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arXiv:1810.08179 · Physical Review Research

Thermodynamics and feature extraction by machine learning

Shotaro Shiba, Dimitrios Giataganas

Published Sep 15, 2020• 47 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevresearch.2.033415

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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 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.
Deep Neural Network (DNN) has been very successful.Especially, unsupervised DNN can extract the features of complicated dataset without any prior knowledge.However, theoretical understanding of how it extracts the features and what it extracts as the features is not clear.We investigate these questions from the theoretical physics point of view by using the Restricted Boltzmann Machine (RBM) and the Autoencoder (AE) trained by spin configurations of the Ising model.In order to explore the features RBM and AE extract, we train them by spin configurations of the 2-dimensional Ising model and construct a flow of its parameters (in particular temperature) generated by the trained RBM and AE.This flow is motivated by the renormalization group flow of statistical model.It is thought that this flow emphasizes the "relevant" features the unsupervised networks learn, and eliminate "irrelevant" information from the dataset.We show that the RBM trained by spin configurations at various temperatures generates a flow on which the temperature approaches the critical temperature T c ≃ 2.27 if the number of visible units is less than hidden units.Unlike the RBM case, the temperature of the fixed points of AE trained sufficiently is lower than the critical temperature if AE is sufficiently trained (5000 epochs).On the other hand, if the learning epochs are 1000, the fixed point temperature monotonically increase with the number of hidden units.We also analyze the weight matrices by singular value decomposition in order to read out the information RBM and AE learn.4 Autoencoder 4.1 Setup . . . . . .
A Boltzmann machine is a stochastic neural network that has been extensively used in the layers of deep architectures for modern machine learning applications. In this paper, we develop a Boltzmann machine that is capable of modeling thermodynamic observables for physical systems in thermal equilibrium. Through unsupervised learning, we train the Boltzmann machine on data sets constructed with spin configurations importance sampled from the partition function of an Ising Hamiltonian at different temperatures using Monte Carlo (MC) methods. The trained Boltzmann machine is then used to generate spin states, for which we compare thermodynamic observables to those computed by direct MC sampling. We demonstrate that the Boltzmann machine can faithfully reproduce the observables of the physical system. Further, we observe that the number of neurons required to obtain accurate results increases as the system is brought close to criticality.
We explore alternative experimental setups for the iterative sampling (flow) from Restricted Boltzmann Machines (RBM) mapped on the temperature space of square lattice Ising models by a neural network thermometer. This framework has been introduced to explore connections between RBM-based deep neural networks and the Renormalization Group (RG). It has been found that, under certain conditions, the flow of an RBM trained with Ising spin configurations approaches in the temperature space a value around the critical one: $ k_B T_c / J \approx 2.269$. In this paper we consider datasets with no information about model topology to argue that a neural network thermometer is not an accurate way to detect whether the RBM has learned scale invariance or not.
Theoretical understanding of how a deep neural network (DNN) extracts features from input images is still unclear, but it is widely believed that the extraction is performed hierarchically through a process of coarse graining. It reminds us of the basic renormalization group (RG) concept in statistical physics. In order to explore possible relations between DNN and RG, we use the restricted Boltzmann machine (RBM) applied to an Ising model and construct a flow of model parameters (in particular, temperature) generated by the RBM. We show that the unsupervised RBM trained by spin configurations at various temperatures from T=0 to T=6 generates a flow along which the temperature approaches the critical value T_{c}=2.27. This behavior is the opposite of the typical RG flow of the Ising model. By analyzing various properties of the weight matrices of the trained RBM, we discuss why it flows towards T_{c} and how the RBM learns to extract features of spin configurations.
Conventionally, the study of phases in statistical mechan- ics is performed with the help of random sampling tools. Among the most powerful are Monte Carlo simulations consisting of a stochastic importance sampling over state space and evaluation of estimators for physical quantities. The ability of modern machine learning techniques to classify, identify, or in- terpret massive data sets provides a complementary paradigm to the above approach to analyze the exponentially large number of states in statistical physics. In this report, it is demonstrated by application on Ising-type models that deep learning has potential wide applications in solving many-body statis- tical physics problems. In application of supervised learning, we showed that the feed-forward neural network can identify phases and phase transitions in the ferromagnetic Ising model and the convolutional neural network (CNN) is extremely powerful in classifying T = 0 and T = ∞ phases in the Ising gauge model; In application of unsupervised learning, we illustrated that a deep auto-encoder constructed by stacked restricted Boltzmann machines (RBM)\nis closely related to the renormalization group (RG) method well understood in modern physics and our reconstruction of Ising spin configurations in the ferromagnetic Ising model is similar to the hand-written digits reconstruction.