Boosting Monte Carlo simulations of spin glasses using autoregressive neural networks

Learning nonequilibrium statistical mechanics and dynamical phase transitions

Nonequilibrium statistical mechanics exhibit a variety of complex phenomena far from equilibrium. It inherits challenges of equilibrium, including accurately describing the joint distribution of a large number of configurations, and also poses new challenges as the distribution evolves over time. Characterizing dynamical phase transitions as an emergent behavior further requires tracking nonequilibrium systems under a control parameter. While a number of methods have been proposed, such as tensor networks for one-dimensional lattices, we lack a method for arbitrary time beyond the steady state and for higher dimensions. Here, we develop a general computational framework to study the time evolution of nonequilibrium systems in statistical mechanics by leveraging variational autoregressive networks, which offer an efficient computation on the dynamical partition function, a central quantity for discovering the phase transition. We apply the approach to prototype models of nonequilibrium statistical mechanics, including the kinetically constrained models of structural glasses up to three dimensions. The approach uncovers the active-inactive phase transition of spin flips, the dynamical phase diagram, as well as new scaling relations. The result highlights the potential of machine learning dynamical phase transitions in nonequilibrium systems.

Cluster percolation in the two-dimensional Ising spin glass

Suitable cluster definitions have allowed researchers to describe many ordering transitions in spin systems as geometric phenomena related to percolation. For spin glasses and some other systems with quenched disorder, however, such a connection has not been fully established, and the numerical evidence remains incomplete. Here we use Monte Carlo simulations to study the percolation properties of several classes of clusters occurring in the Edwards-Anderson Ising spin-glass model in two dimensions. The Fortuin-Kasteleyn-Coniglio-Klein clusters originally defined for the ferromagnetic problem do percolate at a temperature that remains nonzero in the thermodynamic limit. On the Nishimori line, this location is accurately predicted by an argument due to Yamaguchi. More relevant for the spin-glass transition are clusters defined on the basis of the overlap of several replicas. We show that various such cluster types have percolation thresholds that shift to lower temperatures by increasing the system size, in agreement with the zero-temperature spin-glass transition in two dimensions. The overlap is linked to the difference in density of the two largest clusters, thus supporting a picture where the spin-glass transition corresponds to an emergent density difference of the two largest clusters inside the percolating phase.

Analysis of autocorrelation times in neural Markov chain Monte Carlo simulations

We provide a deepened study of autocorrelations in neural Markov chain Monte Carlo (NMCMC) simulations, a version of the traditional Metropolis algorithm which employs neural networks to provide independent proposals. We illustrate our ideas using the two-dimensional Ising model. We discuss several estimates of autocorrelation times in the context of NMCMC, some inspired by analytical results derived for the Metropolized independent sampler (MIS). We check their reliability by estimating them on a small system where analytical results can also be obtained. Based on the analytical results for MIS, we propose a loss function and study its impact on the autocorrelation times. Although, this function's performance is a bit inferior to the traditional Kullback-Leibler divergence, it offers two training algorithms which in some situations may be beneficial. By studying a small 4×4 system, we gain access to the dynamics of the training process, which we visualize using several observables. Furthermore, we quantitatively investigate the impact of imposing global discrete symmetries of the system in the neural network training process on the autocorrelation times. Eventually, we propose a scheme which incorporates partial heat-bath updates, which considerably improves the quality of the training. The impact of the above enhancements is discussed for a 16×16 spin system. The summary of our findings may serve as guidance to the implementation of NMCMC simulations for more complicated models.

