Hopfield Neural Network and Anisotropic Ising Model

Abstract

The probabilistic Hopfield model known also as the Boltzman machine is a basic example in the zoo of artificial neural networks. Initially it was designed as a model of associative memory, but played a fundamental role in understanding the statistical nature of the realm of neural networks. The close relation between the Boltzman machine and the Ising model was a challenging observation in[1]. In this note we go further, we establish another type of structural similarity between these models sharing the methods of the Bethe ansatz family of integrable statistical mechanics. We examine the asymmetric model on the triangular lattice with arbitrary weights. We show that the probability of passing a trajectory in time dynamics obeys the Gibbs distribution with a partition function of the Ising model on the cubic lattice with additional weights on diagonals.