The autoregressive neural network architecture of the Boltzmann distribution of pairwise interacting spins systems

Generative Autoregressive Neural Networks (ARNN) 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 a physical interpretation of the ARNNs by reformulating 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 like the residual connections and a recurrent architecture with clear physical meanings. However, the exponential growth, with system size, of the number of parameters of the hidden layers makes its direct application unfeasible. Nevertheless, its architecture's explicit formulation allows using 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 performances in approximating the Boltzmann distributions of the corresponding physics model compared to other commonly used ARNN architectures. The connection established between the physics of the system and the ARNN architecture provides a way to derive new neural network architectures for different interacting systems and interpret existing ones from a physical perspective.