用图神经网络生成规则格点剖分，可高效采样卡拉比-丘三流形。
Sampling Triangulations and Calabi-Yau Threefolds with Autoregressive GNNs
- 基于带符号电路的图结构，实现自回归生成规则剖分。
- 在40个点以内多边形上采样均匀性优于所有对比方法。
- 模型小、训练快，适用于弦论中高维卡拉比-丘流形采样。
我们提出 dualGNN，一种用于采样格点多面体精细规则剖分的自回归消息传递图神经网络。dualGNN 在剖分对偶图的推广形式上运行，边带有‘带符号电路’标签——源自定向拟阵理论的组合不变量。我们证明这些电路在保留一定幅度信息的前提下，足以确定剖分的正则性。该模型不依赖多面体的点数，且在保持定向对称性（SL(d,Z) ⋉ Z^d）下不变；掩码机制确保每次生成均为精细剖分（二维情形）。在未见多边形上（点数 N_pts ≤ 40），dualGNN 是唯一在所有诊断测试中（KL散度、碰撞次数、样本自相关）与均匀采样一致的采样器。模型规模小（约92k参数），单张消费级显卡训练约7.5小时。我们将 dualGNN 应用于弦论，实现了 h^{1,1}=86 时卡拉比-丘三流形的均匀采样；在 h^{1,1}=128 时也未观察到偏差，但诊断能力较弱。代码、训练脚本和预训练模型可在 https://github.com/natemacfadden/dualGNN（pip install dualgnn）获取，并已集成至 CYTools。
原文摘要 · Abstract (English)
We introduce `dualGNN', an autoregressive message-passing GNN for sampling fine, regular triangulations of lattice polytopes. dualGNN operates on a generalization of the dual graph of a triangulation, with edges labeled by `signed circuits' -- combinatorial invariants from the theory of oriented matroids. We show that these circuits are necessary and sufficient to determine a triangulation's regularity from the graph, provided certain magnitude information is retained. The model is independent of the polytope's point count and invariant under its orientation-preserving symmetries ($\mathrm{SL}(d,\mathbb{Z}) \ltimes \mathbb{Z}^d$), and our masking procedure further guarantees that every rollout produces a fine triangulation (in 2D). On unseen polygons with $N_\mathrm{pts} \leq 40$, dualGNN is the only sampler we tested that is consistent with uniform sampling across all our diagnostics (KL divergence from uniformity, collision counts, and sample autocorrelation). The model is small ($\sim92$k parameters) and trains in $\sim7.5$ hours on a single consumer GPU. We apply dualGNN to string theory, sampling Calabi-Yau threefolds uniformly at $h^{1,1}=86$; we also sample CYs at $h^{1,1}=128$, observing no deviations from uniformity, but our diagnostics are weaker here. Code, training scripts, and pretrained models are available at https://github.com/natemacfadden/dualGNN (pip install dualgnn), and dualGNN is integrated into CYTools.
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