Sampling Triangulations and Calabi-Yau Threefolds with Autoregressive GNNs

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 (SL(d,mathbbZ) ltimes mathbbZd), and our masking procedure further guarantees that every rollout produces a fine triangulation (in 2D). On unseen polygons with Npts ≤ 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 (~92k parameters) and trains in ~7.5 hours on a single consumer GPU. We apply dualGNN to string theory, sampling Calabi-Yau threefolds uniformly at h1,1=86; we also sample CYs at h1,1=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.