用Transformer生成高维几何的特殊三角剖分，助力弦理论研究。
Generating Special Triangulations with Transformers
- 设计适配编码方案的Transformer，生成4维反射多面体的精细三角剖分
- 模型可自生成数据并重训练，提升生成质量与多样性
- 适合弦理论、代数几何与组合数学方向的研究者
三角剖分，即几何对象被分解为类似三角形的结构，在数学和物理多个领域中至关重要。特别是4维反射多面体的精细、正则、星形三角剖分（FRSTs）能构造出光滑的卡拉比-丘三维流形，对弦理论具有重要意义。然而，三角剖分的高维性与组合复杂性使其难以通过传统数值方法或机器学习建模。本文表明，通过合适的编码方案，Transformer可有效训练以代表性生成不同规模多面体的FRSTs。此外，模型可通过自身输出数据重新训练实现自我优化。这为卡拉比-丘流形的分类及物理、组合学和代数几何研究开辟了新路径。
原文摘要 · Abstract (English)
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.
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