A Bayesian generative neural network framework for epidemic inference problems

The reconstruction of missing information in epidemic spreading on contact networks can be essential in the prevention and containment strategies. The identification and warning of infectious but asymptomatic individuals (i.e., contact tracing), the well-known patient-zero problem, or the inference of the infectivity values in structured populations are examples of significant epidemic inference problems. As the number of possible epidemic cascades grows exponentially with the number of individuals involved and only an almost negligible subset of them is compatible with the observations (e.g., medical tests), epidemic inference in contact networks poses incredible computational challenges. We present a new generative neural networks framework that learns to generate the most probable infection cascades compatible with observations. The proposed method achieves better (in some cases, significantly better) or comparable results with existing methods in all problems considered both in synthetic and real contact networks. Given its generality, clear Bayesian and variational nature, the presented framework paves the way to solve fundamental inference epidemic problems with high precision in small and medium-sized real case scenarios such as the spread of infections in workplaces and hospitals.

Adaptive Monte Carlo augmented with normalizing flows

SignificanceMonte Carlo methods, tools for sampling data from probability distributions, are widely used in the physical sciences, applied mathematics, and Bayesian statistics. Nevertheless, there are many situations in which it is computationally prohibitive to use Monte Carlo due to slow "mixing" between modes of a distribution unless hand-tuned algorithms are used to accelerate the scheme. Machine learning techniques based on generative models offer a compelling alternative to the challenge of designing efficient schemes for a specific system. Here, we formalize Monte Carlo augmented with normalizing flows and show that, with limited prior data and a physically inspired algorithm, we can substantially accelerate sampling with generative models.

Power-grid stability predictions using transferable machine learning

Complex network analyses have provided clues to improve power-grid stability with the help of numerical models. The high computational cost of numerical simulations, however, has inhibited the approach, especially when it deals with the dynamic properties of power grids such as frequency synchronization. In this study, we investigate machine learning techniques to estimate the stability of power-grid synchronization. We test three different machine learning algorithms-random forest, support vector machine, and artificial neural network-training them with two different types of synthetic power grids consisting of homogeneous and heterogeneous input-power distribution, respectively. We find that the three machine learning models better predict the synchronization stability of power-grid nodes when they are trained with the heterogeneous input-power distribution rather than the homogeneous one. With the real-world power grids of Great Britain, Spain, France, and Germany, we also demonstrate that the machine learning algorithms trained on synthetic power grids are transferable to the stability prediction of the real-world power grids, which implies the prospective applicability of machine learning techniques on power-grid studies.

Quantum-Inspired Magnetic Hamiltonian Monte Carlo

Hamiltonian Monte Carlo (HMC) is a Markov Chain Monte Carlo algorithm that is able to generate distant proposals via the use of Hamiltonian dynamics, which are able to incorporate first-order gradient information about the target posterior. This has driven its rise in popularity in the machine learning community in recent times. It has been shown that making use of the energy-time uncertainty relation from quantum mechanics, one can devise an extension to HMC by allowing the mass matrix to be random with a probability distribution instead of a fixed mass. Furthermore, Magnetic Hamiltonian Monte Carlo (MHMC) has been recently proposed as an extension to HMC and adds a magnetic field to HMC which results in non-canonical dynamics associated with the movement of a particle under a magnetic field. In this work, we utilise the non-canonical dynamics of MHMC while allowing the mass matrix to be random to create the Quantum-Inspired Magnetic Hamiltonian Monte Carlo (QIMHMC) algorithm, which is shown to converge to the correct steady state distribution. Empirical results on a broad class of target posterior distributions show that the proposed method produces better sampling performance than HMC, MHMC and HMC with a random mass matrix.

Generative neural samplers for the quantum Heisenberg chain

Generative neural samplers offer a complementary approach to Monte Carlo methods for problems in statistical physics and quantum field theory. This paper tests the ability of generative neural samplers to estimate observables for real-world low-dimensional spin systems. It maps out how autoregressive models can sample configurations of a quantum Heisenberg chain via a classical approximation based on the Suzuki-Trotter transformation. We present results for energy, specific heat, and susceptibility for the isotropic XXX and the anisotropic XY chain are in good agreement with Monte Carlo results within the same approximation scheme.

Scalable neural networks for the efficient learning of disordered quantum systems

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.