Machine learning in lattice quantum gravity

TL;DR

This paper investigates the use of seven supervised and seven unsupervised machine-learning models to identify phase transitions in four-dimensional Causal Dynamical Triangulations (CDT) from Monte Carlo data. By mapping 30 purely geometric features of CDT configurations to phase labels, the authors show that supervised models can accurately classify phases and locate transition points with precision often surpassing traditional order-parameter methods, while unsupervised approaches lag unless carefully tuned or restricted to two clusters. The findings demonstrate that automated ML can robustly detect CDT phase boundaries and even suggest sharper signals, offering a promising tool for exploring phase structure in lattice quantum gravity and guiding future studies on larger feature sets and alternative topologies.

Abstract

Using numerical data coming from Monte Carlo simulations of four-dimensional Causal Dynamical Triangulations, we study how automated machine learning algorithms can be used to recognize transitions between different phases of quantum geometries observed in lattice quantum gravity. We tested seven supervised and seven unsupervised machine learning models and found that most of them were very successful in that task, even outperforming standard methods based on order parameters.

Figures (4)

• Figure 1: The phase diagram of the toroidal CDT. Solid lines denote measured phase transition lines, where first-order transitions are shown in blue while higher-order transitions in red. Dashed lines are extrapolations.
• Figure 2: Machine learning analysis of the $A-B$ transition for fixed $\kappa_0=4.8$, the $A - C$ transition for fixed $\Delta=0.6$, and the $B - C_b$ transition for fixed $\kappa_0=2.2$. All data were measured for $\bar{N}_{41}=100\mathrm{k}$. Red points indicate the mean probabilities that Monte Carlo–generated data at a given parameter value belong to one of the phases. The probabilities were computed with the Logistic Regression model, trained on subsets of data from the lowest and highest parameter values (empty dots). The solid black line denotes the standard CDT order parameter $OP_1=N_0/N_{41}$ ($A-B$ transition) or $OP_2=N_{32}/N_{41}$ ($A-C$ and $B-C_b$ transitions), whereas the dashed lines denote the susceptibilities $\chi$ of the probabilities and the order parameters; $\langle OP_1 \rangle$, $\langle OP_2 \rangle$ and $\chi$ were rescaled and shifted to fit in range $(0,1)$. The phase transition region is shaded.
• Figure 3: Summary of the results obtained by different ML models in the study of individual phase transitions. Legend: dark green – the model correctly identifies phase transitions without the need for "manual" hyperparameter optimization; light green – the model correctly identifies phase transitions but requires "manual" hyperparameter optimization; yellow – the model identifies phase transitions, but produces results different from those of other models and standard methods; red – the model fails to work correctly.
• Figure 4: Machine learning analysis of the $A - B$ transition for fixed $\kappa_0=4.8$, the $A - C$ transition for fixed $\Delta=0.6$, and the $B - C_b$ transition for fixed $\kappa_0=2.2$. All data were measured for $\bar{N}_{41}=100\mathrm{k}$. Tested ML models are denoted by different colors. Phase transition region is shaded. In the case of the $A - B$ transition, results of models other than Decision Tree and Random Forest are optically indistinguishable.