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arXiv:2501.05547 · Physical review. E

Deep learning of phase transitions with minimal examples

A. M. Abuali, David A. Clarke, Morten Hjorth-Jensen, Ioannis Konstantinidis, Claudia Ratti, Jianyi Yang

Published Aug 20, 2025• 6 citations
arXivHTMLPDF

Abstract

Over the past several years, there have been many studies demonstrating the ability of deep neural networks to identify phase transitions in many physical systems, notably in classical statistical physics systems. One often finds that the prediction of deep learning methods trained on many ensembles below and above the critical temperature T_{c} behaves similarly to an order parameter, and this analogy has been successfully used to locate T_{c} and estimate universal critical exponents. In this work, we pay particular attention to the ability of a convolutional neural network to capture these critical parameters for the two-dimensional Ising model when the network is trained on configurations at T=0 and T=∞ only. We directly compare its output to the same network trained at multiple temperatures below and above T_{c} to gain understanding of how this extreme restriction of training data can impact a neural network's ability to classify phases. We find that the network trained on two temperatures is still able to identify T_{c} and ν, while the extraction of γ becomes more challenging.

DOI
10.1103/wjvx-5nk7

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Over the past several years, there have been many studies demonstrating the ability of deep neural networks to identify phase transitions in many physical systems, notably in classical statistical physics systems. One often finds that the prediction of deep learning methods trained on many ensembles below and above the critical temperature T_{c} behaves similarly to an order parameter, and this analogy has been successfully used to locate T_{c} and estimate universal critical exponents. In this work, we pay particular attention to the ability of a convolutional neural network to capture these critical parameters for the two-dimensional Ising model when the network is trained on configurations at T=0 and T=∞ only. We directly compare its output to the same network trained at multiple temperatures below and above T_{c} to gain understanding of how this extreme restriction of training data can impact a neural network's ability to classify phases. We find that the network trained on two temperatures is still able to identify T_{c} and ν, while the extraction of γ becomes more challenging.
Abstract We discuss an aspect of neural networks for the purpose of phase transition detection. To this end, we first train the neural network by feeding Ising/Potts configurations with labels of temperature so that it can predict the temperature of the input. We do not explicitly supervise whether the configurations are in the ordered/disordered phase. Nevertheless, we can identify the critical temperature from the parameters (weights and biases) of the trained neural network. We attempt to understand how temperature-supervised neural networks capture information on the phase transition by paying attention to what quantities they learn. Our detailed analyses reveal that they learn different physical quantities depending on how well they are trained. The main observation in this study is how the weights in the trained neural network can have information on the phase transition in addition to temperature.
Abstract Phase transitions play a central role in statistical physics and are well understood in systems at thermal equilibrium. However, their nonequilibrium counterparts remain less explored because of the lack of a unified theoretical framework. In this study, we examine the potential of deep learning, specially convolutional neural networks (CNN), to detect nonequilibrium phase transitions (NEPT) in a 2D Ising model-like square lattice ferromagnetic system under effective parameter h . Without this parameter ( h = 0 ), we generate equilibrium spin configurations using the standard Metropolis algorithm as usual. With h ≠ 0 , we generate nonequilibrium spin configurations using an effective Glauber update rule which breaks detailed balance. While the former is used to train CNN, the later is not subsequently described by the usual Boltzmann-Gibbs distribution rather used to test the model. The study shows that the trained CNN can distinguish between ordered and disordered nonequilibrium steady states and accurately predict the critical temperature of the system undergoing NEPT. Finite size scaling analysis of the model’s output (perdition) reproduces critical exponents that agree with theoretical expectations, indicating that this method provides a strong, data driven approach for examining nonequilibrium critical phenomena.
Abstract We analyze the Ising model with anisotropic interactions in two dimensions using deep learning neural networks (NN) trained to classify paramagnetic (PM) and ferromagnetic (FM) phases. The spin configurations used as training data are labeled as belonging to either PM state or the FM state using analytically known phase transition temperatures. We use the variation of the output function to determine the critical temperature and to estimate the correlation length critical exponent, ν\nu ν ν , in a wide range of the anisotropy parameter. We find that on increasing the anisotropy, the critical exponent remains consistent with the expected value ν=1\nu {=}1ν=1 ν = 1 for the Ising universality class. At the same time, the NN prediction for the critical temperature, TcT_cTc​ T c , is shifted from the analytically known value, with the magnitude of the shift increasing with anisotropy.
Learning with an artificial neural network encodes the system behavior in a feed-forward function with a number of parameters optimized by data-driven training. An open question is whether one can minimize the network complexity without loss of performance to reveal how and why it works. Here we investigate the learning of the phase transition in the Ising model and find that having two hidden neurons can be enough for an accurate prediction of critical temperature. We show that the networks learn the scaling dimension of the order parameter while being trained as a phase classifier, demonstrating how the machine learning exploits the Ising universality to work for different lattices of the same criticality within a single set of trainings in one lattice geometry.
We present an analysis of neural network-based machine learning schemes for phases and phase transitions in theoretical condensed matter research, focusing on neural networks with a single hidden layer. Such shallow neural networks were previously found to be efficient in classifying phases and locating phase transitions of various basic model systems. In order to rationalize the emergence of the classification process and for identifying any underlying physical quantities, it is feasible to examine the weight matrices and the convolutional filter kernels that result from the learning process of such shallow networks. Furthermore, we demonstrate how the learning-by-confusing scheme can be used, in combination with a simple threshold-value classification method, to diagnose the learning parameters of neural networks. In particular, we study the classification process of both fully-connected and convolutional neural networks for the two-dimensional Ising model with extended domain wall configurations included in the low-temperature regime. Moreover, we consider the two-dimensional XY model and contrast the performance of the learning-by-confusing scheme and convolutional neural networks trained on bare spin configurations to the case of preprocessed samples with respect to vortex configurations. We discuss these findings in relation to similar recent investigations and possible further applications.
Conventionally, the study of phases in statistical mechan- ics is performed with the help of random sampling tools. Among the most powerful are Monte Carlo simulations consisting of a stochastic importance sampling over state space and evaluation of estimators for physical quantities. The ability of modern machine learning techniques to classify, identify, or in- terpret massive data sets provides a complementary paradigm to the above approach to analyze the exponentially large number of states in statistical physics. In this report, it is demonstrated by application on Ising-type models that deep learning has potential wide applications in solving many-body statis- tical physics problems. In application of supervised learning, we showed that the feed-forward neural network can identify phases and phase transitions in the ferromagnetic Ising model and the convolutional neural network (CNN) is extremely powerful in classifying T = 0 and T = ∞ phases in the Ising gauge model; In application of unsupervised learning, we illustrated that a deep auto-encoder constructed by stacked restricted Boltzmann machines (RBM)\nis closely related to the renormalization group (RG) method well understood in modern physics and our reconstruction of Ising spin configurations in the ferromagnetic Ising model is similar to the hand-written digits reconstruction.