Machine learning for phase transition has received intensive research interest in recent years. However, its application in percolation still remains challenging. We propose an auxiliary Ising mapping method for the machine learning study of the standard percolation as well as a variety of statistical mechanical systems in correlated percolation representation. We demonstrate that unsupervised machine learning is able to accurately locate the percolation threshold, independent of the spatial dimension of system or the type of phase transition, which can be first-order or continuous. Moreover, we show that, by neural network machine learning, auxiliary Ising configurations for different universalities can be classified with a high confidence level. Our results indicate that the auxiliary Ising mapping method, despite its simplicity, can advance the application of machine learning in statistical and condensed-matter physics.
Machine learning for phase transition has received intensive research interest in recent years. However, its application in percolation still remains challenging. We propose an auxiliary Ising mapping method for the machine learning study of the standard percolation as well as a variety of statistical mechanical systems in correlated percolation representation. We demonstrate that unsupervised machine learning is able to accurately locate the percolation threshold, independent of the spatial dimension of system or the type of phase transition, which can be first-order or continuous. Moreover, we show that, by neural network machine learning, auxiliary Ising configurations for different universalities can be classified with a high confidence level. Our results indicate that the auxiliary Ising mapping method, despite its simplicity, can advance the application of machine learning in statistical and condensed-matter physics.
The identification and characterization of phase transitions is one of the oldest and most central problems in condensed matter physics, traditionally approached through order parameters, free-energy expansions, renormalization-group analysis, and direct numerical simulation. Over the past decade, machine learning (ML) has emerged as a complementary and, in some regimes, transformative methodology for this task. Supervised classifiers can learn to distinguish phases directly from raw configurations without an a priori order parameter; unsupervised and self-supervised techniques such as principal component analysis, autoencoders, and learning-by-confusion can locate transition points with minimal or no labeled data; and generative and variational architectures, including neural-network quantum states, now approximate ground-state wavefunctions of strongly correlated systems well beyond the reach of exact diagonalization. This paper reviews the theoretical foundations of phase transitions relevant to machine-learning applications, surveys the principal families of algorithms used to detect and classify phases, and examines their application across classical spin systems, percolation, topological and quantum phases, strongly correlated electron systems, and experimentally measured spectroscopic data. Representative case studies are discussed, including convolutional and graph-neural-network classifiers of lattice configurations, explainable-ML detection of thermodynamic transitions in photoemission spectra, and deep-learning-guided discovery of high-temperature superconductors. The review closes with a critical discussion of persistent challenges interpretability, data scarcity in correlated-electron experiments, generalization across Hamiltonians, and the reliability of learned order parameters near criticality and outlines directions for future research, including physics-informed architectures, uncertainty quantification, and closed-loop integration with autonomous experimentation.
The detection of phase transitions is a fundamental challenge in condensed matter physics, traditionally addressed through analytical methods and direct numerical simulations. In recent years, machine learning techniques have emerged as powerful tools to complement these standard approaches, offering valuable insights into phase and structure determination. Additionally, they have been shown to enhance the application of traditional methods. In this work, we review recent advancements in this area, with a focus on our contributions to phase and structure determination using supervised and unsupervised learning methods in several systems: (a) 2D site percolation, (b) the 3D Anderson model of localization, (c) the 2D J1-J2 Ising model, and (d) the prediction of large-angle convergent beam electron diffraction patterns.
Abstract Studying percolation phase transitions offers valuable insights into the characteristics of phase transitions, shedding light on the underlying mechanisms that govern the formation of global connectivity within a system. We explore the percolation phase transition in the 3D cubic Ising model by employing two machine learning techniques. Our results demonstrate that machine learning methods can distinguish different phases during the percolation transition. Through the finite-size scaling analysis on the output of the neural networks, the percolation temperature and a correlation length exponent in the geometrical percolation transition are extracted and compared to those in the thermal magnetization phase transition within the 3D Ising model. These findings provide a valuable method for enhancing our understanding of the properties of the QCD critical point, which belongs to the same universality class as the 3D Ising model.
Machine learning (ML) has been well applied to studying equilibrium phase transition models by accurately predicating critical thresholds and some critical exponents. Difficulty will be raised, however, for integrating ML into nonequilibrium phase transitions. The extra dimension in a given nonequilibrium system, namely time, can greatly slow down the procedure toward the steady state. In this paper we find that by using some simple techniques of ML, non-steady-state configurations of directed percolation (DP) suffice to capture its essential critical behaviors in both (1+1) and (2+1) dimensions. With the supervised learning method, the framework of our binary classification neural networks can identify the phase transition threshold, as well as the spatial and temporal correlation exponents. The characteristic time t_{c}, specifying the transition from active phases to absorbing ones, is also a major product of the learning. Moreover, we employ the convolutional autoencoder, an unsupervised learning technique, to extract dimensionality reduction representations and cluster configurations of (1+1) bond DP. It is quite appealing that such a method can yield a reasonable estimation of the critical point.
Neural networks in machine learning are highly capable of classifying data and recognizing graphics, and have been widely used in statistical physics, particularly in phase transitions. In this paper, we review recent developments in machine learning algorithms in the application of primary phase transition models. We first introduce the background of modern machine learning algorithms and related pioneering work in equilibrium phase transitions. Second, by taking directed percolation as an example of a typical non-equilibrium phase transition model, we present some of the latest research findings of our group. The research involves supervised, unsupervised, and semi-supervised learning for classifying phases, predicting critical points, and measuring critical exponents. Thereafter, we elaborate on related work on machine learning algorithms in phase transition research in quantum many-body theory, soft matter physics, and high-energy physics. Lastly, corresponding discussions and prospects are presented.
Abstract Machine learning has been widely applied in physics research. Although unsupervised learning can extract the critical points of phase transitions, the percolation model remains a challenge. Unsupervised learning using the raw configurations of the percolation model fails to capture the critical points. To capture the configuration characteristics of the percolation model, this paper proposes using the maximum cluster as input to the neural network. It is well understood that the order parameter of the percolation model is not simply the particle density, but rather the probability that a given site or bond belongs to the percolating cluster. Additionally, we introduce the use of a Siamese Neural Network (SNN) to detect percolation phase transitions. Unlike unsupervised dimensionality reduction methods or supervised binary classification outputs, the SNN produces a scalar output referred to as similarity. By combining the maximum cluster and the SNN, we not only successfully extract the critical value of the percolation model, but also calculate the correlation exponent via data collapse. We believe that the SNN has great potential in handling phase transition classification problems and can serve as a reference for studying other phase transition systems.