dualGNN

dualGNN is an autoregressive message-passing GNN for sampling fine, regular triangulations (FRTs) of convex lattice polytopes. It operates on a generalization of the dual graph of a triangulation, with edges labeled by "signed circuits" -- combinatorial invariants from oriented matroid theory that are necessary and sufficient for exposing regularity. The model is independent of the number of points in the polytope, invariant under its orientation-preserving symmetries, and guarantees (in 2D) that every rollout is a fine triangulation. On unseen polygons it is the most uniform FRT sampler tested, with ~92k parameters, trained in ~7.5 hours on a single consumer GPU. Applied to string theory, it uniformly samples Calabi-Yau threefolds at $h^{1,1} = 86$ (consistent with uniformity at $h^{1,1} = 128$) -- an order of magnitude beyond previous learned methods with a ~1000x smaller model.

The model was presented in the paper Sampling Triangulations and Calabi-Yau Threefolds with Autoregressive GNNs.

Files

file	what it is
reinforce.pt	D=32, K=16 model after REINFORCE fine-tuning -- the default
D32K16.pt	the same model before fine-tuning (SFT only), for comparison

Usage

The weights here are the same files bundled inside the dualgnn pip package, so the simplest path needs nothing from this page:

# pip install dualgnn
import numpy as np
from dualgnn import sample_frts

pts = np.array([[x, y] for x in range(5) for y in range(5)])  # [0,4]^2
fts = sample_frts(pts, 1000, only_regular=True, seed=0)

To load this repo's checkpoint explicitly:

from huggingface_hub import hf_hub_download
from dualgnn.model import DualGNN

path = hf_hub_download("natemacfadden/dualGNN", "reinforce.pt")
net  = DualGNN.from_ckpt(path)

String Theory Application

Pair the 2D sampler with the NTFE algorithm to sample fine, regular, star triangulations (FRSTs) of a reflexive 4D polytope:

import numpy as np
from cytools import Polytope
from dualgnn.model import DualGNN
from dualgnn.ntfe  import sample_ntfes

verts = [[-1, -1, -1, -1], [-1, -1, -1,  3], [-1, -1,  3, -1], [-1,  3, -1, -1],
         [ 1, -1, -1, -1], [ 1, -1, -1,  3], [ 1, -1,  3, -1], [ 1,  3, -1, -1]]
poly = Polytope(np.array(verts, dtype=np.int64))                           # reflexive, h11 = 86
net  = DualGNN.default()
heights = sample_ntfes(poly, net, N=20, N_face_triangs=1_000, n_workers=4) # (20, npts) float64

Links

• Paper: Sampling Triangulations and Calabi-Yau Threefolds with Autoregressive GNNs (arXiv:2605.27770)
• Code / training scripts: github.com/natemacfadden/dualGNN
• Interactive Demo: Open in Colab
• Benchmark protocol: eval/
• Archive: doi:10.5281/zenodo.20622920

Citation

@article{MacFadden:2605.27770,
  author        = {MacFadden, Nate},
  title         = {Sampling Triangulations and Calabi-{Y}au Threefolds with Autoregressive {GNN}s},
  year          = {2026},
  eprint        = {2605.27770},
  archivePrefix = {arXiv},
  primaryClass  = {hep-th},
  doi           = {10.48550/arXiv.2605.27770},
  url           = {https://arxiv.org/abs/2605.27770},
}