Detection of Phase Transition via Convolutional Neural Networks

Abstract

A Convolutional Neural Network (CNN) is designed to study correlation between the temperature and the spin configuration of the 2 dimensional Ising model. Our CNN is able to find the characteristic feature of the phase transition without prior knowledge. Also a novel order parameter on the basis of the CNN is introduced to identify the location of the critical temperature; the result is found to be consistent with the exact value.Studies of phase transition are connected to various areas among theoretical/experimental physics. 1-7) Calculating order parameters is one of the conventional ways to define phases and phase transitions. However, some phases like topological phases 8) do not have any clear order parameters. Even if there are certain theoretical order parameters like entanglement entropy, 9,10) they are difficult to measure in experiments.Machine learning (ML) techniques are useful to resolve this undesirable situation. In fact, ML techniques have been already applied to various problems in theoretical physics: finding approximate potential surface, 11) a study of transition in glassy liquids, 12) solving mean-field equations 14) and quantum many-body systems, 13, 15) a study of topological phases. 16) Especially, ML techniques based on convolutional neural network (CNN) have been developing since the recent groundbreaking record 17) in ImageNet Large Scale Visual Recognition Challenge 2012 (ILSVRC2012), 18) and it is applied to investigate phases of matters with great successes on classifications of phases in 2D systems [19][20][21] and 3D systems. 22,23) It is even possible to draw phase diagrams. 23) In these previous works, however, one needs some informations of the answers for the problems a priori. For example, to classify phases of a system, the training process requires the values of critical temperatures or the location of phase boundaries. This fact prevents applications of the ML techniques to unknown systems so far.The learning process without any answers is called unsupervised learning. Indeed, there are known results on detecting the phase transitions based on typical unsupervised learning architectures called autoencoder which is equivalent to principal component analysis 32) and its variant called variational autoencoder. ?) These architectures encode informations of given samples to lower dimensional vectors, and it is pointed out that such encoding process is similar to encoding physical state informations to order parameters of the systems. However, it is not evident whether the latent variables provide the critical temperature.We propose a novel but simple prescription to estimate the critical temperature of the system via neural network (NN) based on ML techniques without a priori knowledge of the order parameter. Throughout this letter, we focus on the fer- * akinori.tanaka@riken.jp † akio.tomiya@mail.ccnu.edu.cn m m Fig. 1. '(Color online)' Plots of weight matrix components in convolutional neural network (left) and fully connected neural network (right). Horizontal axis correspon...

Scale-invariant feature extraction of neural network and renormalization group flow

“…Then in this background of the hidden spin configurations, the expectation value of v i can be again easily calculated by using Eq. (18). We write it as v A(1) i…”

“…Two-dimensional Ising model is the simplest statistical model to exhibit the second order phase transition, and there are many previous studies of the Ising model using machine learnings. See e.g.,[16][17][18][19][20][21].…”

Phase transition encoded in neural network

“…Another intriguing approach was proposed in [4] to detect the phase transition by specu-lating that the information of order parameters are encoded in the weights of neural network as a consequence of training. They attempted to identify the critical temperature of the two dimensional Ising model based on supervised machine learning.…”

Unsupervised Machine Learning for Analysis of Coexisting Lipid Phases and Domain Growth in Biological Membranes

“…Currently, machine learning (ML) methods have shown a lot of promise in many fields 11 , and recently some of them have been applied for analysis and detection of classical and quantum phase transition data generated by simulations. Both supervised and unsupervised ML approaches have been used for this purpose [12][13][14][15][16][17][18][19] , but most of these pioneer studies used small Ising-like systems for their investigations. ML has previously been coupled with MD simulations of biomolecular systems in a limited context.…”