We apply the hierarchical autoregressive neural network sampling algorithm to the two-dimensional Q-state Potts model and perform simulations around the phase transition at Q=12. We quantify the performance of the approach in the vicinity of the first-order phase transition and compare it with that of the Wolff cluster algorithm. We find a significant improvement as far as the statistical uncertainty is concerned at a similar numerical effort. In order to efficiently train large neural networks we introduce the technique of pretraining. It allows us to train some neural networks using smaller system sizes and then employ them as starting configurations for larger system sizes. This is possible due to the recursive construction of our hierarchical approach. Our results serve as a demonstration of the performance of the hierarchical approach for systems exhibiting bimodal distributions. Additionally, we provide estimates of the free energy and entropy in the vicinity of the phase transition with statistical uncertainties of the order of 10^{-7} for the former and 10^{-3} for the latter based on a statistics of 10^{6} configurations.
We apply the hierarchical autoregressive neural network sampling algorithm to the two-dimensional Q-state Potts model and perform simulations around the phase transition at Q=12. We quantify the performance of the approach in the vicinity of the first-order phase transition and compare it with that of the Wolff cluster algorithm. We find a significant improvement as far as the statistical uncertainty is concerned at a similar numerical effort. In order to efficiently train large neural networks we introduce the technique of pretraining. It allows us to train some neural networks using smaller system sizes and then employ them as starting configurations for larger system sizes. This is possible due to the recursive construction of our hierarchical approach. Our results serve as a demonstration of the performance of the hierarchical approach for systems exhibiting bimodal distributions. Additionally, we provide estimates of the free energy and entropy in the vicinity of the phase transition with statistical uncertainties of the order of 10^{-7} for the former and 10^{-3} for the latter based on a statistics of 10^{6} configurations.
It was recently proposed that neural networks could be used to approximate many-dimensional probability distributions that appear e.g. in lattice field theories or statistical mechanics. Subsequently they can be used as variational approximators to assess extensive properties of statistical systems, like free energy, and also as neural samplers used in Monte Carlo simulations. The practical application of this approach is unfortunately limited by its unfavourable scaling both of the numerical cost required for training, and the memory requirements with the system size. This is due to the fact that the original proposition involved a neural network of width which scaled with the total number of degrees of freedom, e.g. L 2 in case of a two dimensional L × L lattice. In this work we propose a hierarchical association of physical degrees of freedom, for instance spins, to neurons which replaces it with the scaling with the linear extent L of the system. We demonstrate our approach on the two-dimensional Ising model by simulating lattices of various sizes up to 128 × 128 spins, with time benchmarks reaching lattices of size 512 × 512 . We observe that our proposal improves the quality of neural network training, i.e. the approximated probability distribution is closer to the target that could be previously achieved. As a consequence, the variational free energy reaches a value closer to its theoretical expectation and, if applied in a Markov Chain Monte Carlo algorithm, the resulting autocorrelation time is smaller. Finally, the replacement of a single neural network by a hierarchy of smaller networks considerably reduces the memory requirements.
Abstract Using the techniques of neural networks (NN), we study the three-dimensional (3D) five-state ferromagnetic Potts model on the cubic lattice as well as the two-dimensional (2D) three-state antiferromagnetic Potts model on the square lattice. Unlike the conventional approach, here we follow the idea employed by Li et al (2018 Ann. Phys., NY 391 312–331). Specifically, instead of numerically generating numerous objects for the training, the whole or part of the theoretical ground state configurations of the studied models are considered as the training sets. Remarkably, our investigation of these two models provides convincing evidence for the effectiveness of the method of preparing training sets used in this study. In particular, the results of the 3D model obtained here imply that the NN approach is as efficient as the traditional method since the signal of a first order phase transition, namely tunneling between two channels, determined by the NN method is as strong as that calculated with the Monte Carlo technique. Furthermore, the outcomes associated with the considered 2D system indicate even little partial information of the ground states can lead to conclusive results regarding the studied phase transition. The achievements reached in our investigation demonstrate that the performance of NN, using certain amount of the theoretical ground state configurations as the training sets, is impressive.
A universal (supervised) neural network (NN), which is only trained once on a one-dimensional lattice of 200 sites, is employed to study the phase transition of the two-dimensional (2D) 5-state ferromagnetic Potts model on the square lattice. In particular, the NN is obtained by using merely two artificially made configurations as the training set. Due to the elegant features of the considered NN, results associated with systems consisting of over 4000000 spins can be obtained with ease, and convincing evidence showing the investigated phase transition is weakly first order is reached. The outcomes demonstrated here can hardly be achieved with the standard NNs that are commonly used in the literature.
Efficient sampling of complex high-dimensional probability distributions is a central task in computational science.Machine learning methods like autoregressive neural networks, used with Markov chain Monte Carlo sampling, provide good approximations to such distributions but suffer from either intrinsic bias or high variance.In this Letter, we propose a way to make this approximation unbiased and with low variance.Our method uses physical symmetries and variable-size cluster updates which utilize the structure of autoregressive factorization.We test our method for first-and second-order phase transitions of classical spin systems, showing its viability for critical systems and in the presence of metastable states.
Conventionally, the training of a neural network for learning phases of matter uses real physical quantities as the training set. However, it has been demonstrated in several studies that this may not be required. Here we investigate the phase transitions of the two-dimensional (2D) q-state Potts models on the square lattice using a pre-trained neural network (NN). The employed NN was trained previously using two artifically made configurations as the training set. Hence no training is conducted for the present study. Remarkably, the used NN not only calculates the critical points of the considered phase transitions precisely, but also determines the nature of these phase transition definitely without ambiguity. Our results as well as the outcomes found previously suggest that training a NN using real physical quantities as the training set is not required to obtain a working NN. These unconventional studies also imply that if the configuration space of a system can be classified into two categories, then the NN considered here is likely applicable to study the phase transition of that system. Comparison between our approach and other known NN methods for studying the 2D q-state Potts models is briefly discussed as well.
A neural network study of the phase transitions of the two-dimensional antiferromagnetic q-state Potts models on the square lattice
The critical phenomena of the two-dimensional antiferromagnetic q-state Potts model on the square lattice with q=2,3,4 are investigated using the techniques of neural networks (NN). In particular, an unconventional supervised NN which is trained using no information about the physics of the considered systems is employed. In addition, conventional unsupervised autoencoders (AECs) are used in our study as well. Remarkably, while the conventional AECs fail to uncover the critical phenomena of the systems investigated here, our unconventional supervised NN correctly identifies the critical behaviors of all three considered antiferromagnetic q-state models. The results obtained in this study suggest convincingly that the applicability of our unconventional supervised NN is broader than one anticipates. In particular, when a new system is studied with our NN, it is likely that it is not necessary to conduct any training, and one only needs to examine whether an appropriate reduced representation of the original raw configurations exists, so that the same already trained NN can be employed to explore the related phase transition efficiently.