# Lecture Notes on Normalizing Flows for Lattice Quantum Field Theories

arXiv: https://arxiv.org/abs/2504.18126 · HTML: https://mlsanity.com/paper/2504.18126/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 5 of its 40 references that are in the map.
- First posted on arXiv 25 Apr 2025.

## Builds on (most cited first, 5 of 5)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Geometric Deep Learning and Equivariant Neural Networks](https://mlsanity.com/paper/2105.13926/) (2021)
- [Multi-Resolution Continuous Normalizing Flows](https://mlsanity.com/paper/2106.08462/) (2021)
- [Non-Perturbative Trivializing Flows for Lattice Gauge Theories](https://mlsanity.com/paper/2410.13161/) (2024)
- [Multi-Lattice Sampling of Quantum Field Theories via Neural Operator-based Flows](https://mlsanity.com/paper/2401.00828/) (2024)

## Built on by (most cited first, 4 of 4)

- [SESaMo: Symmetry-Enforcing Stochastic Modulation for Normalizing Flows](https://mlsanity.com/paper/2505.19619/) (2025)
- [Analytic Bijections for Smooth and Interpretable Normalizing Flows](https://mlsanity.com/paper/2601.10774/) (2026)
- [Weight-Space Physics: Interpretable Hypernetworks for Lattice Quantum Field Theories](https://mlsanity.com/paper/2607.07127/) (2026)
- [What Neural Network Field Theory Can and Cannot Realise on a Computer](https://mlsanity.com/paper/2608.21523/) (2026)

## Abstract (arXiv)

Numerical simulations of quantum field theories on lattices serve as a fundamental tool for studying the non-perturbative regime of the theories, where analytic tools often fall short. Challenges arise when one takes the continuum limit or as the system approaches a critical point, especially in the presence of non-trivial topological structures in the theory. Rapid recent advances in machine learning provide a promising avenue for progress in this area. These lecture notes aim to give a brief account of lattice field theories, normalizing flows, and how the latter can be applied to study the former. The notes are based on the lectures given by the first author in various recent research schools.