Detecting Homeomorphic 3-manifolds via Graph Neural Networks
<|reference_start|>Detecting Homeomorphic 3-manifolds via Graph Neural Networks: Motivated by the enumeration of the BPS spectra of certain 3d $\mathcal{N}=2$ supersymmetric quantum field theories, obtained from the compactification of 6d superconformal field theories on three-manifolds, we study the homeomorphism problem for a class of graph-manifolds using Graph Neural Network techniques. Utilizing the JSJ decomposition, a unique representation via a plumbing graph is extracted from a graph-manifold. Homeomorphic graph-manifolds are related via a sequence of von Neumann moves on this graph; the algorithmic application of these moves can determine if two graphs correspond to homeomorphic graph-manifolds in super-polynomial time. However, by employing Graph Neural Networks (GNNs), the same problem can be addressed, at the cost of accuracy, in polynomial time. We build a dataset composed of pairs of plumbing graphs, together with a hidden label encoding whether the pair is homeomorphic. We train and benchmark a variety of network architectures within a supervised learning setting by testing different combinations of two convolutional layers (GEN, GCN, GAT, NNConv), followed by an aggregation layer and a classification layer. We discuss the strengths and weaknesses of the different GNNs for this homeomorphism problem.<|reference_end|>
@article{lawrie2024detecting,
    title={Detecting Homeomorphic 3-manifolds via Graph Neural Networks},
    author={Craig Lawrie, Lorenzo Mansi},
    journal={arXiv preprint arXiv:2409.02126},
    year={2024},
    number={DESY-24-122},
    archivePrefix={arXiv},
    eprint={2409.02126},
    primaryClass={cs.LG hep-th}
}