## Deep Learning the Ising Model Near Criticality

*Alan Morningstar, Roger G. Melko*; 18(163):1−17, 2018.

### Abstract

It is well established that neural networks with deep
architectures perform better than shallow networks for many
tasks in machine learning. In statistical physics, while there
has been recent interest in representing physical data with
generative modelling, the focus has been on shallow neural
networks. A natural question to ask is whether deep neural
networks hold any advantage over shallow networks in
representing such data. We investigate this question by using
unsupervised, generative graphical models to learn the
probability distribution of a two-dimensional Ising system. Deep
Boltzmann machines, deep belief networks, and deep restricted
Boltzmann networks are trained on thermal spin configurations
from this system, and compared to the shallow architecture of
the restricted Boltzmann machine. We benchmark the models,
focussing on the accuracy of generating energetic observables
near the phase transition, where these quantities are most
difficult to approximate. Interestingly, after training the
generative networks, we observe that the accuracy essentially
depends only on the number of neurons in the first hidden layer
of the network, and not on other model details such as network
depth or model type. This is evidence that shallow networks are
more efficient than deep networks at representing physical
probability distributions associated with Ising systems near
criticality.

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