# Numerical determination of the width and shape of the effective string using Stochastic Normalizing Flows

arXiv: https://arxiv.org/abs/2409.15937 · HTML: https://mlsanity.com/paper/2409.15937/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 60 references that are in the map.
- First posted on arXiv 24 Sep 2024.

## Builds on (most cited first, 12 of 12)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [Stochastic Normalizing Flows](https://mlsanity.com/paper/2002.06707/) (2020)
- [Flow-based generative models for Markov chain Monte Carlo in lattice field theory](https://mlsanity.com/paper/1904.12072/) (2019)
- [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)
- [Detecting and Mitigating Mode-Collapse for Flow-based Sampling of Lattice Field Theories](https://mlsanity.com/paper/2302.14082/) (2023)
- [Scaling Up Machine Learning For Quantum Field Theory with Equivariant Continuous Flows](https://mlsanity.com/paper/2110.02673/) (2021)
- [Advances in machine-learning-based sampling motivated by lattice quantum chromodynamics](https://mlsanity.com/paper/2309.01156/) (2023)
- [Sampling the lattice Nambu-Goto string using Continuous Normalizing Flows](https://mlsanity.com/paper/2307.01107/) (2023)

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

- [Scaling flow-based approaches for topology sampling in SU(3) gauge theory](https://mlsanity.com/paper/2510.25704/) (2025)
- [Group-Equivariant Diffusion Models for Lattice Field Theory](https://mlsanity.com/paper/2510.26081/) (2025)
- [Toward Scalable Normalizing Flows for the Hubbard Model](https://mlsanity.com/paper/2601.18273/) (2026)
- [A scalable flow-based approach to mitigate topological freezing](https://mlsanity.com/paper/2601.20708/) (2026)
- [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)

## Abstract (arXiv)

Flow-based architectures have recently proved to be an efficient tool for numerical simulations of Effective String Theories regularized on the lattice that otherwise cannot be efficiently sampled by standard Monte Carlo methods. In this work we use Stochastic Normalizing Flows, a state-of-the-art deep learning architecture based on non-equilibrium Monte Carlo simulations, to study different effective string models. After testing the reliability of this approach through a comparison with exact results for the Nambu-Got=o model, we discuss results on observables that are challenging to study analytically, such as the width of the string and the shape of the flux density. Furthermore, we perform a novel numerical study of Effective String Theories with terms beyond the Nambu-Got=o action, including a broader discussion on their significance for lattice gauge theories. The combination of these findings enables a quantitative description of the fine details of the confinement mechanism in different lattice gauge theories. The results presented in this work establish the reliability and feasibility of flow-based samplers for Effective String Theories and pave the way for future applications on more complex models.