Unsupervised learning of phase transitions: From principal component analysis to variational autoencoders

Abstract

We employ unsupervised machine learning techniques to learn latent parameters which best describe states of the two-dimensional Ising model and the three-dimensional XY model. These methods range from principal component analysis to artificial neural network based variational autoencoders. The states are sampled using a Monte-Carlo simulation above and below the critical temperature. We find that the predicted latent parameters correspond to the known order parameters. The latent representation of the states of the models in question are clustered, which makes it possible to identify phases without prior knowledge of their existence or the underlying Hamiltonian. Furthermore, we find that the reconstruction loss function can be used as a universal identifier for phase transitions.

Latent Space Cartography: Visual Analysis of Vector Space Embeddings

“…The research literature involving latent spaces often employs visualizations for qualitative evaluation. Most of these visualizations are 2D scatter plots, using two intrinsic latent dimensions directly [Wet17, MNG17, YHSBK17], axes from dimensionality reduction techniques such as t‐SNE ( e.g ., [TWR∗17, JYY∗16, HSSQ17, MKS∗15, YSD∗18]) or PCA ( e.g ., [SRM∗16, MSSW16,UFDR16,FSBL17,GBWD∗18]), or axes of custom projections [BCZ∗16a, BCZ∗16b]. Alternatively, some papers show a 2D grid of reconstructed examples [DTD∗18, JBJ18, ZSE17, KPHL17].…”

“…The research literature involving latent spaces often employs visualizations for qualitative evaluation. Most of these visualizations are 2D scatter plots, using two intrinsic latent dimensions directly [Wet17,MNG17,YHSBK17], axes from dimen- Researchers have also developed interactive visual analysis tools for latent spaces [STN * 16, JSL * 17, HG18, LBT * 18, LNH * 18]. Some tools focus on a subset of tasks [STN * 16, LBT * 18] in word embeddings, which we extend and bring to a broader range of latent spaces.…”

Visualizing a neural network that develops quantum perturbation theory

“…The former is important in atomic and molecular physics, as well as quantum chemistry. The latter is related to machine learning phases of matter, which has already become a topic of intensive interest [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20]. Our understanding of how neural network works in this simple problem can shed light on understanding more sophisticated problems better.…”

Learning phase transitions from dynamics

“…Machine learning is emerging as a novel tool for identifying phases of matter [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15]. At its core, this problem can be cast as a classification problem in which data obtained from physical systems are assigned a class (i.e., a phase) using machine learning methods.…”

“…At its core, this problem can be cast as a classification problem in which data obtained from physical systems are assigned a class (i.e., a phase) using machine learning methods. This approach has enabled the autonomous detection of order parameters [2,5,6], phase transitions [1,3], and entire phase diagrams [4,7,16,17]. Simultaneous research effort at the interface between machine learning and many-body physics has focused on the use of neural networks for efficient representations of quantum wave functions [18][19][20][21][22][23][24][25][26], drawing a parallel between deep networks and the renormalization group [27][28][29].…”