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Utilizing GNNs to predict 3D convex polyhedra from 2D planes
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This thesis tackles the folding problem: starting from a 2D template of rigid, hinge-linked faces, pre-dict which face pairs should be adjacent in the final structure, purely topologically, so that the tem-plate closes into a convex 3D polyhedron. The main challenge is the large combinatorial search space, which grows quadratically with face count, making rule-based approaches difficult to scale. This motivates investigating data-driven methods, specifically Graph Neural Networks (GNNs),to assess their suitability for folding. The work constructs a data set pairing polyhedral with their unique 2D unfoldings, spanning 2039 convex polyhedral and about 1.6M unfoldings. Building on this, I present FINE-GNN (Fixed-nodeI Ncremental EdgeGNN), an adapted GNN architecture that per-forms sequential, node-centric edge predictions. At each step, it decides whether and where to add an edge, providing a general architecture for edge prediction on fixed node sets that extends beyond folding. Using this architecture, I inject degree-budget constraints derived from the face’s shape. This simple topological signal cuts false positives, lifts Precision, and enables exact-match recon-structions at lower face counts (predicted adjacency equals ground truth), something prior GNNs fail to achieve. I further study node-ordering policies, which determine the next face to process. While macro metrics (Precision, Recall) differ little, per-graph analyses reveal clear distinctions: Structured policies consistently outperform a random order. The strongest model configuration in this thesis, using constraints with a cyclic order policy, achieves 37.2% Precision overall, with much higher val-ues at lower face counts (78.3% at 6 faces, 61.2% at 7, 47.0% at 8).These results show that GNNs are a promising fit for folding at small to moderate sizes and offer a practical foundation for scaling to larger, more complex polyhedra. To my knowledge, this is the first data-driven study to directly predict face adjacencies for polyhedral folding, establishing a base line for subsequent research.
Descrição
Tese de mestrado, Ciência de Dados, 2026, Universidade de Lisboa, Faculdade de Ciências
Palavras-chave
GNNs Sequential prediction Adjacency Prediction Polyhedra 2D-to-3DFolding