Distinguishing Calabi-Yau Topology using Machine Learning

TL;DR

This work investigates distinguishing Calabi-Yau topology by predicting refined topological invariants from CICY data using a Google Inception-based network. It focuses on four divisibility invariants $d_1$, $d_2$, $d_3$, and $d_p$ derived from triple intersection numbers to identify topology, outperforming traditional regressors with around 90% cross-validated accuracy. The study demonstrates high-accuracy predictions across outputs, indicating that deep learning can capture intricate algebraic-geometric structures beyond Hodge-number predictions. It also outlines future directions, including extracting conjectural formulas and extending to additional invariants such as Gromov–Witten numbers, signaling a broader potential for AI-assisted mathematics in geometry.

Abstract

While the earliest applications of AI methodologies to pure mathematics and theoretical physics began with the study of Hodge numbers of Calabi-Yau manifolds, the topology type of such manifold also crucially depend on their intersection theory. Continuing the paradigm of machine learning algebraic geometry, we here investigate the triple intersection numbers, focusing on certain divisibility invariants constructed therefrom, using the Inception convolutional neural network. We find $\sim90\%$ accuracies in prediction in a standard fivefold cross-validation, signifying that more sophisticated tasks of identification of manifold topologies can also be performed by machine learning.

Figures (5)

• Figure 1: Model Structure
• Figure 2: This depicts comparisons between actual values and model's predictions across CICY triple intersection numbers $d_1$, $d_2$, $d_3$ and $d_p$. We use 80% of the data as the training set and 20% of the data as the testing set.
• Figure 3: This residual scatter plots comparing true and predicted CICY triple intersection numbers $d_1$, $d_2$, $d_3$, and $d_p$, based on an 80%/20% train–test split.
• Figure 4: The accuracy and MSE of different regressors across CICY triple intersection numbers $d_1$, $d_2$, $d_3$ and $d_p$, including SVM regressor, Random Forest regressor, XGBoost regressor and our Inception Network model: 80% of the data are used as the training set and 20% of the data as the testing set.
• Figure 5: This presents a comparative analysis between our Inception Network model and conventional machine learning regressors across various fractions of training data. The first figure presents a line graph measuring the accuracy of different regressors in different fractions of training data while the second figure presents MSE.

Theorems & Definitions (1)

• Theorem 2.1: C. T. C. Wall wall1966classification