# What Neural Network Field Theory Can and Cannot Realise on a Computer

arXiv: https://arxiv.org/abs/2608.21523 · HTML: https://mlsanity.com/paper/2608.21523/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 17 of its 39 references that are in the map.
- First posted on arXiv 21 Aug 2026.

## Builds on (most cited first, 17 of 17)

- [Deep Neural Networks as Gaussian Processes](https://mlsanity.com/paper/1711.00165/) (2017)
- [Gaussian Process Behaviour in Wide Deep Neural Networks](https://mlsanity.com/paper/1804.11271/) (2018)
- [Flow-based generative models for Markov chain Monte Carlo in lattice field theory](https://mlsanity.com/paper/1904.12072/) (2019)
- [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)
- [Neural Networks and Quantum Field Theory](https://mlsanity.com/paper/2008.08601/) (2020)
- [Learning Lattice Quantum Field Theories with Equivariant Continuous Flows](https://mlsanity.com/paper/2207.00283/) (2022)
- [Symmetry-via-Duality: Invariant Neural Network Densities from Parameter-Space Correlators](https://mlsanity.com/paper/2106.00694/) (2021)
- [Building Quantum Field Theories Out of Neurons](https://mlsanity.com/paper/2112.04527/) (2021)
- [Advances in machine-learning-based sampling motivated by lattice quantum chromodynamics](https://mlsanity.com/paper/2309.01156/) (2023)
- [Neural Network Field Theories: Non-Gaussianity, Actions, and Locality](https://mlsanity.com/paper/2307.03223/) (2023)
- [Conformal Fields from Neural Networks](https://mlsanity.com/paper/2409.12222/) (2024)
- [Bayesian RG Flow in Neural Network Field Theories](https://mlsanity.com/paper/2405.17538/) (2024)
- [Wilsonian Renormalization of Neural Network Gaussian Processes](https://mlsanity.com/paper/2405.06008/) (2024)
- [Quantum Mechanics and Neural Networks](https://mlsanity.com/paper/2504.05462/) (2025)
- [Lecture Notes on Normalizing Flows for Lattice Quantum Field Theories](https://mlsanity.com/paper/2504.18126/) (2025)
- [Spinning Conformal Correlators from Neural Networks](https://mlsanity.com/paper/2608.15001/) (2026)

## Abstract (arXiv)

One aim of neural network field theory is to put a quantum or effective field theory on a computer, with the network ensemble itself as the theory. We ask how far that aim can be pushed for a function class regular enough to be computed with. Our main result is a no-go theorem with assumptions that hold for standard network architectures. We use it to separate four versions of neural network field theory, according to whether the defining object is the finite width ensemble or its infinite width limit, and whether the target we want to compute is a quantum or an effective field theory. Neither finite width interpretation is straightforwardly consistent. For finite width ensembles with finite variance at each point, the QFT interpretation fails reflection positivity, while the EFT interpretation establishes no scale separation by which the positivity violation can be placed outside its domain of validity. Of the two limit versions, one can be simulated in full and the other only in part, as only its smeared correlators are computable with a controlled error. As such, at the level of a controlled numerical computation, the QFT and EFT versions cannot be distinguished. One dimension escapes the obstruction, yet reflection positivity is shown to still fail there at every finite width for the cosine network. Two escapes from the theorem remain, giving up either finite variance at a point or exact rotation invariance, and we discuss both of these possibilities.