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arXiv:2504.19795 · Progress of Theoretical and Experimental Physics

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

Shang-Wei Li, Yuan-Heng Tseng, Ming-Che Hsieh, Fu-Jiun Jiang

Published Oct 23, 2025• 2 citations
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Abstract

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 $J_1 < 0$ 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 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.

DOI
10.1093/ptep/ptaf151

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Abstract The minimal training set to train a working convolutional neural network (CNN) is explored in detail. The model under consideration is the frustrated J1J_1J1​-J2J_2J2​ Ising model on the square lattice. Here J1<0J_1 < 0J1​<0 and J2>0J_2 > 0J2​>0 are the nearest and next-to-nearest neighboring couplings, respectively. We train the CNN using the configurations of g=defJ2/∣J1∣=0.7g \stackrel{\text{def}}{=} J_2/|J_1| = 0.7g=defJ2​/∣J1​∣=0.7 and employ the resulting CNN to study the phase transition of g=0.8g = 0.8g=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 TcT_cTc​ of g=0.7g=0.7g=0.7, are needed to accurately determine the TcT_cTc​ of g=0.8g=0.8g=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 TcT_cTc​ of g=0.7g=0.7g=0.7. Otherwise, the performance of the obtained CNN is not of high quality, and hence cannot determine the TcT_cTc​ of g=0.8g=0.8g=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.
This paper presents the investigation of convolutional neural network (CNN) prediction successfully recognizing the temperature of the nonequilibrium phase transitions in two-dimensional (2D) Ising spins on a square lattice. The model uses image snapshots of ferromagnetic 2D spin configurations as an input shape to provide the average output predictions. By considering supervised machine learning techniques, we perform Metropolis Monte Carlo (MC) simulations to generate the configurations. In the equilibrium Ising model, the Metropolis algorithm respects detailed balance condition (DBC), while its nonequilibrium version violates DBC. Violating the DBC of the algorithm is characterized by a parameter −8<ε<8. We find the exact result of the transition temperature Tc(ε) in terms of ε. If we set ε=0, the usual single spin-flip algorithm can be restored, and the equilibrium configurations generated with such a set up are used to train our model. For ε≠0, the system attains the nonequilibrium steady states (NESS), and the modified algorithm generates NESS configurations (test dataset). The trained model is successfully tested on the test dataset. Our result shows that CNN can determine Tc(ε≠0) for various ε values, consistent with the exact result.
Abstract We study the phase transitions of the 2D antiferromagnetic Ising model with nearest J1J_1J1​ and next-to-nearest J2J_2J2​ interactions on the triangular lattice for J2/J1=0.1,0.5J_2/J_1 = 0.1, 0.5J2​/J1​=0.1,0.5, and 1.0. The method of supervised neural networks (NNs) is employed for the investigation. Although supervised NN is used, no real spin configurations are needed for the training. In addition, two kinds of configurations which have their spins arranged in a staggered pattern are considered as the training set. Remarkably, with this unconventional training strategy, not only are the critical temperatures of the studied J2/J1J_2/J_1J2​/J1​ computed accurately by the resulting NN, but also the nature of the investigated phase transitions is determined correctly. Specifically, the phase transitions associated with J2/J1=0.1,0.5J_2/J_1 = 0.1, 0.5J2​/J1​=0.1,0.5, and 1.0 are first order. These conclusions are consistent with the known results obtained by other methods. Since the training strategy is simple, the NN calculations are highly efficient. It remains to be examined whether the unconventional training approach considered in this study can be used to investigate other models with untypical phase transitions or with nontrivial ground-state configurations.
Recently, there has been significant progress in solving quantum many-particle problems via machine learning based on the restricted Boltzmann machine. However, it is still highly challenging to solve frustrated models via machine learning, which has not been demonstrated so far. In this paper, we design a brand new convolutional neural network (CNN) to solve such quantum many-particle problems. We demonstrate, for the first time, solving the highly frustrated spin-1/2 ${J}_{1}\ensuremath{-}{J}_{2}$ antiferromagnetic Heisenberg model on square lattices via CNN. The energy per site achieved by the CNN is even better than previous string-bond-state calculations. Our work therefore opens up a new routine to solve challenging frustrated quantum many-particle problems using machine learning.
We study the phase transitions of the two-dimensional antiferromagnetic Ising model with nearest J1J_1J1​ and next-to-nearest J2J_2J2​ interactions on the triangular lattice for J2/J1=0.1,0.5J_2/J_1 = 0.1, 0.5J2​/J1​=0.1,0.5 and 1.0. The method of supervised neural networks (NN) is employed for the investigation. While supervised NN is used, no real spin configurations are needed for the training. In addition, two kinds of configurations having their spins be arranged in a staggered pattern are considered as the training set. Remarkably, with this unconventional training strategy, not only the critical temperatures of the studied J2/J1J_2/J_1J2​/J1​ are computed accurately by the resulting NN, but also the nature of the investigated phase transitions are determined correctly. Specifically, the phase transitions associated with J2/J1=0.1,0.5J_2/J_1 = 0.1, 0.5J2​/J1​=0.1,0.5 and 1.0 are first order. These conclusions are consistent with the known results obtained by other methods. Since the training strategy is simple, the NN calculations is highly efficient. It remains to examine whether the unconventional training approach considered in this study can be used to investigate other models with untypical phase transitions or with nontrivial ground state configurations.
Abstract Employing a deep convolutional neural network (deep CNN) trained on spin configurations of the 2D Ising model and the temperatures, we examine whether the deep CNN can detect the phase transition of the 2D qqq-state Potts model. To this end, we generate binarized images of spin configurations of the qqq-state Potts model (q≥3q\ge 3q≥3) by replacing the spin variables {0,1,…,⌊q/2⌋−1}\{0,1,\ldots,\lfloor q/2\rfloor-1\}{0,1,…,⌊q/2⌋−1} and {⌊q/2⌋,…,q−1}\{\lfloor q/2\rfloor,\ldots,q-1\}{⌊q/2⌋,…,q−1} with {0}\{0\}{0} and {1}\{1\}{1}, respectively. Then, we input these images to the trained CNN to output the predicted temperatures. The binarized images of the qqq-state Potts model are entirely different from Ising spin configurations, particularly at the transition temperature. Moreover, our CNN model is not trained on the information about whether phases are ordered/disordered but is naively trained by Ising spin configurations labeled with temperatures at which they are generated. Nevertheless, the deep CNN can detect the transition point with high accuracy, regardless of the type of transition. We also find that, in the high-temperature region, the CNN outputs the temperature based on the internal energy, whereas, in the low-temperature region, the output depends on the magnetization and possibly the internal energy as well. However, in the vicinity of the transition point, the CNN may use more general factors to detect the transition point.
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