Machine learning of explicit order parameters: From the Ising model to SU(2) lattice gauge theory

We present a solution to the problem of interpreting neural networks classifying phases of matter. We devise a procedure for reconstructing the decision function of an artificial neural network as a simple function of the input, provided the decision function is sufficiently symmetric. In this case one can easily deduce the quantity by which the neural network classifies the input. The method is applied to the Ising model and SU(2) lattice gauge theory. In both systems we deduce the explicit expressions of the order parameters from the decision functions of the neural networks. We assume no prior knowledge about the Hamiltonian or the order parameters except Monte Carlo–sampled configurations.

Figure 1

The correlation probing neural network consists of three types of neural networks stacked on top of each other. The localization network is a fully convolutional neural network which prohibits connections outside of the receptive field of each output neuron and thus only recognizes correlations in the receptive field. The averaging layer averages over the input from the localization network, similarly to how the magnetization averages over all spins. The prediction network is a fully connected neural network, which transforms the output of the averaging layer to a prediction probability.

Figure 2

Results of the correlation probing neural network applied to the Ising model. The latent prediction $\xi$ is the argument of the $\text{sigmoid}$ function in the last layer of the prediction network. (a) The latent prediction is perfectly correlated with the absolute value of the average of spins, i.e., the magnetization in the $1\times 1$ network, for all sampled configurations. (b) The latent prediction of the $1\times 2$ network is perfectly correlated with the average product of neighbors, i.e., the expected energy per site.

Figure 3

Results of the correlation probing network applied to SU(2) lattice gauge theory. (a),(b) The latent prediction shows a strong correlation with the Polyakov loop in both the $2\times 8\times 8\times 8$ network and the $2\times 1\times 1\times 1$ network. (c) The average prediction probability of the two networks.

Figure 4

Results of the correlation probing neural network applied to the Ising model. (a) The latent prediction is perfectly correlated with the absolute value of the average of spins, i.e., the magnetization in the $1\times 1$ network. (b) The latent prediction of the $1\times 2$ network is not correlated with the absolute value of the average of spins. (d) The latent prediction of the $1\times 2$ network is perfectly correlated with the average product of neighbors, i.e., the expected energy per site. (c) The latent prediction of the $1\times 1$ network is not correlated with the average of neighbors.

Figure 5

(a) Finding a possible phase transition in SU(2) lattice gauge theory with PCA. Green solid: The average mean squared error reconstruction loss as a function of temperature is a universal identifier for a phase transition. It was calculated in 100 independent incremental PCA runs with two principal components (PC), measured in units of $\times {10}^{-5}$ and shifted by the value at $\beta =3.5$. Blue dotted: The average norm of the PC also indicates a phase transition. (b) There is no correlation between the principal components and the Polyakov loop.