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StatPhys 27 Main Conference

Discriminative and generative machine learning for spin systems based on physically interpretable features

Corneel Casert, Kyle Mills, Jannes Nys, J. Ryckebusch, Isaac Tamblyn, Tom Vieijra

Published Apr 17, 2019• 0 citations
OpenAlex

Abstract

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.

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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.
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.
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.
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.
The identification of phases of matter is a challenging task, especially in quantum mechanics, where the complexity of the ground state appears to grow exponentially with the size of the system. Traditionally, physicists have to identify the relevant order parameters for the classification of the different phases. We here follow a radically different approach: we address this problem with a state-of-the-art deep learning technique, adversarial domain adaptation. We derive the phase diagram of the whole parameter space starting from a fixed and known subspace using unsupervised learning. This method has the advantage that the input of the algorithm can be directly the ground state without any ad hoc feature engineering. Furthermore, the dimension of the parameter space is unrestricted. More specifically, the input data set contains both labeled and unlabeled data instances. The first kind is a system that admits an accurate analytical or numerical solution, and one can recover its phase diagram. The second type is the physical system with an unknown phase diagram. Adversarial domain adaptation uses both types of data to create invariant feature extracting layers in a deep learning architecture. Once these layers are trained, we can attach an unsupervised learner to the network to find phase transitions. We show the success of this technique by applying it on several paradigmatic models: the Ising model with different temperatures, the Bose-Hubbard model, and the Su-Schrieffer-Heeger model with disorder. The method finds unknown transitions successfully and predicts transition points in close agreement with standard methods. This study opens the door to the classification of physical systems where the phase boundaries are complex such as the many-body localization problem or the Bose glass phase.
Machine learning methods are powerful in distinguishing different phases of matter in an automated way and provide a new perspective on the study of physical phenomena.We train a restricted Boltzmann machine (RBM) on data constructed with spin configurations sampled from the Ising Hamiltonian at different values of temperature and external magnetic field using Monte Carlo methods.From the trained machine we obtain the flow of iterative reconstruction of spin state configurations to faithfully reproduce the observables of the physical system.We find that the flow of the trained RBM approaches the spin configurations of the maximal possible specific heat which resemble the near-criticality region of the Ising model.In the special case of the vanishing magnetic field the trained RBM converges to the critical point of the renormalization group (RG) flow of the lattice model.Our results suggest an explanation of how the machine identifies the physical phase transitions, by recognizing certain properties of the configuration like the maximization of the specific heat, instead of associating directly the recognition procedure with the RG flow and its fixed points.Then from the reconstructed data we deduce the critical exponent associated with the magnetization to find satisfactory agreement with the actual physical value.We assume no prior knowledge about the criticality of the system and its Hamiltonian.
The application of unsupervised machine learning to the study of physical problems is an emerging field aimed at leveraging their outstanding capabilities to distill essential information from complex data sets. Here we use Generative Adversarial Networks (GANs), deep learning algorithms mainly used to synthesize realistic images, for the analysis of a physical problem: the propagation of interacting quantum particles on disordered chains, characterized by the emergence of complex quantum correlations. We find that GANs can learn these correlations and identify the physical control parameters in a completely unsupervised manner. After training on a data set of unlabeled examples, the algorithm can generate, without further calculations, a much larger number of unseen yet physically correct new instances. Most notably, the knowledge distilled in the algorithm's latent space identifies the relevant control parameter, the level of disorder in our case, even though this was not highlighted in the training set. This allows tuning the disorder level in the generated samples to values the algorithm was not explicitly trained on, thereby uncovering a fundamental physical phenomenon---the crossover from nearly free propagation to localization. These results demonstrate the ability of deep neural networks to learn and generalize physical rules by scanning examples and suggest their use in physical analysis and discovery.