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 provably necessary and empirically 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 we 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 ~1300x 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

Limitations

• The fineness guarantee holds in 2D; regularity is not guaranteed per rollout (only_regular=True filters by rejection).
• K=16 message-passing rounds cap the effective graph diameter; very large polygons may exceed it.
• Uniformity is validated to h^{1,1}=86; at h^{1,1}=128 diagnostics are consistent with uniformity but weaker.
• Paper figures are not reproducible from the shipped inference code alone (see the repo README).

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},
}