We develop the neural network based “learning from regression uncertainty” approach for the automatic detection of phases of matter in nonequilibrium active systems. Taking the flocking phase transition of self-propelled active particles described by the Vicsek model for example, we find that after training a neural network for solving the inverse statistical problem, i.e. for performing the regression task of reconstructing the noise level from given samples of such a nonequilibrium many-body complex system’s steady state configurations, the uncertainty of regression results obtained by the well-trained network can actually be utilized to reveal possible phase transitions in the system under study. The noise level dependence of regression uncertainty is assumed to be in a non-trivial M-shape, and its valley appears at the critical point of the flocking phase transition. By directly comparing this regression-based approach with the widely-used classification-based “learning by confusion” and “learning with blanking” approaches, we show that our approach has practical effectiveness, efficiency, good generality for various physical systems across interdisciplinary fields, and a greater possibility of being interpretable via conventional notions of physics. These approaches can complement each other to serve as a promising generic toolbox for investigating rich critical phenomena and providing data-driven evidence on the existence of various phase transitions, especially for those complex scenarios associated with first-order phase transitions or nonequilibrium active systems where traditional research methods in physics could face difficulties.
We develop the neural network based “learning from regression uncertainty” approach for the automatic detection of phases of matter in nonequilibrium active systems. Taking the flocking phase transition of self-propelled active particles described by the Vicsek model for example, we find that after training a neural network for solving the inverse statistical problem, i.e. for performing the regression task of reconstructing the noise level from given samples of such a nonequilibrium many-body complex system’s steady state configurations, the uncertainty of regression results obtained by the well-trained network can actually be utilized to reveal possible phase transitions in the system under study. The noise level dependence of regression uncertainty is assumed to be in a non-trivial M-shape, and its valley appears at the critical point of the flocking phase transition. By directly comparing this regression-based approach with the widely-used classification-based “learning by confusion” and “learning with blanking” approaches, we show that our approach has practical effectiveness, efficiency, good generality for various physical systems across interdisciplinary fields, and a greater possibility of being interpretable via conventional notions of physics. These approaches can complement each other to serve as a promising generic toolbox for investigating rich critical phenomena and providing data-driven evidence on the existence of various phase transitions, especially for those complex scenarios associated with first-order phase transitions or nonequilibrium active systems where traditional research methods in physics could face difficulties.
Abstract For performing regression tasks involved in various physics problems, enhancing the precision or equivalently reducing the uncertainty of regression results is undoubtedly one of the central goals. Here, somewhat surprisingly, the unfavorable regression uncertainty in performing the regression tasks of inverse statistical problems is found to contain hidden information concerning the phase transitions of the system under consideration. By utilizing this hidden information, a new unsupervised machine learning approach was developed in this work for automated detection of phases of matter, dubbed learning from regression uncertainty. This is achieved by revealing an intrinsic connection between regression uncertainty and response properties of the system, thus making the outputs of this machine learning approach directly interpretable via conventional notions of physics. It is demonstrated by identifying the critical points of the ferromagnetic Ising model and the three-state clock model, and revealing the existence of the intermediate phase in the six-state and seven-state clock models. Comparing to the widely-used classification-based approaches developed so far, although successful, their recognized classes of patterns are essentially abstract, which hinders their straightforward relation to conventional notions of physics. These challenges persist even when one employs the state-of-the-art deep neural networks (NNs) that excel at classification tasks. In contrast, with the core working horse being an NN performing regression tasks, our new approach is not only practically more efficient, but also paves the way towards intriguing possibilities for unveiling new physics via machine learning in a physically interpretable manner.
We study the first-order flocking transition of birds flying in low-visibility conditions by employing three different representative types of neural network (NN) based machine architectures that are trained via either an unsupervised approach called learning by confusion or a widely used supervised approach. We find that after the training via either the unsupervised approach or the supervised one, all of these three different representative types of NNs, namely, the fully-connected NN, the convolutional NN, and the residual NN, are able to successfully identify the first-order flocking transition point of this nonequilibrium many-body system. This indicates that NN based machine can be employed as a promising generic tool to investigate rich physics in scenarios associated to first-order phase transitions and nonequilibrium many-body systems.
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.
Identifying phase transitions and classifying phases of matter is central to understanding the properties and behavior of a broad range of material systems. In recent years, machine-learning (ML) techniques have been successfully applied to perform such tasks in a data-driven manner. However, the success of this approach notwithstanding, we still lack a clear understanding of ML methods for detecting phase transitions, particularly of those that utilize neural networks (NNs). In this work, we derive analytical expressions for the optimal output of three widely used NN-based methods for detecting phase transitions. These optimal predictions correspond to the results obtained in the limit of high model capacity. Therefore, in practice they can, for example, be recovered using sufficiently large, well-trained NNs. The inner workings of the considered methods are revealed through the explicit dependence of the optimal output on the input data. By evaluating the analytical expressions, we can identify phase transitions directly from experimentally accessible data without training NNs, which makes this procedure favorable in terms of computation time. Our theoretical results are supported by extensive numerical simulations covering, e.g., topological, quantum, and many-body localization phase transitions. We expect similar analyses to provide a deeper understanding of other classification tasks in condensed matter physics.
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.
Detection of phase transitions is a critical task in statistical physics, traditionally pursued through analytic methods and direct numerical simulations. Recently, machine-learning techniques have emerged as promising tools in this context, with a particular focus on supervised and unsupervised learning methods, along with nonlearning approaches. In this paper, we study the performance of unsupervised learning in detecting phase transitions in the J_{1}-J_{2} Ising model on the square lattice. The model is chosen due to its simplicity and complexity, thus providing an understanding of the application of machine-learning techniques in both straightforward and challenging scenarios. We propose a simple method based on a direct comparison of configurations. The reconstruction error, defined as the mean-squared distance between two configurations, is used to determine the critical temperatures. The results from the comparison of configurations are contrasted with those of the configurations generated by variational autoencoders. Our findings highlight that for certain systems a simpler method can yield results comparable to more complex neural networks. This paper contributes to the broader understanding of machine-learning applications in statistical physics and introduces an efficient approach to the detection of phase transitions using machine determination techniques.