Distinguishing Elliptic Fibrations with AI

We survey some recent applications of machine learning to problems in geometry and theoretical physics. Pure mathematical data has been compiled over the last few decades by the community and experiments in supervised, semi-supervised and unsupervised machine learning have found surprising success. We thus advocate the programme of machine learning mathematical structures, and formulating conjectures via pattern recognition, in other words using artificial intelligence to hel...

We propose machine learning inspired methods for computing numerical Calabi-Yau (Ricci flat K\"ahler) metrics, and implement them using Tensorflow/Keras. We compare them with previous work, and find that they are far more accurate for manifolds with little or no symmetry. We also discuss issues such as overparameterization and choice of optimization methods.

We present a new machine learning library for computing metrics of string compactification spaces. We benchmark the performance on Monte-Carlo sampled integrals against previous numerical approximations and find that our neural networks are more sample- and computation-efficient. We are the first to provide the possibility to compute these metrics for arbitrary, user-specified shape and size parameters of the compact space and observe a linear relation between optimization of...

Algebraic varieties are the geometric shapes defined by systems of polynomial equations; they are ubiquitous across mathematics and science. Amongst these algebraic varieties are Q-Fano varieties: positively curved shapes which have Q-factorial terminal singularities. Q-Fano varieties are of fundamental importance in geometry as they are "atomic pieces" of more complex shapes - the process of breaking a shape into simpler pieces in this sense is called the Minimal Model Progr...

We consider the construction of Calabi-Yau varieties recently generalized to where the defining equations may have negative degrees over some projective space factors in the embedding space. Within such "generalized complete intersection" Calabi-Yau ("gCICY") three-folds, we find several sequences of distinct manifolds. These include both novel elliptic and K3-fibrations and involve Hirzebruch surfaces and their higher dimensional analogues. En route, we generalize the standa...

We describe how simple machine learning methods successfully predict geometric properties from Hilbert series (HS). Regressors predict embedding weights in projective space to ${\sim}1$ mean absolute error, whilst classifiers predict dimension and Gorenstein index to $>90\%$ accuracy with ${\sim}0.5\%$ standard error. Binary random forest classifiers managed to distinguish whether the underlying HS describes a complete intersection with high accuracies exceeding $95\%$. Neura...

In this work we study genus one fibrations in Calabi-Yau three-folds with a non-trivial first fundamental group. The manifolds under consideration are constructed as smooth quotients of complete intersection Calabi-Yau three-folds (CICYs) by a freely acting, discrete automorphism. By probing the compatibility of symmetries with genus one fibrations (that is, discrete group actions which preserve a local decomposition of the manifold into fiber and base) we find fibrations tha...

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 $\sim99\%$ accuracy, in particular for the lopsided amoeba of $F_0$ with positive coefficients which we place primary focus. Using weights...

We provide the first estimate of the number of fine, regular, star triangulations of the four-dimensional reflexive polytopes, as classified by Kreuzer and Skarke (KS). This provides an upper bound on the number of Calabi-Yau threefold hypersurfaces in toric varieties. The estimate is performed with deep learning, specifically the novel equation learner (EQL) architecture. We demonstrate that EQL networks accurately predict numbers of triangulations far beyond the $h^{1,1}$ t...

We present new invariant machine learning models that approximate the Ricci-flat metric on Calabi-Yau (CY) manifolds with discrete symmetries. We accomplish this by combining the $\phi$-model of the cymetric package with non-trainable, $G$-invariant, canonicalization layers that project the $\phi$-model's input data (i.e. points sampled from the CY geometry) to the fundamental domain of a given symmetry group $G$. These $G$-invariant layers are easy to concatenate, provided o...