Machine-Learning Dessins d'Enfants: Explorations via Modular and Seiberg-Witten Curves

Abstract

We apply machine-learning to the study of dessins d'enfants. Specifically, we investigate a class of dessins which reside at the intersection of the investigations of modular subgroups, Seiberg-Witten curves and extremal elliptic K3 surfaces. A deep feed-forward neural network with simple structure and standard activation functions without prior knowledge of the underlying mathematics is established and imposed onto the classification of extension degree over the rationals, known to be a difficult problem. The classifications reached 0.92 accuracy with 0.03 standard error relatively quickly. The Seiberg-Witten curves for those with rational coefficients are also tabulated.

arXiv comment60 pages, 197 figures. Acknowledgements updated to reflect thanks to the group at UoAugsburg for highlighting a data analysis problem, that lead authors to identify the dessin d'enfant representation subtlety and use the improved cyclic edge list representation, as in version 3

References in corpus (16)

• Monopole Condensation, And Confinement In N=2 Supersymmetric Yang-Mills Theory
• N=2 dualities
• Gauge Theories from Toric Geometry and Brane Tilings
• Machine Learning of Calabi-Yau Volumes
• Machine Learning in the String Landscape
• Evolving neural networks with genetic algorithms to study the String Landscape
• Machine Learning CICY Threefolds
• Machine Learning Line Bundle Cohomology
• Quiver Mutations, Seiberg Duality and Machine Learning
• Toric CFTs, Permutation Triples, and Belyi Pairs
• N=2 Gauge Theories: Congruence Subgroups, Coset Graphs and Modular Surfaces
• Modular Subgroups, Dessins d'Enfants and Elliptic K3 Surfaces
• Nongeometric heterotic strings and dual F-theory with enhanced gauge groups
• Invariants of Toric Seiberg Duality
• The Beta Ansatz: A Tale of Two Complex Structures
• Dessins d'Enfants in $\mathcal{N}=2$ Generalised Quiver Theories

Cited by in corpus (16)

• Machine Learning Calabi-Yau Hypersurfaces
• Hilbert Series, Machine Learning, and Applications to Physics
• Neurons on Amoebae
• Brain Webs for Brane Webs
• Integrality, Duality and Finiteness in Combinatoric Topological Strings
• Machine learning Sasakian and $G_2$ topology on contact Calabi-Yau $7$-manifolds
• Polytopes and Machine Learning
• Cluster Algebras: Network Science and Machine Learning
• Dessins d'Enfants, Seiberg-Witten Curves and Conformal Blocks
• Calabi-Yau Four/Five/Six-folds as ${\mathbb{P}}_{\text{w}}^n$ Hypersurfaces: Machine Learning, Approximation, and Generation
• Four-dimensional $N=1$ theories, S-fold constraints on T-branes, and behaviors in IR and UV
• Machine Learning Clifford invariants of ADE Coxeter elements
• Reflexions on Mahler: Dessins, Modularity and Gauge Theories
• Learning 3-Manifold Triangulations
• Metaheuristic Generation of Brane Tilings
• Seiberg-Witten geometry, modular rational elliptic surfaces and BPS quivers