Sampling Triangulations and Calabi-Yau Threefolds with Autoregressive GNNs

We introduce `dualGNN', an autoregressive message-passing GNN for sampling fine, regular triangulations (FRTs) of convex polytopes. dualGNN operates on a generalization of the dual graph of a triangulation, with edges labeled by `signed circuits' -- combinatorial invariants from oriented matroid theory which we show are both necessary and sufficient for exposing regularity. The model is independent of the number of points in the polytope and invariant under the polytope's orientation-preserving symmetries ($\mathrm{SL}(d,\mathbb{Z}) \ltimes \mathbb{Z}^d$). When implemented with a certain masking procedure, one can also guarantee that every rollout produces a fine triangulation (in $2$D). On unseen polygons with $N_\mathrm{pts} \leq 40$, dualGNN is the most uniform FRT sampler we tested, and even a model trained on a single polygon generalizes well to other polygons. The model is small ($\sim92$k parameters), trains in $\sim7.5$ hours on a single consumer GPU, and runs without modification on an M1 MacBook Pro. We apply dualGNN to string theory, uniformly sampling Calabi-Yau threefolds at $h^{1,1}=86$ and consistent with uniformity at $h^{1,1}=128$. This is an order of magnitude beyond previous learned methods with a model $\sim1000\times$ smaller. Code, training scripts, and pretrained models are available at https://github.com/natemacfadden/dualGNN .