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Deep neural networks as lattice gauge theories

arXiv 2608.19331 ↗First posted 19 Aug 2026hep-th cond-mat.dis-nn cs.LGcited by 0

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Abstract (arXiv)

We modify the NN/QFT duality [1] to incorporate the layerwise permutation symmetry of the network, resulting in a (0!+!1)-dimensional lattice gauge theory, in which each layer of N neurons acts as an N-component lattice site, and the weight matrices play the role of gauge fields living on the links. In this framework, we compute the tree-level neuron-neuron propagator which describes the evolution of layer variance in the network, and develop the Feynman diagram machinery to compute interactions in the perturbative expansion in 1/N. In particular, we obtain a recursive expression for all corrections to the exact propagator at O(1), representing statistical fluctuations in the ensemble of networks, including infinitely-many loop diagrams mediating the interactions from previous layers. We also present a preliminary analysis of neuron scattering amplitudes that contribute order-by-order in 1/N, which provides a field-theoretic framework for studying higher-point correlations, and by extension information propagation, in deep networks. We remark on some interesting directions for future work at the intersection of neural networks and quantum field theory.

Key facts. No citing papers in the map yet (data through 7 Oct 2026). Builds on 34 of its 54 references that are in the map. First posted on arXiv 19 Aug 2026.

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References · 34

  1. 2015Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shiftcited by 6,979
  2. 2016Group Equivariant Convolutional Networkscited by 1,154
  3. 2022Git Re-Basin: Merging Models modulo Permutation Symmetriescited by 460
  4. 2021The Role of Permutation Invariance in Linear Mode Connectivity of Neural Networkscited by 285
  5. 2016Exponential expressivity in deep neural networks through transient chaoscited by 260
  6. 2016Deep Information Propagationcited by 194
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  9. 2023Energy Transformercited by 69
  10. 2019Weight-space symmetry in deep networks gives rise to permutation saddles, connected by equal-loss valleys across the loss landscapecited by 61
  11. 2020Neural Networks and Quantum Field Theorycited by 58
  12. 2023Dynamics of Finite Width Kernel and Prediction Fluctuations in Mean Field Neural Networkscited by 42
22 more
  1. 2021Building Quantum Field Theories Out of Neuronscited by 27
  2. 2021Nonperturbative renormalization for the neural network-QFT correspondencecited by 24
  3. 2021The edge of chaos: quantum field theory and deep neural networkscited by 23
  4. 2023Neural Network Field Theories: Non-Gaussianity, Actions, and Localitycited by 21
  5. 2026There Will Be a Scientific Theory of Deep Learningcited by 21
  6. 2021Unified field theoretical approach to deep and recurrent neuronal networkscited by 20
  7. 2023Renormalizing Diffusion Modelscited by 16
  8. 2025Beyond the Permutation Symmetry of Transformers: The Role of Rotation for Model Fusioncited by 16
  9. 2025Theory of Scaling Laws for In-Context Regression: Depth, Width, Context and Timecited by 15
  10. 2023Bayesian Renormalizationcited by 13
  11. 2024Understanding Diffusion Models by Feynman's Path Integralcited by 13
  12. 2019Covariance in Physics and Convolutional Neural Networkscited by 12
  13. 2024Conformal Fields from Neural Networkscited by 10
  14. 2024Wilsonian Renormalization of Neural Network Gaussian Processescited by 8
  15. 2025Applications of Statistical Field Theory in Deep Learningcited by 8
  16. 2025Fermions and Supersymmetry in Neural Network Field Theoriescited by 5
  17. 2026Disordered Dynamics in High Dimensions: Connections to Random Matrices and Machine Learningcited by 4
  18. 2023Criticality versus uniformity in deep neural networkscited by 3
  19. 2025Random Matrix Theory for Deep Learning: Beyond Eigenvalues of Linear Modelscited by 3
  20. 2026Towards Worst-Case Guarantees with Scale-Aware Interpretabilitycited by 3
  21. 2026Critical Organization of Deep Neural Networks, and p-Adic Statistical Field Theoriescited by 1
  22. 2026Spontaneous symmetry breaking and Goldstone modes for deep information propagationcited by 1

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