Sampling Triangulations and Calabi-Yau Threefolds with Autoregressive GNNs

Code

• github.com/natemacfadden/dualgnn
  OfficialIn paper★ 0

Abstract

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,Z) Z^d), and our masking procedure further guarantees that every rollout produces a fine triangulation (in 2D). On unseen polygons with N_pts 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 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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