ID: 2110.12483

Machine Learning Line Bundle Connections

October 24, 2021

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Holomorphic feedforward networks

May 9, 2021

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Michael R. Douglas
Complex Variables
Numerical Analysis
Numerical Analysis

A very popular model in machine learning is the feedforward neural network (FFN). The FFN can approximate general functions and mitigate the curse of dimensionality. Here we introduce FFNs which represent sections of holomorphic line bundles on complex manifolds, and ask some questions about their approximating power. We also explain formal similarities between the standard approach to supervised learning and the problem of finding numerical Ricci flat K\"ahler metrics, which...

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Deep learning complete intersection Calabi-Yau manifolds

November 20, 2023

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Harold Erbin, Riccardo Finotello
Machine Learning
Algebraic Geometry

We review advancements in deep learning techniques for complete intersection Calabi-Yau (CICY) 3- and 4-folds, with the aim of understanding better how to handle algebraic topological data with machine learning. We first discuss methodological aspects and data analysis, before describing neural networks architectures. Then, we describe the state-of-the art accuracy in predicting Hodge numbers. We include new results on extrapolating predictions from low to high Hodge numbers,...

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Deep multi-task mining Calabi-Yau four-folds

August 4, 2021

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Harold Erbin, Riccardo Finotello, ... , Tamaazousti Mohamed
Machine Learning
Algebraic Geometry

We continue earlier efforts in computing the dimensions of tangent space cohomologies of Calabi-Yau manifolds using deep learning. In this paper, we consider the dataset of all Calabi-Yau four-folds constructed as complete intersections in products of projective spaces. Employing neural networks inspired by state-of-the-art computer vision architectures, we improve earlier benchmarks and demonstrate that all four non-trivial Hodge numbers can be learned at the same time using...

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Calabi-Yau Metrics, Energy Functionals and Machine-Learning

December 20, 2021

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Anthony Ashmore, Lucille Calmon, ... , Ovrut Burt A.
Machine Learning
Algebraic Geometry

We apply machine learning to the problem of finding numerical Calabi-Yau metrics. We extend previous work on learning approximate Ricci-flat metrics calculated using Donaldson's algorithm to the much more accurate "optimal" metrics of Headrick and Nassar. We show that machine learning is able to predict the K\"ahler potential of a Calabi-Yau metric having seen only a small sample of training data.

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The Calabi-Yau Landscape: from Geometry, to Physics, to Machine-Learning

December 7, 2018

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Yang-Hui He
Algebraic Geometry
Mathematical Physics
Machine Learning

We present a pedagogical introduction to the recent advances in the computational geometry, physical implications, and data science of Calabi-Yau manifolds. Aimed at the beginning research student and using Calabi-Yau spaces as an exciting play-ground, we intend to teach some mathematics to the budding physicist, some physics to the budding mathematician, and some machine-learning to both. Based on various lecture series, colloquia and seminars given by the author in the past...

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Machine learning for complete intersection Calabi-Yau manifolds: a methodological study

July 30, 2020

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Harold Erbin, Riccardo Finotello
Machine Learning
Algebraic Geometry

We revisit the question of predicting both Hodge numbers $h^{1,1}$ and $h^{2,1}$ of complete intersection Calabi-Yau (CICY) 3-folds using machine learning (ML), considering both the old and new datasets built respectively by Candelas-Dale-Lutken-Schimmrigk / Green-H\"ubsch-Lutken and by Anderson-Gao-Gray-Lee. In real world applications, implementing a ML system rarely reduces to feed the brute data to the algorithm. Instead, the typical workflow starts with an exploratory dat...

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Moduli-dependent Calabi-Yau and SU(3)-structure metrics from Machine Learning

December 8, 2020

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Lara B. Anderson, Mathis Gerdes, James Gray, Sven Krippendorf, ... , Ruehle Fabian
High Energy Physics - Theory

We use machine learning to approximate Calabi-Yau and SU(3)-structure metrics, including for the first time complex structure moduli dependence. Our new methods furthermore improve existing numerical approximations in terms of accuracy and speed. Knowing these metrics has numerous applications, ranging from computations of crucial aspects of the effective field theory of string compactifications such as the canonical normalizations for Yukawa couplings, and the massive string...

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Lectures on the Calabi-Yau Landscape

January 5, 2020

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Jiakang Bao, Yang-Hui He, ... , Pietromonaco Stephen
Mathematical Physics

In these lecture notes, we survey the landscape of Calabi-Yau threefolds, and the use of machine learning to explore it. We begin with the compact portion of the landscape, focusing in particular on complete intersection Calabi-Yau varieties (CICYs) and elliptic fibrations. Non-compact Calabi-Yau manifolds are manifest in Type II superstring theories, they arise as representation varieties of quivers, used to describe gauge theories in the bulk familiar four dimensions. Final...

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Numerical Hermitian Yang-Mills Connections and Kahler Cone Substructure

March 15, 2011

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Lara B. Anderson, Volker Braun, Burt A. Ovrut
High Energy Physics - Theory

We further develop the numerical algorithm for computing the gauge connection of slope-stable holomorphic vector bundles on Calabi-Yau manifolds. In particular, recent work on the generalized Donaldson algorithm is extended to bundles with Kahler cone substructure on manifolds with h^{1,1}>1. Since the computation depends only on a one-dimensional ray in the Kahler moduli space, it can probe slope-stability regardless of the size of h^{1,1}. Suitably normalized error measures...

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Learning Size and Shape of Calabi-Yau Spaces

November 2, 2021

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Magdalena Larfors, Andre Lukas, ... , Schneider Robin
Machine Learning

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...

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