Comments on the minimal training set for CNN: a case study of the frustrated $J_1$-$J_2$ Ising model on the square lattice

TL;DR

This work probes the minimal, real-data training set needed for CNNs to detect phase transitions in the frustrated $J_1$-$J_2$ Ising model on a square lattice. By training three CNN architectures on configurations at $g=0.7$ and transferring to $g=0.8$, the authors demonstrate that using configurations from just two temperatures (one below and one above the $T_c$ of $g=0.7$) can yield accurate estimates of the critical temperature, provided the training temps are not too far from $T_c(g=0.7)$. They also show that including low-temperature data can be beneficial but may introduce stronger finite-size effects, and that the optimal training set depends on capturing nearly all relevant states of the system. The findings offer practical guidelines for efficient ML studies of critical phenomena and implications for experiments where data are limited, illustrating how transfer learning and careful training-set design can still yield precise critical points. All mathematical notation is presented within $...$ delimiters.

Abstract

The minimal training set to train a working CNN is explored in detail. The considered model is the frustrated $J_1$-$J_2$ Ising model on the square lattice. Here $J_1 < 0$ and $J_2 > 0$ are the nearest and next-to-nearest neighboring couplings, respectively. We train the CNN using the configurations of $g \stackrel{\text{def}}{=} J_2/|J_1| = 0.7$ and employ the resulting CNN to study the phase transition of $g = 0.8$. We find that this transfer learning is successful. In particular, only configurations of two temperatures, one is below and one is above the critical temperature $T_c$ of $g=0.7$, are needed to reach accurately determination of the $T_c$ of $g=0.8$. However, it may be subtle to use this strategy for the training. Specifically, for the considered model, due to the inefficiency of the single spin flip algorithm used in sampling the configurations at the low-temperature region, the two temperatures associated with the training set should not be too far away from the $T_c$ of $g=0.7$, otherwise, the performance of the obtained CNN is not of high quality, hence cannot determine the $T_c$ of $g=0.8$ accurately. For the considered model, we also uncover the condition for training a successful CNN when only configurations of two temperatures are considered as the training set.

Figures (22)

• Figure 1: The frustrated $J_1$-$J_2$ Ising model on the square lattice studied here. The figure is taken from Ref. Li24.
• Figure 2: The first CNN used here. The numbers at the bottom of the figure are for $L=32$. The figure is created based on a figure of the submitted version of Ref. Tse24.
• Figure 3: The second CNN used here. The figure is taken from the submitted version of Ref. Tse24. The numbers at the bottom of the figure are for $L=64$.
• Figure 4: The values of loss function during the training for conducting 10 epochs using CNN2.
• Figure 5: The transfer learning results. (Left) The NN outputs as a function of $T$. The horizontal dashed line is 0.5. (Right) The associated standard deviations (STD) as a function of $T$. In both panels, the vertical solid lines are the theoretical $T_c$ of $g=0.8$. These outcomes are associated with CNN1.