In recent years, significant attention has been paid to using end-to-end neural networks for analyzing Monte Carlo data. However, the exploration of non-end-to-end generative adversarial neural networks remains limited. Here, we investigate classical many-body systems using generative adversarial neural networks. We employ the conditional generative adversarial network with an auxiliary classifier (AC-GAN) and integrate self-attention layers into the generator. This modification enables the network learn the distribution of the two-dimensional (2D) XY model’s spin configurations and the physical quantities of interest. Utilizing the symmetry of the systems, we discover that AC-GAN can be trained with a very small raw dataset. This approach allows us to obtain reliable measurements for models typically demanding large samples, such as the large-sized 2D XY and the 3D constrained Heisenberg models. Moreover, we demonstrate the capability of AC-GAN to identify the phase transition points by quantifying the distribution changes in the spin configurations of the systems.
In recent years, significant attention has been paid to using end-to-end neural networks for analyzing Monte Carlo data. However, the exploration of non-end-to-end generative adversarial neural networks remains limited. Here, we investigate classical many-body systems using generative adversarial neural networks. We employ the conditional generative adversarial network with an auxiliary classifier (AC-GAN) and integrate self-attention layers into the generator. This modification enables the network learn the distribution of the two-dimensional (2D) XY model’s spin configurations and the physical quantities of interest. Utilizing the symmetry of the systems, we discover that AC-GAN can be trained with a very small raw dataset. This approach allows us to obtain reliable measurements for models typically demanding large samples, such as the large-sized 2D XY and the 3D constrained Heisenberg models. Moreover, we demonstrate the capability of AC-GAN to identify the phase transition points by quantifying the distribution changes in the spin configurations of the systems.
In this work, we study generative adversarial networks (GANs) as a tool to learn the distribution of spin configurations and to generate samples, conditioned on external tuning parameters or other quantities associated with individual configurations. For concreteness, we focus on two examples of conditional variables---the temperature of the system and the energy of the samples. We show that temperature-conditioned models can not only be used to generate samples across thermal phase transitions, but also be employed as unsupervised indicators of transitions. To this end, we introduce a GAN-fidelity measure that captures the model’s susceptibility to external changes of parameters. The proposed energy-conditioned models are integrated with Monte Carlo simulations to perform over-relaxation steps, which break the Markov chain and reduce auto-correlations. We propose ways of efficiently representing the physical states in our network architectures, e.g., by exploiting symmetries, and to minimize the correlations between generated samples. A detailed evaluation, using the two-dimensional XY model as an example, shows that these incorporations bring in considerable improvements over standard machine-learning approaches. We further study the performance of our architectures when no training data is provided near the critical region.
Recently, much effort has been devoted to studying whether machine learning methods are capable of recognizing phase boundaries in spin systems. This is typically done using deep neural networks, trained on system configurations at fixed control parameter values, but without receiving any a priori knowledge of physical features. Opening the ‘black-box algorithms’ and uncovering which features are captured by the neural network is a crucial and oft-overlooked step. Without this additional step, one cannot guarantee that the algorithm's decision on the phase boundaries is based on physically relevant features, or on less relevant characteristics—which would limit its applicability. We use the example of exploring the two-dimensional phase diagram of a non-equilibrium spin system (active Ising model) to highlight the importance of scrutinizing the internal representation of a neural network. By only training networks on a small slice of the phase diagram, we show that some networks capture the relevant physics to complete the remainder of the phase diagram. Other networks fail in doing so—even though they are perfectly capable of reaching their training objective. We demonstrate that by highlighting on which input regions networks base their decision, we can select the relevant networks and show that they can capture physical features such as emergent magnetization patterns [Casert, C., Vieijra, T., Nys, J., & Ryckebusch, J. (2019). Physical Review E, 99(2), 023304]. An additional test of whether deep neural networks can recognize physical characteristics is in its potential to generate additional system configurations. We show that a generative adversarial network (GAN) can create spin configurations that are indistinguishable from the training data [Mills, K., & Tamblyn, I. (2017). arXiv preprint arXiv:1710.08053]. By conditioning the GAN on physical quantities, we show it can accurately learn how to use this information in its generation procedure, and create configurations at a requested value of an observable, e.g. at a fixed energy. Furthermore, we show how we can use GANs to create configurations at system sizes much larger than the training data, allowing for highly efficient sampling of arbitrarily large configurations.
