Sampling Triangulations and Calabi-Yau Threefolds with Autoregressive GNNs

Abstract: 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 bysigned 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 .

• The paper introduces dualGNN, an autoregressive GNN that uniformly samples fine triangulations via circuit-based message passing.
• It leverages built-in symmetry invariance to generalize across unseen polygon sizes and outperforms MCMC and transformer methods in efficiency and uniformity.
• The method enables practical Calabi–Yau threefold enumeration by reducing sample bias and computational overhead in high Hodge number models.

Autoregressive GNNs for Uniform Triangulation Sampling and Calabi-Yau Threefold Enumeration

Introduction and Problem Setting

The paper "Sampling Triangulations and Calabi-Yau Threefolds with Autoregressive GNNs" (2605.27770) addresses a core challenge in combinatorial geometry: uniformly sampling fine, regular triangulations (FRTs) of lattice polytopes. This problem is of particular interest in the context of enumerating Calabi-Yau (CY) threefolds in string theory, where FRTs underlie the generation of distinct geometric configurations critical for classification and physical model searches.

The core difficulties in this domain involve the exponential scaling of triangulation counts (reaching upwards of $10^{180}$ for relevant polygons), intricate local and global regularity constraints, and strong symmetry requirements under $SL(d,\mathbb{Z})\ltimes\mathbb{Z}^d$. Previous approaches—both classical algorithms based on combinatorial geometry and transformer-based deep learning generative models—succumb to scalability limits, statistical bias, or lack generalization across polytopes.

The dualGNN Architecture

Circuit-based Message Passing

dualGNN introduces a message-passing graph neural network incorporating autoregressive sampling on a graph whose nodes correspond to candidate simplices of a polytope. Critically, edge features are labeled by ‘signed circuits’ derived from oriented matroid theory—minimal dependencies among lattice points—that are both necessary and sufficient to expose regularity constraints in triangulations. These circuits serve as invariants under orientation-preserving polytope symmetries, ensuring that model outputs respect the geometry's intrinsic structure.

Symmetry and Generalization

Unlike prior transformer-based approaches, dualGNN's architectural invariance to $SL(d,\mathbb{Z})\ltimes\mathbb{Z}^d$ transformations is built at the representation level, obviating the need for extensive data augmentation or retraining per polygon. The model’s independence from the size of the point set enables zero-shot generalization to unseen polytopes up to at least 40 points, a property unattainable with standard sequence modeling architectures.

Autoregressive Triangulation Generation

Leveraging message-passing, dualGNN performs $K$ rounds of propagation followed by a softmax over candidate simplices, autoregressively sampling simplices and recursively masking out invalid overlaps. This masking guarantees that every rollout in 2D produces a fine triangulation. Regularity is learned through the circuit features, with the model further fine-tuned using REINFORCE for entropy maximization to approach true uniformity over FRTs.

Empirical Performance and Numerical Results

Uniformity and Scalability

Exhaustive benchmarks demonstrate that dualGNN is the most uniform FRT sampler among all tested baselines. Even when trained on a single polygon, it generalizes effectively to unseen, larger polygons. For moderate-sized polygons—where full enumeration allows direct computation of KL divergence and collision statistics—the model achieves lower deviation from the uniform distribution than flip-walk Markov Chain Monte Carlo (MCMC), pushing, fast, and grow2d baselines. Sample autocorrelation analysis confirms the generation of nearly independent triangulations, a property that MCMC-based approaches fail to achieve due to mixing time and flip-graph obstructions (bipartite graphs).

For large polygons beyond exhaustive enumeration—where theoretical analysis becomes probabilistic—the model consistently matches the empirical statistics expected of a true uniform sampler, such as expected collision rates. Importantly, dualGNN is the only method showing no repeated samples (collisions) in high-$N_{pts}$ settings within feasible sampling budgets.

Computational Efficiency

Despite the detailed combinatorial encoding, dualGNN is highly efficient in both training (∼7.5 hours on a single consumer GPU, 92k parameters) and inference. Unlike flip-walk and similar MCMC methods, each dualGNN sample is independent, sidestepping the issue of long-mixing chains. The model is significantly smaller and faster to train than contemporary transformer-based models such as CYTransformer, and requires no model specialization per polygon size.

Application to Calabi-Yau Threefold Enumeration

dualGNN is applied to the uniform sampling of Calabi–Yau threefolds via Batyrev's construction, where FRTs of the 2-faces of 4D reflexive polytopes define the topological diversity of CYs. The work leverages a prior reduction showing that uniform sampling of 2D FRTs directly induces uniform sampling over the equivalence classes of CYs, bypassing exponential redundancy present in FRST enumeration.

Empirical evaluation at high Hodge numbers ($h^{1,1}=86$ and up to $h^{1,1}=128$) demonstrates that dualGNN is capable of uniformly sampling CYs at magnitudes previously unreachable by learned methods, outperforming CYTransformer by at least an order of magnitude and with a model approximately 1000× smaller. Uniformity diagnostics—KL divergence, collision statistics, and flop-distance distributions—demonstrate that dualGNN produces markedly more diverse and uniform CY samples when compared to the dominant random triangulations fast method.

Comparison to Baseline and Prior Methods

The construction and evaluation of dualGNN reveal several critical advantages over classical and modern machine learning baselines:

• Classical Samplers: Methods such as pushing and grow2d are computationally efficient but introduce significant bias due to incompleteness in regular triangulation coverage or randomness in local choices.
• MCMC-based Methods (flip-walk): Show substantial correlation between samples (violating the independence desirable for statistical uniformity), sensitivity to walk length, and failure modes in bipartite flip-graphs.
• Prior Deep Generative Models (CYTransformer): Require retraining per polygon size, are not polymorphic across symmetries, and underperform on uniformity and scale, especially at high Hodge numbers relevant for string theory.

Limitations and Theoretical Implications

While dualGNN offers strong empirical performance and broad generalization, its fineness guarantee holds strictly in 2D; for higher dimensions, the corresponding geometric properties required for guaranteed fine completions no longer apply. The regularity constraint, while well-exposed via circuit features, is enforced only statistically, not deterministically, in the autoregressive sampler. The empirical uniformity lacks a formal theoretical guarantee—no known efficient algorithms provide this either.

The combinatorial encoding via oriented matroid circuits opens possibilities for more general realizable matroid modeling with GNNs, extending applicability to other domains such as convex polyhedral theory, optimization, and broader mathematical combinatorics.

Conclusion

dualGNN establishes a new standard for uniform, generalizable, and scalable sampling of fine, regular triangulations in convex lattice polytopes, with immediate application to CY threefold enumeration in string theory. The architecture’s explicit encoding of combinatorial invariants, symmetry invariance, and efficient autoregressive sampling enable advances unattainable by previous classical or deep learning approaches. Its demonstrated empirical uniformity and efficiency on practical hardware open new avenues for large-scale exploration of geometrically and physically motivated combinatorial spaces.

Further directions include architectural extensions for enforcing regularity in higher dimensions, theoretical uniformity analyses, and adaptation to related combinatorial geometric enumeration problems exploiting the oriented matroid framework.