The GINN framework: a stochastic QED correspondence for stability and chaos in deep neural networks

Abstract: The development of a Euclidean stochastic field-theoretic approach that maps deep neural networks (DNNs) to quantum electrodynamics (QED) with local U(1) symmetry is presented. Neural activations and weights are represented by fermionic matter and gauge fields, with a fictitious Langevin time enabling covariant gauge fixing. This mapping identifies the gauge parameter with kernel design choices in wide DNNs, relating stability thresholds to gauge-dependent amplification factors. Finite-width fluctuations correspond to loop corrections in QED. As a proof of concept, we validate the theoretical predictions through numerical simulations of standard multilayer perceptrons and, in parallel, propose a gauge-invariant neural network (GINN) implementation using magnitude--phase parameterization of weights. Finally, a double-copy replica approach is shown to unify the computation of the largest Lyapunov exponent in stochastic QED and wide DNNs.