# Learning Lattice Quantum Field Theories with Equivariant Continuous Flows

arXiv: https://arxiv.org/abs/2207.00283 · HTML: https://mlsanity.com/paper/2207.00283/

## Key facts

- Cited by 37 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 49 references that are in the map.
- First posted on arXiv 1 Jul 2022.

## Builds on (most cited first, 14 of 14)

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domains](https://mlsanity.com/paper/2006.10739/) (2020)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [Equivariant Flows: Exact Likelihood Generative Learning for Symmetric Densities](https://mlsanity.com/paper/2006.02425/) (2020)
- [Flow-based generative models for Markov chain Monte Carlo in lattice field theory](https://mlsanity.com/paper/1904.12072/) (2019)
- [Normalizing Flows on Tori and Spheres](https://mlsanity.com/paper/2002.02428/) (2020)
- [Equivariant flow-based sampling for lattice gauge theory](https://mlsanity.com/paper/2003.06413/) (2020)
- [Sampling using SU(N) gauge equivariant flows](https://mlsanity.com/paper/2008.05456/) (2020)
- [Introduction to Normalizing Flows for Lattice Field Theory](https://mlsanity.com/paper/2101.08176/) (2021)
- [Towards Novel Insights in Lattice Field Theory with Explainable Machine Learning](https://mlsanity.com/paper/2003.01504/) (2020)
- [Path-Gradient Estimators for Continuous Normalizing Flows](https://mlsanity.com/paper/2206.09016/) (2022)
- [Machine Learning of Thermodynamic Observables in the Presence of Mode Collapse](https://mlsanity.com/paper/2111.11303/) (2021)
- [Towards reduction of autocorrelation in HMC by machine learning](https://mlsanity.com/paper/1712.03893/) (2017)
- [Classifying Topological Charge in SU(3) Yang-Mills Theory with Machine Learning](https://mlsanity.com/paper/1909.06238/) (2019)

## Built on by (most cited first, 30 of 37)

- [Does equivariance matter at scale?](https://mlsanity.com/paper/2410.23179/) (2024)
- [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)
- [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)
- [Efficient mapping of phase diagrams with conditional Boltzmann Generators](https://mlsanity.com/paper/2406.12378/) (2024)
- [Renormalizing Diffusion Models](https://mlsanity.com/paper/2308.12355/) (2023)
- [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)
- [On learning higher-order cumulants in diffusion models](https://mlsanity.com/paper/2410.21212/) (2024)
- [Bayesian RG Flow in Neural Network Field Theories](https://mlsanity.com/paper/2405.17538/) (2024)
- [Physics-Conditioned Diffusion Models for Lattice Gauge Theory](https://mlsanity.com/paper/2502.05504/) (2025)
- [Fluctuation without dissipation: Microcanonical Langevin Monte Carlo](https://mlsanity.com/paper/2303.18221/) (2023)
- [Energy based diffusion generator for efficient sampling of Boltzmann distributions](https://mlsanity.com/paper/2401.02080/) (2024)
- [Geometrical aspects of lattice gauge equivariant convolutional neural networks](https://mlsanity.com/paper/2303.11448/) (2023)
- [Numerical determination of the width and shape of the effective string using Stochastic Normalizing Flows](https://mlsanity.com/paper/2409.15937/) (2024)
- [Multilevel Generative Samplers for Investigating Critical Phenomena](https://mlsanity.com/paper/2503.08918/) (2025)
- [Scaling flow-based approaches for topology sampling in SU(3) gauge theory](https://mlsanity.com/paper/2510.25704/) (2025)
- [Neural Flow Samplers with Shortcut Models](https://mlsanity.com/paper/2502.07337/) (2025)
- [Rao-Blackwell Gradient Estimators for Equivariant Denoising Diffusion](https://mlsanity.com/paper/2502.09890/) (2025)
- [Demonstrating Real Advantage of Machine-Learning-Enhanced Monte Carlo for Combinatorial Optimization](https://mlsanity.com/paper/2510.19544/) (2025)
- [Fixed point actions from convolutional neural networks](https://mlsanity.com/paper/2311.17816/) (2023)
- [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)
- [Machine learning a fixed point action for SU(3) gauge theory with a gauge equivariant convolutional neural network](https://mlsanity.com/paper/2401.06481/) (2024)
- [SESaMo: Symmetry-Enforcing Stochastic Modulation for Normalizing Flows](https://mlsanity.com/paper/2505.19619/) (2025)
- [Viability of perturbative expansion for quantum field theories on neurons](https://mlsanity.com/paper/2508.03810/) (2025)
- [Generative Models and Statistical Validation](https://mlsanity.com/paper/2605.30453/) (2026)
- [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)

## Abstract (arXiv)

We propose a novel machine learning method for sampling from the high-dimensional probability distributions of Lattice Field Theories, which is based on a single neural ODE layer and incorporates the full symmetries of the problem. We test our model on the φ4 theory, showing that it systematically outperforms previously proposed flow-based methods in sampling efficiency, and the improvement is especially pronounced for larger lattices. Furthermore, we demonstrate that our model can learn a continuous family of theories at once, and the results of learning can be transferred to larger lattices. Such generalizations further accentuate the advantages of machine learning methods.