TriSearch: Learning to Optimize Triangulations via Bistellar Flips

Triangulations, i.e., well-structured decompositions of geometric objects into triangle-like pieces, are central objects in many domains of mathematics and physics. In particular, fine, regular, and star triangulations (FRSTs) of 4D reflexive polytopes give rise to smooth Calabi-Yau threefolds, which are of significant interest in string theory. However, the high dimensionality and combinatorial complexity of triangulations make them particularly challenging to model with classical numerical methods or machine learning. In this work, we show that transformers, equipped with an appropriate encoding scheme, can be effectively trained to representatively generate new FRSTs across a range of polytope sizes. Moreover, these models can also self-improve through retraining on their own output. This opens the door to both concrete applications to the classification of Calabi-Yau manifolds and further research in physics, combinatorics and algebraic geometry.

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 ($\mathrm{SL}(d,\mathbb{Z}) \ltimes \mathbb{Z}^d$), and our masking procedure further guarantees that every rollout produces a fine triangulation (in 2D). On unseen polygons with $N_\mathrm{pts} \leq 40$, 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 ($\sim92$k parameters) and trains in $\sim7.5$ 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.

3D policy learning promises superior generalization and cross-embodiment transfer, but progress has been hindered by training instabilities and severe overfitting, precluding the adoption of powerful 3D perception models. In this work, we systematically diagnose these failures, identifying the omission of 3D data augmentation and the adverse effects of Batch Normalization as primary causes. We propose a new architecture coupling a scalable transformer-based 3D encoder with a diffusion decoder, engineered specifically for stability at scale and designed to leverage large-scale pre-training. Our approach significantly outperforms state-of-the-art 3D baselines on challenging manipulation benchmarks, establishing a new and robust foundation for scalable 3D imitation learning. Project Page: https://r3d-policy.github.io/