# Neural Networks and Quantum Field Theory

arXiv: https://arxiv.org/abs/2008.08601 · HTML: https://mlsanity.com/paper/2008.08601/

## Key facts

- Cited by 61 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 19 Aug 2020.

## Built on by (most cited first, 30 of 61)

- [Towards Understanding Grokking: An Effective Theory of Representation Learning](https://mlsanity.com/paper/2205.10343/) (2022)
- [Representation Learning via Quantum Neural Tangent Kernels](https://mlsanity.com/paper/2111.04225/) (2021)
- [Symmetry-via-Duality: Invariant Neural Network Densities from Parameter-Space Correlators](https://mlsanity.com/paper/2106.00694/) (2021)
- [Nonperturbative renormalization for the neural network-QFT correspondence](https://mlsanity.com/paper/2108.01403/) (2021)
- [The edge of chaos: quantum field theory and deep neural networks](https://mlsanity.com/paper/2109.13247/) (2021)
- [Contrasting random and learned features in deep Bayesian linear regression](https://mlsanity.com/paper/2203.00573/) (2022)
- [Neural Network Field Theories: Non-Gaussianity, Actions, and Locality](https://mlsanity.com/paper/2307.03223/) (2023)
- [The Limitations of Large Width in Neural Networks: A Deep Gaussian Process Perspective](https://mlsanity.com/paper/2106.06529/) (2021)
- [Unified field theoretical approach to deep and recurrent neuronal networks](https://mlsanity.com/paper/2112.05589/) (2021)
- [Emergent Equivariance in Deep Ensembles](https://mlsanity.com/paper/2403.03103/) (2024)
- [Quantum field-theoretic machine learning](https://mlsanity.com/paper/2102.09449/) (2021)
- [A theory of representation learning gives a deep generalisation of kernel methods](https://mlsanity.com/paper/2108.13097/) (2021)
- [Towards quantifying information flows: relative entropy in deep neural networks and the renormalization group](https://mlsanity.com/paper/2107.06898/) (2021)
- [Bayesian Renormalization](https://mlsanity.com/paper/2305.10491/) (2023)
- [Structures of Neural Network Effective Theories](https://mlsanity.com/paper/2305.02334/) (2023)
- [Rigor with Machine Learning from Field Theory to the Poincar'e Conjecture](https://mlsanity.com/paper/2402.13321/) (2024)
- [Conformal Fields from Neural Networks](https://mlsanity.com/paper/2409.12222/) (2024)
- [A duality connecting neural network and cosmological dynamics](https://mlsanity.com/paper/2202.11104/) (2022)
- [Laziness, Barren Plateau, and Noise in Machine Learning](https://mlsanity.com/paper/2206.09313/) (2022)
- [On learning higher-order cumulants in diffusion models](https://mlsanity.com/paper/2410.21212/) (2024)
- [From Kernels to Features: A Multi-Scale Adaptive Theory of Feature Learning](https://mlsanity.com/paper/2502.03210/) (2025)
- [On the Dynamics of Inference and Learning](https://mlsanity.com/paper/2204.12939/) (2022)
- [Metric Flows with Neural Networks](https://mlsanity.com/paper/2310.19870/) (2023)
- [Bayesian RG Flow in Neural Network Field Theories](https://mlsanity.com/paper/2405.17538/) (2024)
- [NCoder -- A Quantum Field Theory approach to encoding data](https://mlsanity.com/paper/2402.00944/) (2024)
- [Wilsonian Renormalization of Neural Network Gaussian Processes](https://mlsanity.com/paper/2405.06008/) (2024)
- [Renormalization in the neural network-quantum field theory correspondence](https://mlsanity.com/paper/2212.11811/) (2022)
- [Physics-Driven Learning for Inverse Problems in Quantum Chromodynamics](https://mlsanity.com/paper/2501.05580/) (2025)
- [Field theory for optimal signal propagation in ResNets](https://mlsanity.com/paper/2305.07715/) (2023)
- [Fermions and Supersymmetry in Neural Network Field Theories](https://mlsanity.com/paper/2511.16741/) (2025)

## Abstract (arXiv)

We propose a theoretical understanding of neural networks in terms of Wilsonian effective field theory. The correspondence relies on the fact that many asymptotic neural networks are drawn from Gaussian processes, the analog of non-interacting field theories. Moving away from the asymptotic limit yields a non-Gaussian process and corresponds to turning on particle interactions, allowing for the computation of correlation functions of neural network outputs with Feynman diagrams. Minimal non-Gaussian process likelihoods are determined by the most relevant non-Gaussian terms, according to the flow in their coefficients induced by the Wilsonian renormalization group. This yields a direct connection between overparameterization and simplicity of neural network likelihoods. Whether the coefficients are constants or functions may be understood in terms of GP limit symmetries, as expected from 't Hooft's technical naturalness. General theoretical calculations are matched to neural network experiments in the simplest class of models allowing the correspondence. Our formalism is valid for any of the many architectures that becomes a GP in an asymptotic limit, a property preserved under certain types of training.