Identifying phase transitions in complex many-body systems traditionally necessitates the definition of specific order parameters, a task often requiring prior knowledge of the statistical model and the symmetry-breaking mechanism. In this work, we propose a framework for detecting phase transitions directly from raw (experimental) data without requiring knowledge of the underlying model Hamiltonian, parameters, or pre-defined labels. Inspired by generative modeling in machine learning, our method utilizes autoregressive networks to estimate the normalized probability distribution of the system from raw configuration data. We then quantify the intrinsic sensitivity of this learned distribution to control parameters (such as temperature) to construct a robust indicator of phase transitions. This indicator is based on the expectation of the change in absolute logarithmic probability, derived entirely from the raw data. Our approach is purely data-driven: it takes raw data across varying control parameters as input and outputs the most likely estimate of the phase transition point. To validate our approach, we conduct extensive numerical experiments on the 2D Ising model on both triangular and square lattices, and on the Sherrington-Kirkpatrick (SK) model utilizing raw data generated via Markov Chain Monte Carlo and Tensor Network methods. The results demonstrate that our generative approach accurately identifies phase transitions using only raw data. Our framework provides a general tool for exploring critical phenomena in model systems, with the potential to be extended to realistic experimental data where theoretical descriptions remain incomplete.
The advancement of diverse generative deep learning models and their variants has furnished substantial insights for investigating quantum many-body problems. In this work, we design two models based on the foundational architecture of generative adversarial networks (GANs) to investigate the ground-state properties and phase transition characteristics of the two-dimensional quantum rotor model (QRM). Within a semi-supervised learning framework, we incorporate multiple layers of transposed convolutions in the generator, enabling the conditional GAN to more efficiently extract low-dimensional encoded information. Analysis of one-dimensional latent variables associated with ground-state samples for different system sizes allows us to pinpoint the location of the critical point. In addition, we introduce dynamically adaptive weighting factors related to the distributional characteristics into the loss function of the deep convolutional GAN, and utilize upsampling techniques to enlarge the generated sample sizes. Comparisons of the optimization processes for mean magnetization and potential energy density across different magnetization regimes of QRM demonstrate that our model can efficiently generate valid ground-state samples, significantly reducing computational time. Our results highlight the promising potential of generative deep learning in quantum phase transition research, especially in critical point identification and the auxiliary generation of simulation data for quantum many-body models.
Abstract In this work, we explore the statistical system's configuration generation with generative models from Deep Learning to go beyond conventional Monte Carlo methods. Specifically, we devise a conditional generative adversarial network (cGAN) for the Ising system spin configuration generation, and we demonstrate it's working outside the training range of temperature for the ensemble of configurations. Being different from the original GAN design, we add a further recognizer network for the constraint on the conditional parameters (in our case temperature) and also provide a diversity benefit for the generative model. We showed that the newly proposed cGAN here can learn the distribution of the Ising model for different temperatures, which can generate spin configuration efficiently with correct (within a probability distribution) temperature estimation for the microscopic configurations. Moreover, without information about criticality provided inside the training data set, it is also shown that the developed cGAN can generate Ising spin configurations around the phase transition point with the order parameter (mean magnetization) reasonably well-match to the conventional MCMC simulation but with parallel sampling advantage. We also compared typical spin configurations from cGAN with specified conditional temperature to be the critical temperature with samples simulated by MCMC, which visibly is not distinguishable. This thus can possibly help to avoid critical slowing-down as shown in the traditional Monte Carlo method.
The detection of phase transitions in quantum many-body systems with lowest possible prior knowledge of their details is among the most rousing goals of the flourishing application of machine-learning techniques to physical questions. Here, we train a Generative Adversarial Network (GAN) with the Entanglement Spectrum of a system bipartition, as extracted by means of Matrix Product States ansätze. We are able to identify gapless-to-gapped phase transitions in different one-dimensional models by looking at the machine inability to reconstruct outsider data with respect to the training set. We foresee that GAN-based methods will become instrumental in anomaly detection schemes applied to the determination of phase-diagrams.