The minimal training set to train a working convolutional neural network (CNN) is explored in detail. The model under consideration is the frustrated <inline-formula><tex-math id='TM0001' notation='LaTeX'>$J_1$</tex-math></inline-formula>- <inline-formula><tex-math id='TM0002' notation='LaTeX'>$J_2$</tex-math></inline-formula> Ising model on the square lattice. Here <inline-formula><tex-math id='TM0003' notation='LaTeX'>$J_1 &lt; 0$</tex-math></inline-formula> and <inline-formula><tex-math id='TM0004' notation='LaTeX'>$J_2 &gt; 0$</tex-math></inline-formula> are the nearest and next-to-nearest neighboring couplings, respectively. We train the CNN using the configurations of <inline-formula><tex-math id='TM0005' notation='LaTeX'>$g \stackrel{\text{def}}{=} J_2/|J_1| = 0.7$</tex-math></inline-formula> and employ the resulting CNN to study the phase transition of <inline-formula><tex-math id='TM0006' notation='LaTeX'>$g = 0.8$</tex-math></inline-formula>. We find that this transfer learning is successful. In particular, only configurations of two temperatures, one below and one above the critical temperature <inline-formula><tex-math id='TM0007' notation='LaTeX'>$T_c$</tex-math></inline-formula> of <inline-formula><tex-math id='TM0008' notation='LaTeX'>$g=0.7$</tex-math></inline-formula>, are needed to accurately determine the <inline-formula><tex-math id='TM0009' notation='LaTeX'>$T_c$</tex-math></inline-formula> of <inline-formula><tex-math id='TM0010' notation='LaTeX'>$g=0.8$</tex-math></inline-formula>. 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 <inline-formula><tex-math id='TM0011' notation='LaTeX'>$T_c$</tex-math></inline-formula> of <inline-formula><tex-math id='TM0012' notation='LaTeX'>$g=0.7$</tex-math></inline-formula>. Otherwise, the performance of the obtained CNN is not of high quality, and hence cannot determine the <inline-formula><tex-math id='TM0013' notation='LaTeX'>$T_c$</tex-math></inline-formula> of <inline-formula><tex-math id='TM0014' notation='LaTeX'>$g=0.8$</tex-math></inline-formula> 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.

Taiwan

Shang-Wei Li

Yuan-Heng Tseng

Ming-Che Hsieh

Fu-Jiun Jiang

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