients should stay on Path: Better Estimators of the Reverse- and Forward KL Divergence for Normalizing Flows](https://mlsanity.com/paper/2207.08219/) (2022)
- [Advances in machine-learning-based sampling motivated by lattice quantum chromodynamics](https://mlsanity.com/paper/2309.01156/) (2023)
- [Local-Global MCMC kernels: the best of both worlds](https://mlsanity.com/paper/2111.02702/) (2021)
- [On Sampling with Approximate Transport Maps](https://mlsanity.com/paper/2302.04763/) (2023)
- [Applications of Machine Learning to Lattice Quantum Field Theory](https://mlsanity.com/paper/2202.05838/) (2022)
- [Generative Diffusion Models for Lattice Field Theory](https://mlsanity.com/paper/2311.03578/) (2023)
- [Sampling the lattice Nambu-Goto string using Continuous Normalizing Flows](https://mlsanity.com/paper/2307.01107/) (2023)
- [Analysis of autocorrelation times in Neural Markov Chain Monte Carlo simulations](https://mlsanity.com/paper/2111.10189/) (2021)
- [Machine Learning of Thermodynamic Observables in the Presence of Mode Collapse](https://mlsanity.com/paper/2111.11303/) (2021)
- [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)
- [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)
- [Balanced Training of Energy-Based Models with Adaptive Flow Sampling](https://mlsanity.com/paper/2306.00684/) (2023)
- [Learning to Sample Better](https://mlsanity.com/paper/2310.11232/) (2023)
- [The NFLikelihood: an unsupervised DNNLikelihood from Normalizing Flows](https://mlsanity.com/paper/2309.09743/) (2023)
- [Numerical and geometrical aspects of flow-based variational quantum Monte Carlo](https://mlsanity.com/paper/2203.14824/) (2022)
- [NeuMC -- a package for neural sampling for lattice field theories](https://mlsanity.com/paper/2503.11482/) (2025)
- [Locality-constrained autoregressive cum conditional normalizing flow for lattice field theory simulations](https://mlsanity.com/paper/2304.01798/) (2023)
- [An AI-powered Technology Stack for Solving Many-Electron Field Theory](https://mlsanity.com/paper/2403.18840/) (2024)
- [Operator-Guided Model Reduction for Generative Sampling in Lattice Field Theory](https://mlsanity.com/paper/2605.11199/) (2026)

## Abstract (arXiv)

Recent results have demonstrated that samplers constructed with flow-based generative models are a promising new approach for configuration generation in lattice field theory. In this paper, we present a set of training- and architecture-based methods to construct flow models for targets with multiple separated modes (i.e.~vacua) as well as targets with extended/continuous modes. We demonstrate the application of these methods to modeling two-dimensional real and complex scalar field theories in their symmetry-broken phases. In this context we investigate different flow-based sampling algorithms, including a composite sampling algorithm where flow-based proposals are occasionally augmented by applying updates using traditional algorithms like HMC.