使用自回归图神经网络采样三角剖分和卡拉比-丘三维流形
Sampling Triangulations and Calabi-Yau Threefolds with Autoregressive GNNs
AI总结：
提出dualGNN，一种自回归消息传递图神经网络，用于采样凸多面体的精细正则三角剖分，并应用于弦论中均匀采样卡拉比-丘三维流形，模型参数少、训练快、泛化能力强。
摘要：
我们引入了`dualGNN`，一种自回归消息传递图神经网络，用于采样凸多面体的精细正则三角剖分（FRT）。dualGNN 在三角剖分对偶图的推广上操作，边由`有向电路`标记——来自定向拟阵理论的组合不变量，我们证明这些对于揭示正则性既是必要的也是充分的。该模型独立于多面体中的点数，并在多面体的保向对称群（$\mathrm{SL}(d,\mathbb{Z}) \ltimes \mathbb{Z}^d$）下不变。当使用某种掩码程序实现时，还可以保证每次 rollout 都产生一个精细三角剖分（在二维中）。在 $N_\mathrm{pts} \leq 40$ 的未见多边形上，dualGNN 是我们测试过的最均匀的 FRT 采样器，甚至仅在单个多边形上训练的模型也能很好地泛化到其他多边形。该模型很小（约 92k 参数），在单个消费级 GPU 上训练约 7.5 小时，并且无需修改即可在 M1 MacBook Pro 上运行。我们将 dualGNN 应用于弦论，在 $h^{1,1}=86$ 处均匀采样卡拉比-丘三维流形，并在 $h^{1,1}=128$ 处与均匀性一致。这比之前的学习方法提升了一个数量级，而模型小了约 1000 倍。代码、训练脚本和预训练模型可在 https://github.com/natemacfadden/dualGNN 获取。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
