  Generative Autoregressive Neural Networks (ARNNs) have recently demonstrated
exceptional results in image and language generation tasks, contributing to the
growing popularity of generative models in both scientific and commercial
applications. This work presents an exact mapping of the Boltzmann distribution
of binary pairwise interacting systems into autoregressive form. The resulting
ARNN architecture has weights and biases of its first layer corresponding to
the Hamiltonian's couplings and external fields, featuring widely used
structures such as the residual connections and a recurrent architecture with
clear physical meanings. Moreover, its architecture's explicit formulation
enables the use of statistical physics techniques to derive new ARNNs for
specific systems. As examples, new effective ARNN architectures are derived
from two well-known mean-field systems, the Curie-Weiss and
Sherrington-Kirkpatrick models, showing superior performance in approximating
the Boltzmann distributions of the corresponding physics model compared to
other commonly used architectures. The connection established between the
physics of the system and the neural network architecture provides a means to
derive new architectures for different interacting systems and interpret
existing ones from a physical perspective.
