Fig. 2: Results for Curie–Weiss (CW) model. | Communications Physics

Fig. 2: Results for Curie–Weiss (CW) model.

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

Fig. 2

The CW model considered has J = 1 and h = 0 (see eq. (14)). The system undergoes a second-order phase transition at β = 1 where a spontaneous magnetization appears37. Six different architectures, the 1P, CW∞, 1L, CWN, MADE21, MADE22, are represented in the panels in the figure in, respectively, orange (or orange-circle), light-blue (or light-blue-circle), green, yellow, blue and red. a–c Relative error in the estimation of the free energy for different system sizes with respect to the analytic solution. The CWN architecture has its parameters fixed and precomputed analytically, and the error is too small to be seen at this scale. The y-axis is plotted on a logarithmic scale down to 10−4 and then linearly to zero. d The dependence on N of the mean and maximum relative error of the two smaller architectures, 1P and CW∞, both of which scale linearly with the size of the system. e Distribution of the overlaps of the samples generated by the ARNNs for the CW system with N = 200 variables and β = 1.3.

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