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arXiv:2210.06203 · Entropy

Thermodynamics of the Ising Model Encoded in Restricted Boltzmann Machines

Jing Gu, Kai Zhang

Published Nov 22, 2022• 5 citations
arXivHTMLPDF

Abstract

The restricted Boltzmann machine (RBM) is a two-layer energy-based model that uses its hidden-visible connections to learn the underlying distribution of visible units, whose interactions are often complicated by high-order correlations. Previous studies on the Ising model of small system sizes have shown that RBMs are able to accurately learn the Boltzmann distribution and reconstruct thermal quantities at temperatures away from the critical point Tc. How the RBM encodes the Boltzmann distribution and captures the phase transition are, however, not well explained. In this work, we perform RBM learning of the 2d and 3d Ising model and carefully examine how the RBM extracts useful probabilistic and physical information from Ising configurations. We find several indicators derived from the weight matrix that could characterize the Ising phase transition. We verify that the hidden encoding of a visible state tends to have an equal number of positive and negative units, whose sequence is randomly assigned during training and can be inferred by analyzing the weight matrix. We also explore the physical meaning of the visible energy and loss function (pseudo-likelihood) of the RBM and show that they could be harnessed to predict the critical point or estimate physical quantities such as entropy.

DOI
10.3390/e24121701

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The restricted Boltzmann machine (RBM) is a two-layer energy-based model that uses its hidden-visible connections to learn the underlying distribution of visible units, whose interactions are often complicated by high-order correlations. Previous studies on the Ising model of small system sizes have shown that RBMs are able to accurately learn the Boltzmann distribution and reconstruct thermal quantities at temperatures away from the critical point Tc. How the RBM encodes the Boltzmann distribution and captures the phase transition are, however, not well explained. In this work, we perform RBM learning of the 2d and 3d Ising model and carefully examine how the RBM extracts useful probabilistic and physical information from Ising configurations. We find several indicators derived from the weight matrix that could characterize the Ising phase transition. We verify that the hidden encoding of a visible state tends to have an equal number of positive and negative units, whose sequence is randomly assigned during training and can be inferred by analyzing the weight matrix. We also explore the physical meaning of the visible energy and loss function (pseudo-likelihood) of the RBM and show that they could be harnessed to predict the critical point or estimate physical quantities such as entropy.
Abstract A restricted Boltzmann machine (RBM) is a two-layer neural-network machine-learning model comprising a visible layer that represents observed data and a hidden layer that captures correlations among the visible units. Rooted in ideas from statistical mechanics, RBMs provide a probabilistic framework for modeling complex, high-dimensional distributions through an energy function and a corresponding Boltzmann distribution. This review offers an accessible introduction to RBMs from a physics-oriented perspective, focusing on their core concepts and training principles. We survey key theoretical developments that interpret RBMs as many-body systems, highlighting their equilibrium/non-equilibrium thermodynamics, learning dynamics, mean-field properties, phase diagrams, and free energy calculation. We further discuss recent applications of RBMs to a range of physics problems, including quantum-mechanical wavefunction representation, statistical-mechanical Boltzmann distribution modeling, and biological protein sequence analysis.
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We study the features extracted by the Restricted Boltzmann Machine (RBM) when it is trained with spin configurations of Ising model at various temperatures. Using the trained RBM, we obtain the flow of iterative reconstructions (RBM flow) of the spin configurations and find that in some cases the flow approaches the phase transition point T=TcT=T_cT=Tc​ in Ising model. Since the extracted features are emphasized in the reconstructed configurations, the configurations at such a fixed point should describe nothing but the extracted features. Then we investigate the dependence of the fixed point on various parameters and conjecture the condition where the fixed point of the RBM flow is at the phase transition point. We also provide supporting evidence for the conjecture by analyzing the weight matrix of the trained RBM.
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.
We train a set of restricted Boltzmann machines (RBMs) on one- and two-dimensional Ising spin configurations at various values of temperature, generated using Monte Carlo simulations. We validate the training procedure by monitoring several estimators, including measurements of the log likelihood, with the corresponding partition functions estimated using annealed importance sampling. The effects of various choices of hyperparameters on training the RBM are discussed in detail, with a generic prescription provided. Finally, we present a closed-form expression for extracting the values of couplings, for every nnn-point interaction between the visible nodes of an RBM, in a binary system such as the Ising model. We aim at using this study as the foundation for further investigations of less well-known systems.
Detecting the topological Kosterlitz-Thouless (KT) transition in the prototypical two-dimensional (2D) XY model using unsupervised machine learning methods has long been a challenging problem due to the lack of suitable order parameters. To address this issue, we begin with a conventional real-valued restricted Boltzmann machine (RBM) (RBM-xy), which uses exponential conditional probabilities to generate visible units. We then develop a novel real-valued RBM (RBM-CosSin) featuring nonlinear cos/sincos/sincos/sin activation, whose visible units follow the von Mises distribution. Our findings reveal that RBM-CosSin effectively learns the underlying Boltzmann distribution of 2D XY systems and generates authentic XY configurations that accurately capture both thermodynamics and topological order (vortex). Furthermore, we demonstrate that it is possible to extract phase transition information, including the KT transition, from the weight matrices without relying on prior physics knowledge.