# Quantum field-theoretic machine learning

arXiv: https://arxiv.org/abs/2102.09449 · HTML: https://mlsanity.com/paper/2102.09449/

## Key facts

- Cited by 16 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 72 references that are in the map.
- First posted on arXiv 18 Feb 2021.

## Builds on (most cited first, 15 of 15)

- [Gaussian Process Behaviour in Wide Deep Neural Networks](https://mlsanity.com/paper/1804.11271/) (2018)
- [Bayesian Deep Convolutional Networks with Many Channels are Gaussian Processes](https://mlsanity.com/paper/1810.05148/) (2018)
- [Deep Convolutional Networks as shallow Gaussian Processes](https://mlsanity.com/paper/1808.05587/) (2018)
- [Statistical physics of inference: Thresholds and algorithms](https://mlsanity.com/paper/1511.02476/) (2015)
- [Dynamics of stochastic gradient descent for two-layer neural networks in the teacher-student setup](https://mlsanity.com/paper/1906.08632/) (2019)
- [Equivariant flow-based sampling for lattice gauge theory](https://mlsanity.com/paper/2003.06413/) (2020)
- [Neural Networks and Quantum Field Theory](https://mlsanity.com/paper/2008.08601/) (2020)
- [Discovering Phase Transitions with Unsupervised Learning](https://mlsanity.com/paper/1606.00318/) (2016)
- [Lattice gauge equivariant convolutional neural networks](https://mlsanity.com/paper/2012.12901/) (2020)
- [Phase Diagram of Restricted Boltzmann Machines and Generalised Hopfield Networks with Arbitrary Priors](https://mlsanity.com/paper/1702.05882/) (2017)
- [Phase transitions in Restricted Boltzmann Machines with generic priors](https://mlsanity.com/paper/1612.03132/) (2016)
- [Towards Novel Insights in Lattice Field Theory with Explainable Machine Learning](https://mlsanity.com/paper/2003.01504/) (2020)
- [Extending machine learning classification capabilities with histogram reweighting](https://mlsanity.com/paper/2004.14341/) (2020)
- [Adding machine learning within Hamiltonians: Renormalization group transformations, symmetry breaking and restoration](https://mlsanity.com/paper/2010.00054/) (2020)
- [Topological defects and confinement with machine learning: the case of monopoles in compact electrodynamics](https://mlsanity.com/paper/2006.09113/) (2020)

## Built on by (most cited first, 16 of 16)

- [Stochastic normalizing flows as non-equilibrium transformations](https://mlsanity.com/paper/2201.08862/) (2022)
- [Nonperturbative renormalization for the neural network-QFT correspondence](https://mlsanity.com/paper/2108.01403/) (2021)
- [Towards quantifying information flows: relative entropy in deep neural networks and the renormalization group](https://mlsanity.com/paper/2107.06898/) (2021)
- [Applications of Machine Learning to Lattice Quantum Field Theory](https://mlsanity.com/paper/2202.05838/) (2022)
- [Structures of Neural Network Effective Theories](https://mlsanity.com/paper/2305.02334/) (2023)
- [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)
- [On the Dynamics of Inference and Learning](https://mlsanity.com/paper/2204.12939/) (2022)
- [Bayesian RG Flow in Neural Network Field Theories](https://mlsanity.com/paper/2405.17538/) (2024)
- [Physics-Driven Learning for Inverse Problems in Quantum Chromodynamics](https://mlsanity.com/paper/2501.05580/) (2025)
- [Geometrical aspects of lattice gauge equivariant convolutional neural networks](https://mlsanity.com/paper/2303.11448/) (2023)
- [Machine learning with quantum field theories](https://mlsanity.com/paper/2109.07730/) (2021)
- [Quantum field theories, Markov random fields and machine learning](https://mlsanity.com/paper/2110.10928/) (2021)
- [p-Adic Statistical Field Theory and Deep Belief Networks](https://mlsanity.com/paper/2207.13877/) (2022)
- [The neural networks with tensor weights and emergent fermionic Wick rules in the large-width limit](https://mlsanity.com/paper/2507.05303/) (2025)
- [Quantitative analysis of phase transitions in two-dimensional XY models using persistent homology](https://mlsanity.com/paper/2109.10960/) (2021)

## Abstract (arXiv)

We derive machine learning algorithms from discretized Euclidean field theories, making inference and learning possible within dynamics described by quantum field theory. Specifically, we demonstrate that the φ4 scalar field theory satisfies the Hammersley-Clifford theorem, therefore recasting it as a machine learning algorithm within the mathematically rigorous framework of Markov random fields. We illustrate the concepts by minimizing an asymmetric distance between the probability distribution of the φ4 theory and that of target distributions, by quantifying the overlap of statistical ensembles between probability distributions and through reweighting to complex-valued actions with longer-range interactions. Neural network architectures are additionally derived from the φ4 theory which can be viewed as generalizations of conventional neural networks and applications are presented. We conclude by discussing how the proposal opens up a new research avenue, that of developing a mathematical and computational framework of machine learning within quantum field theory.