Finding the deconfinement temperature in lattice Yang-Mills theories from outside the scaling window with machine learning

We study the machine learning techniques applied to the lattice gauge theory’s critical behavior, particularly to the confinement/deconfinement phase transition in the SU(2) and SU(3) gauge theories. We find that the neural network, trained on lattice configurations of gauge fields at an unphysical value of the lattice parameters as an input, builds up a gauge-invariant function, and finds correlations with the target observable that is valid in the physical region of the parameter space. In particular, we show that the algorithm may be trained to build up the Polyakov loop which serves an order parameter of the deconfining phase transition. The machine learning techniques can thus be used as a numerical analog of the analytical continuation from easily accessible but physically uninteresting regions of the coupling space to the interesting but potentially not accessible regions.

Figure 1

The Polyakov loop in SU(2) gauge theory at the $N_t=2$ and $N_s=8$, 16, 32 lattices. The Monte-Carlo (MC) simulation, shown by the blue line, and the prediction of the machine-learning (ML) algorithm, shown by the orange line, overlap within the error bars. The vertical dashed line shows the critical value of $\beta$ obtained with the fits (11) of the Polyakov loop susceptibility (9). We use 100 configurations for all three lattice sizes.

Figure 2

Learning curves for training and validation at the point $\beta =4$ of the SU(2) gauge theory on ${16}^3\times 4$ lattice with the mean squared error (MSE) used as a loss function. The MSE normalized on the value of the order parameter squared, $⟨|L|{⟩}^2$, gives qualitatively the same picture.

Figure 3

The results for the Polyakov loop for SU(2) gauge theory at $N_t=4$ coming from the Monte Carlo (MC) simulations compared with the prediction of the machine-learning (ML) algorithm. The notations are same as in Fig. 1.

Figure 4

The degree of the gauge dependence in the prediction of the order parameter by the ML algorithm. The predicted order parameter along with the prediction uncertainty vs the number of the gauge randomization steps of the initial ${16}^3\times 4$ gluon configuration at $\beta =2.5$.

Figure 5

The results for the Polyakov loop for SU(3) gauge theory at $N_t=2$ obtained with the Monte Carlo simulations as compared to the neural network prediction. The absolute value, the real and imaginary parts of the loop are shown. The ML value of $|L|$ restored from ML predictions of $|\mathrm{Re}[L]|$ and $|\mathrm{Im}[L]|$.

Figure 6

The same results as in Fig. 5, but for the temporal extension $N_t=4$ of SU(3) gauge theory.