Generating Special Triangulations with Transformers
For string theorists and algebraic geometers, this provides a machine learning approach to generate triangulations that are otherwise computationally prohibitive, enabling progress in Calabi-Yau classification.
Transformers with a tailored encoding scheme can generate fine, regular, and star triangulations (FRSTs) of 4D reflexive polytopes, which are crucial for constructing smooth Calabi-Yau threefolds in string theory. The model can also self-improve via retraining on its own output.
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.