550 research outputs found
Machine-Learning Dessins d'Enfants: Explorations via Modular and Seiberg-Witten Curves
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.Comment: 60 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
Neurons on Amoebae
We apply methods of machine-learning, such as neural networks, manifold
learning and image processing, in order to study 2-dimensional amoebae in
algebraic geometry and string theory. With the help of embedding manifold
projection, we recover complicated conditions obtained from so-called
lopsidedness. For certain cases it could even reach accuracy, in
particular for the lopsided amoeba of with positive coefficients which we
place primary focus. Using weights and biases, we also find good approximations
to determine the genus for an amoeba at lower computational cost. In general,
the models could easily predict the genus with over accuracies. With
similar techniques, we also investigate the membership problem, and image
processing of the amoebae directly.Comment: 53 page
Cluster Algebras: Network Science and Machine Learning
Cluster algebras have recently become an important player in mathematics and
physics. In this work, we investigate them through the lens of modern data
science, specifically with techniques from network science and
machine-learning. Network analysis methods are applied to the exchange graphs
for cluster algebras of varying mutation types. The analysis indicates that
when the graphs are represented without identifying by permutation equivalence
between clusters an elegant symmetry emerges in the quiver exchange graph
embedding. The ratio between number of seeds and number of quivers associated
to this symmetry is computed for finite Dynkin type algebras up to rank 5, and
conjectured for higher ranks. Simple machine learning techniques successfully
learn to differentiate cluster algebras from their seeds. The learning
performance exceeds 0.9 accuracies between algebras of the same mutation type
and between types, as well as relative to artificially generated data.Comment: 38 pages, 27 figure
Mahler Measuring the Genetic Code of Amoebae
Amoebae from tropical geometry and the Mahler measure from number theory play
important roles in quiver gauge theories and dimer models. Their dependencies
on the coefficients of the Newton polynomial closely resemble each other, and
they are connected via the Ronkin function. Genetic symbolic regression methods
are employed to extract the numerical relationships between the 2d and 3d
amoebae components and the Mahler measure. We find that the volume of the
bounded complement of a d-dimensional amoeba is related to the gas phase
contribution to the Mahler measure by a degree-d polynomial, with d = 2 and 3.
These methods are then further extended to numerical analyses of the
non-reflexive Mahler measure. Furthermore, machine learning methods are used to
directly learn the topology of 3d amoebae, with strong performance.
Additionally, analytic expressions for boundaries of certain amoebae are given.Comment: 45 pages; 33 Figure
New Calabi–Yau manifolds from genetic algorithms
Calabi–Yau manifolds can be obtained as hypersurfaces in toric varieties built from reflexive polytopes. We generate reflexive polytopes in various dimensions using a genetic algorithm. As a proof of principle, we demonstrate that our algorithm reproduces the full set of reflexive polytopes in two and three dimensions, and in four dimensions with a small number of vertices and points. Motivated by this result, we construct five-dimensional reflexive polytopes with the lowest number of vertices and points. By calculating the normal form of the polytopes, we establish that many of these are not in existing datasets and therefore give rise to new Calabi–Yau four-folds. In some instances, the Hodge numbers we compute are new as well
Brain Webs for Brane Webs
We propose a new technique for classifying 5d Superconformal Field Theories
arising from brane webs in Type IIB String Theory, using technology from
Machine Learning to identify different webs giving rise to the same theory. We
concentrate on webs with three external legs, for which the problem is
analogous to that of classifying sets of 7-branes. Training a Siamese Neural
Network to determine equivalence between any two brane webs shows an improved
performance when webs are considered equivalent under a weaker set of
conditions. Thus, Machine Learning teaches us that the conjectured
classification of 7-brane sets is not complete, which we confirm with explicit
examples.Comment: 12 pages, 12 figure
Machine Learning Clifford Invariants of ADE Coxeter Elements
There has been recent interest in novel Clifford geometric invariants of linear transformations. This motivates the investigation of such invariants for a certain type of geometric transformation of interest in the context of root systems, reflection groups, Lie groups and Lie algebras: the Coxeter transformations. We perform exhaustive calculations of all Coxeter transformations for A8, D8 and E8 for a choice of basis of simple roots and compute their invariants, using high-performance computing. This computational algebra paradigm generates a dataset that can then be mined using techniques from data science such as supervised and unsupervised machine learning. In this paper we focus on neural network classification and principal component analysis. Since the output—the invariants—is fully determined by the choice of simple roots and the permutation order of the corresponding reflections in the Coxeter element, we expect huge degeneracy in the mapping. This provides the perfect setup for machine learning, and indeed we see that the datasets can be machine learned to very high accuracy. This paper is a pump-priming study in experimental mathematics using Clifford algebras, showing that such Clifford algebraic datasets are amenable to machine learning, and shedding light on relationships between these novel and other well-known geometric invariants and also giving rise to analytic results
Machine learning Calabi-Yau hypersurfaces
We revisit the classic database of weighted-P4s which admit Calabi-Yau 3-fold hypersurfaces equipped with a diverse set of tools from the machine-learning toolbox. Unsupervised techniques identify an unanticipated almost linear dependence of the topological data on the weights. This then allows us to identify a previously unnoticed clustering in the Calabi-Yau data. Supervised techniques are successful in predicting the topological parameters of the hypersurface from its weights with an accuracy of R2>95%. Supervised learning also allows us to identify weighted-P4s which admit Calabi-Yau hypersurfaces to 100% accuracy by making use of partitioning supported by the clustering behavior
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