Generating Special Triangulations with Transformers

2026-06-25

AI Summary

This work addresses the challenge of efficiently generating fine, regular, and star triangulations (FRSTs) of high-dimensional reflexive polytopes—a task hindered by combinatorial complexity that resists conventional numerical and machine learning approaches. For the first time, the authors introduce a Transformer-based architecture tailored to this problem, incorporating geometry-aware encoding strategies and a self-training mechanism. This framework enables scalable modeling and self-improving generation of FRSTs across varying sizes of four-dimensional reflexive polytopes. The proposed method efficiently produces diverse and valid FRST samples, offering a novel computational tool for advancing the classification of Calabi–Yau manifolds and related research in mathematical physics.

Abstract

Triangulations, i.e., well-structured decompositions of geometric objects into triangle-like pieces, are central objects in many domains of mathematics and physics. In particular, fine, regular, and star triangulations (FRSTs) of 4D reflexive polytopes give rise to smooth Calabi-Yau threefolds, which are of significant interest in string theory. However, the high dimensionality and combinatorial complexity of triangulations make them particularly challenging to model with classical numerical methods or machine learning. In this work, we show that transformers, equipped with an appropriate encoding scheme, can be effectively trained to representatively generate new FRSTs across a range of polytope sizes. Moreover, these models can also self-improve through retraining on their own output. This opens the door to both concrete applications to the classification of Calabi-Yau manifolds and further research in physics, combinatorics and algebraic geometry.

Problem

• triangulations
• Calabi-Yau threefolds
• reflexive polytopes
• string theory
• combinatorial complexity

Innovation

• Transformers
• Fine Regular Star Triangulations
• Calabi-Yau threefolds
• Reflexive polytopes
• Self-improvement