Emergent field theories from neural networks

TL;DR

This paper uncovers a duality between Hamiltonian systems and neural networks, enabling the modeling of field theories through neural activation and learning dynamics, revealing structural parallels with physical gauge fields.

Contribution

It introduces a novel duality framework linking Hamiltonian physics to neural network dynamics, facilitating the modeling of field theories with neural networks.

Findings

Klein-Gordon fields modeled with symmetric weight tensors

Dirac fields modeled with anti-symmetric weight tensors

Weight and bias dynamics correspond to gauge field components

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.