# A scalable flow-based approach to mitigate topological freezing

arXiv: https://arxiv.org/abs/2601.20708 · HTML: https://mlsanity.com/paper/2601.20708/

## Key facts

- Cited by 1 papers in the map (data through 7 Oct 2026).
- Builds on 20 of its 57 references that are in the map.
- First posted on arXiv 28 Jan 2026.

## Builds on (most cited first, 20 of 20)

- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [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)
- [Equivariant flow-based sampling for lattice gauge theory](https://mlsanity.com/paper/2003.06413/) (2020)
- [Lattice gauge equivariant convolutional neural networks](https://mlsanity.com/paper/2012.12901/) (2020)
- [Gauge-equivariant flow models for sampling in lattice field theories with pseudofermions](https://mlsanity.com/paper/2207.08945/) (2022)
- [Flow-based sampling for fermionic lattice field theories](https://mlsanity.com/paper/2106.05934/) (2021)
- [Stochastic normalizing flows as non-equilibrium transformations](https://mlsanity.com/paper/2201.08862/) (2022)
- [Diffusion Models as Stochastic Quantization in Lattice Field Theory](https://mlsanity.com/paper/2309.17082/) (2023)
- [Normalizing flows for lattice gauge theory in arbitrary space-time dimension](https://mlsanity.com/paper/2305.02402/) (2023)
- [Flow-based sampling in the lattice Schwinger model at criticality](https://mlsanity.com/paper/2202.11712/) (2022)
- [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)
- [Numerical determination of the width and shape of the effective string using Stochastic Normalizing Flows](https://mlsanity.com/paper/2409.15937/) (2024)
- [Scaling flow-based approaches for topology sampling in SU(3) gauge theory](https://mlsanity.com/paper/2510.25704/) (2025)
- [Multilevel Generative Samplers for Investigating Critical Phenomena](https://mlsanity.com/paper/2503.08918/) (2025)
- [Combining complex Langevin dynamics with score-based and energy-based diffusion models](https://mlsanity.com/paper/2510.01328/) (2025)
- [Toward Scalable Normalizing Flows for the Hubbard Model](https://mlsanity.com/paper/2601.18273/) (2026)

## Built on by (most cited first, 1 of 1)

- [Machine learning for four-dimensional SU(3) lattice gauge theories](https://mlsanity.com/paper/2604.12416/) (2026)

## Abstract (arXiv)

As lattice gauge theories with non-trivial topological features are driven towards the continuum limit, standard Markov Chain Monte Carlo simulations suffer for topological freezing, i.e., a dramatic growth of autocorrelations in topological observables. A widely used strategy is the adoption of Open Boundary Conditions (OBC), which restores ergodic sampling of topology but at the price of breaking translation invariance and introducing unphysical boundary artifacts. In this contribution we summarize a scalable, exact flow-based strategy to remove them by transporting configurations from a prior with a OBC defect to a fully periodic ensemble, and apply it to 4d SU(3) Yang--Mills theory. The method is based on a Stochastic Normalizing Flow (SNF) that alternates non-equilibrium Monte Carlo updates with localized, gauge-equivariant defect coupling layers implemented via masked parametric stout smearing. Training is performed by minimizing the average dissipated work, equivalent to a Kullback--Leibler divergence between forward and reverse non-equilibrium path measures, to achieve more reversible trajectories and improved efficiency. We discuss the scaling with the number of degrees of freedom affected by the defect and show that defect SNFs achieve better performances than purely stochastic non-equilibrium methods at comparable cost. Finally, we validate the approach by reproducing reference results for the topological susceptibility.