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 $q$ -state Potts model. To this end, we generate binarized images of spin configurations of the $q$ -state Potts model ( $q\ge 3$ ) by replacing the spin variables $\{0,1,\ldots,\lfloor q/2\rfloor-1\}$ and $\{\lfloor q/2\rfloor,\ldots,q-1\}$ with $\{0\}$ and $\{1\}$ , respectively. Then, we input these images to the trained CNN to output the predicted temperatures. The binarized images of the $q$ -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.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 $q$-state Potts model. To this end, we generate binarized images of spin configurations of the $q$-state Potts model ($q\ge 3$) by replacing the spin variables $\{0,1,\ldots,\lfloor q/2\rfloor-1\}$ and $\{\lfloor q/2\rfloor,\ldots,q-1\}$ with $\{0\}$ and $\{1\}$, respectively. Then, we input these images to the trained CNN to output the predicted temperatures. The binarized images of the $q$-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.

arXiv:2104.03632Progress of Theoretical and Experimental PhysicsApr 30, 202115 citations

Detection of Phase Transition via Convolutional Neural Networks

Akinori Tanaka, Akio Tomiya

We design a Convolutional Neural Network (CNN) which studies correlation between discretized inverse temperature and spin configuration of 2D Ising model and show that it can find a feature of the phase transition without teaching any a priori information for it. We also define a new order parameter via the CNN and show that it provides well approximated critical inverse temperature. In addition, we compare the activation functions for convolution layer and find that the Rectified Linear Unit (ReLU) is important to detect the phase transition of 2D Ising model.We design a Convolutional Neural Network (CNN) which studies correlation between discretized inverse temperature and spin configuration of 2D Ising model and show that it can find a feature of the phase transition without teaching any a priori information for it. We also define a new order parameter via the CNN and show that it provides well approximated critical inverse temperature. In addition, we compare the activation functions for convolution layer and find that the Rectified Linear Unit (ReLU) is important to detect the phase transition of 2D Ising model.

arXiv:1609.09087Journal of the Physical Society of JapanApr 28, 2017164 citations

Monte Carlo simulation of the three-dimensional Potts model

Hans Jürgen Herrmann

The European Physical Journal BJun 1, 197968 citations

Applications of neural networks to the studies of phase transitions of two-dimensional Potts models

C.-D. Li, D.-R. Tan, F.-J. Jiang

arXiv:1703.02369Annals of PhysicsFeb 23, 201863 citations

Phase Diagrams of Three-Dimensional Anderson and Quantum Percolation Models Using Deep Three-Dimensional Convolutional Neural Network

Tomohiro Mano, Tomi Ohtsuki

The three-dimensional Anderson model is a well-studied model of disordered electron systems that shows the delocalization--localization transition. As in our previous papers on two- and three-dimensional (2D, 3D) quantum phase transitions [J. Phys. Soc. Jpn. {\bf 85}, 123706 (2016), {\bf 86}, 044708 (2017)], we used an image recognition algorithm based on a multilayered convolutional neural network. However, in contrast to previous papers in which 2D image recognition was used, we applied 3D image recognition to analyze entire 3D wave functions. We show that a full phase diagram of the disorder-energy plane is obtained once the 3D convolutional neural network has been trained at the band center. We further demonstrate that the full phase diagram for 3D quantum bond and site percolations can be drawn by training the 3D Anderson model at the band center.The three-dimensional Anderson model is a well-studied model of disordered electron systems that shows the delocalization--localization transition. As in our previous papers on two- and three-dimensional (2D, 3D) quantum phase transitions [J. Phys. Soc. Jpn. {\bf 85}, 123706 (2016), {\bf 86}, 044708 (2017)], we used an image recognition algorithm based on a multilayered convolutional neural network. However, in contrast to previous papers in which 2D image recognition was used, we applied 3D image recognition to analyze entire 3D wave functions. We show that a full phase diagram of the disorder-energy plane is obtained once the 3D convolutional neural network has been trained at the band center. We further demonstrate that the full phase diagram for 3D quantum bond and site percolations can be drawn by training the 3D Anderson model at the band center.

arXiv:1709.00812Journal of the Physical Society of JapanOct 27, 201729 citations

q

-state Potts model. To this end, we generate binarized images of spin configurations of the

q

-state Potts model (

q\ge 3

) by replacing the spin variables

\{0,1,\ldots,\lfloor q/2\rfloor-1\}

and

\{\lfloor q/2\rfloor,\ldots,q-1\}

with

\{0\}

and

\{1\}

, respectively. Then, we input these images to the trained CNN to output the predicted temperatures. The binarized images of the

q

-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.