Emergent field theories from neural networks

TL;DR

The paper introduces a duality between Hamiltonian dynamics and neural-network learning dynamics, showing that activation and learning updates can reproduce Hamilton's equations for a given Hamiltonian ${\cal H}(\phi,\beta)$. By enforcing symmetry and commutation conditions, it constructs a dual loss $H(\phi,\pi)$ and demonstrates that emergent field theories arise from neural dynamics: a Klein-Gordon field from a symmetric weight structure on a lattice, and a Dirac field from antisymmetric tensor factors, with spinor assembly emerging from coupled state variables. It further shows that promoting global $U(1)$ symmetry to a local gauge symmetry requires a dynamical gauge field $A_{\mu}$, enabling covariant derivatives and a minimally coupled Dirac equation $i\eta_{\mu\nu}\gamma^{\mu}(\partial^{\nu}+iA^{\nu})\psi - m\psi=0$, with gauge dynamics potentially arising from learning dynamics. The work positions itself as a conceptual bridge between physics and learning, suggesting avenues for emergent-field descriptions while acknowledging the framework's classical scope and open questions about quantum or gravitational extensions.

Abstract

We establish a duality relation between Hamiltonian systems and neural network-based learning systems. We show that the Hamilton's equations for position and momentum variables correspond to the equations governing the activation dynamics of non-trainable variables and the learning dynamics of trainable variables. The duality is then applied to model various field theories using the activation and learning dynamics of neural networks. For Klein-Gordon fields, the corresponding weight tensor is symmetric, while for Dirac fields, the weight tensor must contain an anti-symmetric tensor factor. The dynamical components of the weight and bias tensors correspond, respectively, to the temporal and spatial components of the gauge field.