# Scaling Up Machine Learning For Quantum Field Theory with Equivariant Continuous Flows

arXiv: https://arxiv.org/abs/2110.02673 · HTML: https://mlsanity.com/paper/2110.02673/

## Key facts

- Cited by 24 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 24 references that are in the map.
- First posted on arXiv 6 Oct 2021.

## Builds on (most cited first, 13 of 13)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [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)
- [Glow: Generative Flow with Invertible 1x1 Convolutions](https://mlsanity.com/paper/1807.03039/) (2018)
- [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)
- [Machine learning for molecular simulation](https://mlsanity.com/paper/1911.02792/) (2019)
- [Equivariant flow-based sampling for lattice gauge theory](https://mlsanity.com/paper/2003.06413/) (2020)
- [Provably efficient machine learning for quantum many-body problems](https://mlsanity.com/paper/2106.12627/) (2021)
- [Introduction to Normalizing Flows for Lattice Field Theory](https://mlsanity.com/paper/2101.08176/) (2021)
- [Flow-based sampling for fermionic lattice field theories](https://mlsanity.com/paper/2106.05934/) (2021)
- [Flow-based sampling for multimodal and extended-mode distributions in lattice field theory](https://mlsanity.com/paper/2107.00734/) (2021)

## Built on by (most cited first, 24 of 24)

- [Continual Repeated Annealed Flow Transport Monte Carlo](https://mlsanity.com/paper/2201.13117/) (2022)
- [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)
- [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)
- [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)
- [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)
- [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)
- [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)
- [Demonstrating Real Advantage of Machine-Learning-Enhanced Monte Carlo for Combinatorial Optimization](https://mlsanity.com/paper/2510.19544/) (2025)
- [Non-Perturbative Trivializing Flows for Lattice Gauge Theories](https://mlsanity.com/paper/2410.13161/) (2024)
- [Neural Simulated Annealing](https://mlsanity.com/paper/2203.02201/) (2022)
- [Amortized Bayesian Inference of GISAXS Data with Normalizing Flows](https://mlsanity.com/paper/2210.01543/) (2022)
- [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)
- [The critical slowing down in training diffusion models](https://mlsanity.com/paper/2605.12597/) (2026)
- [Lattice gauge symmetry in neural networks](https://mlsanity.com/paper/2111.04389/) (2021)
- [Accurate Sampling from Diffusion Models](https://mlsanity.com/paper/2609.29902/) (2026)

## Abstract (arXiv)

We propose a continuous normalizing flow for sampling from the high-dimensional probability distributions of Quantum Field Theories in Physics. In contrast to the deep architectures used so far for this task, our proposal is based on a shallow design and incorporates the symmetries of the problem. We test our model on the φ4 theory, showing that it systematically outperforms a realNVP baseline in sampling efficiency, with the difference between the two increasing for larger lattices. On the largest lattice we consider, of size 32× 32, we improve a key metric, the effective sample size, from 1% to 66% w.r.t. the realNVP baseline.