Minimal Training Set for Training a Successful CNN: A Case Study of the Frustrated <i>J</i> 1- <i>J</i> 2 Ising Model on the Square Lattice

Abstract The minimal training set to train a working convolutional neural network (CNN) is explored in detail. The model under consideration is the frustrated $J_1$-$J_2$ Ising model on the square lattice. Here $J_1 &amp;lt; 0$ and $J_2 &amp;gt; 0$ 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 below and one above the critical temperature $T_c$ of $g=0.7$, are needed to accurately determine the $T_c$ of $g=0.8$. However, it may be subtle to use this strategy for the training. Specifically, for the model under consideration, due to the inefficiency of the single spin-flip algorithm used in sampling the configurations in 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, and hence cannot determine the $T_c$ of $g=0.8$ accurately. For the model under consideration, we also uncover the condition for training a successful CNN when only configurations of two temperatures are considered as the training set.