Abstract We introduce the use of neural networks as classifiers on classical disordered systems with no spatial ordering. In this study, we propose a framework of design objectives for learning tasks on disordered systems. Based on our framework, we implement a convolutional neural network trained to identify the spin-glass state in the three-dimensional Edwards–Anderson Ising spin-glass model from an input of Monte Carlo sampled configurations at a given temperature. The neural network is designed to be flexible with the input size and can accurately perform inference over a small sample of the instances in the test set. We examine and discuss the use of the neural network in classifying instances from three-dimensional Edwards–Anderson Ising spin-glass in a (random) field.
Abstract We introduce the use of neural networks as classifiers on classical disordered systems with no spatial ordering. In this study, we propose a framework of design objectives for learning tasks on disordered systems. Based on our framework, we implement a convolutional neural network trained to identify the spin-glass state in the three-dimensional Edwards–Anderson Ising spin-glass model from an input of Monte Carlo sampled configurations at a given temperature. The neural network is designed to be flexible with the input size and can accurately perform inference over a small sample of the instances in the test set. We examine and discuss the use of the neural network in classifying instances from three-dimensional Edwards–Anderson Ising spin-glass in a (random) field.
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.
We review recent developments of machine learning algorithms pertinent to the inverse renormalization group, which was originally established as a generative numerical method by Ron-Swendsen-Brandt via the implementation of compatible Monte Carlo simulations. Inverse renormalization group methods enable the iterative generation of configurations for increasing lattice size without the critical slowing down effect. We discuss the construction of inverse renormalization group transformations with the use of convolutional neural networks and present applications in models of statistical mechanics, lattice field theory, and disordered systems. We highlight the case of the three-dimensional Edwards-Anderson spin glass, where the inverse renormalization group can be employed to construct configurations for lattice volumes that have not yet been accessed by dedicated supercomputers.
Spin glasses are disordered magnets with random interactions that are, generally, in conflict with each other. Finding the ground states of spin glasses is not only essential for the understanding of the nature of disordered magnetic and other physical systems, but also useful to solve a broad array of hard combinatorial optimization problems across multiple disciplines. Despite decades-long efforts, an algorithm with both high accuracy and high efficiency is still lacking. Here we introduce DIRAC - a deep reinforcement learning framework, which can be trained purely on small-scale spin glass instances and then applied to arbitrarily large ones. DIRAC displays better scalability than other methods and can be leveraged to enhance any thermal annealing method. Extensive calculations on 2D, 3D and 4D Edwards-Anderson spin glass instances demonstrate the superior performance of DIRAC over existing methods. As many hard combinatorial optimization problems have Ising spin glass formulations, our results suggest a promising tool in solving these hard problems. Moreover, the presented algorithm will help us better understand the nature of the low-temperature spin-glass phase, which is a fundamental challenge in statistical physics.
Supervised machine learning is emerging as a powerful computational tool to predict the properties of complex quantum systems at a limited computational cost. In this article, we quantify how accurately deep neural networks can learn the properties of disordered quantum systems as a function of the system size. We implement a scalable convolutional network that can address arbitrary system sizes. This network is compared with a recently introduced extensive convolutional architecture [Mills et al., Chem. Sci. 10, 4129 (2019)2041-652010.1039/C8SC04578J] and with conventional dense networks with all-to-all connectivity. The networks are trained to predict the exact ground-state energies of various disordered systems, namely, a continuous-space single-particle Hamiltonian for cold-atoms in speckle disorder, and different setups of a quantum Ising chain with random couplings, including one with only short-range interactions and one augmented with a long-range term. In all testbeds we consider, the scalable network retains high accuracy as the system size increases. Furthermore, we demonstrate that the network scalability enables a transfer-learning protocol, whereby a pretraining performed on small systems drastically accelerates the learning of large-system properties, allowing reaching high accuracy with small training sets. In fact, with the scalable network one can even extrapolate to sizes larger than those included in the training set, accurately reproducing the results of state-of-the-art quantum Monte Carlo simulations.
We investigate the two-dimensional frustrated quantum Heisenberg model with bond disorder on nearest-neighbor couplings using the recently introduced Foundation Neural-Network Quantum States framework, which enables accurate and efficient computation of disorder-averaged observables with a single variational optimization. Simulations on large lattices reveal an extended region of the phase diagram where conventional magnetic long-range order vanishes in the thermodynamic limit, while the Edwards-Anderson order parameter remains finite, signaling the emergence of a quantum spin-glass phase. These findings, supported by a semiclassical analysis based on a large-spin expansion, provide compelling evidence that the spin glass-order is stable against quantum fluctuations, unlike the classical case where it disappears at any finite temperature.
In recent years Machine Learning has proved to be successful in many technological applications and scientific tasks, such as image and speech recognition, natural language
understanding and more. In this work we apply both supervised and unsupervised Machine Learning to three key problems in statistical physics: design of Hamiltonian from data,
phase recognition and study of critical properties of a system undergoing phase transition.
As an example of model design, we use a dataset with spin configurations and corresponding energies randomly sampled from a one dimensional Ising model. We then make two
guesses for the correct Hamiltonian: the former involving interactions of spins with an external local field, the latter involving two body interactions among spins.
The coupling constants are determined with a linear penalized regression, comparing the effects of L1 and L2 penalization terms of the cost function. We pay specific attention
to the problem of overfitting and to the validation process, which is critical for accepting or rejecting the proposed model.
The phase recognition problem is faced with the two dimensional Ising and XY models as examples. After showing the limits of a simple softmax regression for this task,
we build suitable neural networks to overcome these limits. In particular, a feed forward network is built and the learning process is investigated for the Ising model;
while a more sophisticated convolutional network is proposed for the XY model in order to detect local topological structures.
The last part of the work is dedicated to the unsupervised study of phase transitions, and the determination of critical properties (order parameter, critical
temperature, critical exponents). The discussed techniques are Principal Component Analysis (PCA) for dimensional reduction and K-means clustering for organizing data into
subsets with specific properties. PCA is applied to the two dimensional Ising, Potts and XY models, and it is used to find suitable order parameters. The study of the
proposed order parameter as a function of temperature provides evidence of phase transition, and a finite size scaling allows to extrapolate both the critical temperature
in the thermodynamic limit, and the ν critical exponent for the correlation length. K-means clustering is applied to equilibrium configurations of the two dimensional
Ising model, before and after the dimensional reduction, and the critical temperature is estimated. Moreover, a clustering of relaxation curves of magnetization in a
Monte Carlo dynamics is used to build a phase diagram on the parameter space of temperature and magnetic field.
A synergy of C++, Python and Wolfram Mathematica 12.0 is used throughout this work in order to sample input datasets and to build and control customized neural networks
and learning tools. The most relevant codes are provided in the appendix.