用强化学习优化多面体三角剖分，高效搜索超大空间
TriSearch: Learning to Optimize Triangulations via Bistellar Flips
- 用环路支持的局部剖分表示可行翻转，实现无维度依赖的策略学习
- 3D上达到最优度量性能，4D中发现更多不同类型复形
- 零样本泛化到更大多面体，适合高维几何与代数几何研究者
我们提出TriSearch，一种基于强化学习的框架，通过双星翻转优化多面体的三角剖分目标。核心思想是采用环路支持的子剖分动作表示：可行翻转由其支撑环路和局部子剖分定义，使学习策略能基于局部几何与组合特征对翻转进行排序。该方法提供无维度依赖的接口，无需显式枚举整个剖分空间即可高效遍历翻转图。在3D和4D中实例化后，模型实现零样本泛化，从小型训练实例推广至具有指数级更大搜索空间的大型多面体。在3D中取得最佳度量性能；在4D中，在固定预算下发现的反射多面体的细、正规、星型三角剖分数目超过现有采样器。
原文摘要 · Abstract (English)
We introduce TriSearch, a reinforcement learning framework for optimizing objectives over triangulations of a polytope via bistellar flips. The key idea is a circuit-supported subtriangulation action representation: feasible flips are encoded by their supporting circuit and realized local subtriangulation, enabling a learned policy to rank them using local geometric and combinatorial features. This yields a dimension-agnostic interface and enables efficient traversal of the flip graph without explicit enumeration of the full triangulation space. Instantiated in 3D and 4D, TriSearch generalizes zero-shot from small training instances to larger polytopes with exponentially larger search spaces. It achieves top performance on metric objectives in 3D and, in 4D, discovers more distinct Fine, Regular, Star triangulations of reflexive polytopes, corresponding to Calabi-Yau threefolds, than existing samplers under a fixed budget.
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