Machine Learning in Pure Mathematics and Theoretical Physics

The following sections are included:

• Preface
• About the Editor
• About the Contributors
• Acknowledgments
• Contents

Empirical analysis is often the first step towards the birth of a conjecture. This is the case of the Birch–Swinnerton-Dyer (BSD) Conjecture describing the rational points on an elliptic curve, one of the most celebrated unsolved problems in mathematics. Here, we extend the original empirical approach to the analysis of the Cremona database of quantities relevant to BSD, inspecting more than 2.5 million elliptic curves by means of the latest techniques in data science, machine learning and topological data analysis.

Key quantities such as rank, Weierstraß coefficients, period, conductor, Tamagawa number, regulator and order of the Tate–Shafarevich group give rise to a high-dimensional point cloud whose statistical properties we investigate. We reveal patterns and distributions in the rank versus Weierstraß coefficients, as well as the Beta distribution of the BSD ratio of the quantities. Via gradient-boosted trees, machine learning is applied in finding inter-correlation among the various quantities. We anticipate that our approach will spark further research on the statistical properties of large datasets in Number Theory and more in general in pure Mathematics.

Statistical inference is the process of determining a probability distribution over the space of parameters of a model given a dataset. As more data become available, this probability distribution becomes updated via the application of Bayes’ theorem. We present a treatment of this Bayesian updating process as a continuous dynamical system. Statistical inference is then governed by a first-order differential equation describing a trajectory or flow in the information geometry determined by a parametric family of models. We solve this equation for some simple models and show that when the Cramér–Rao bound is saturated the learning rate is governed by a simple 1/T power law, with T a time-like variable denoting the quantity of data. The presence of hidden variables can be incorporated in this setting, leading to an additional driving term in the resulting flow equation. We illustrate this with both analytic and numerical examples based on Gaussians and Gaussian Random Processes and inference of the coupling constant in the 1D Ising model. Finally, we compare the qualitative behavior exhibited by Bayesian flows to the training of various neural networks on benchmarked datasets such as MNIST and CIFAR10 and show how that for networks exhibiting small final losses the simple power law is also satisfied.

We use machine learning to predict the dimension of a lattice polytope directly from its Ehrhart series. This is highly effective, achieving almost 100% accuracy. We also use machine learning to recover the volume of a lattice polytope from its Ehrhart series and to recover the dimension, volume and quasi-period of a rational polytope from its Ehrhart series. In each case, we achieve very high accuracy, and we propose mathematical explanations for why this should be so.

The goal of identifying the Standard Model of particle physics and its extensions within string theory has been one of the principal driving forces in string phenomenology. Recently, the incorporation of artificial intelligence in string theory and certain theoretical advancements have brought to light unexpected solutions to mathematical hurdles that have so far hindered progress in this direction. In this review, we focus on model-building efforts in the context of the E₈ × E₈ heterotic string compactified on smooth Calabi–Yau threefolds and discuss several areas in which machine learning is expected to make a difference.

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 network 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, and conversely.

We propose a paradigm to deep-learn the ever-expanding databases which have emerged in mathematical physics and particle phenomenology, as diverse as the statistics of string vacua or combinatorial and algebraic geometry. As concrete examples, we establish multi-layer neural networks as both classifiers and predictors and train them with a host of available data ranging from Calabi–Yau manifolds and vector bundles, to quiver representations for gauge theories. We find that even a relatively simple neural network can learn many significant quantities to astounding accuracy in a matter of minutes and can also predict hithertofore unencountered results. This paradigm should prove a valuable tool in various investigations in landscapes in physics as well as pure mathematics.

We apply techniques in natural language processing, computational linguistics and machine-learning to investigate papers in hep-th and four related sections of the arXiv: hep-ph, hep-lat, gr-qc and math-ph. All of the titles of papers in each of these sections, from the inception of the arXiv until the end of 2017, are extracted and treated as a corpus which we use to train the neural network Word2Vec. A comparative study of common n-grams, linear syntactical identities, word cloud and word similarities is carried out. We find notable scientific and sociological differences between the fields. In conjunction with support vector machines, we also show that the syntactic structures of the titles in different subfields of high energy and mathematical physics are sufficiently different that a neural network can perform a binary classification of formal versus phenomenological sections with 87.1% accuracy and can perform a finer fivefold classification across all sections with 65.1% accuracy.

Parameter space and function space provide two different duality frames in which to study neural networks. We demonstrate that symmetries of network densities may be determined via dual computations of network correlation functions, even when the density is unknown and the network is not equivariant. Symmetry-via-duality relies on invariance properties of the correlation functions, which stem from the choice of network parameter distributions. Input and output symmetries of neural network densities are determined, which recover known Gaussian process results in the infinite width limit. The mechanism may also be utilized to determine symmetries during training, when parameters are correlated, as well as symmetries of the Neural Tangent Kernel. We demonstrate that the amount of symmetry in the initialization density affects the accuracy of networks trained on Fashion-MNIST and that symmetry breaking helps only when it is in the direction of ground truth.

We survey some recent implementations of supervised learning techniques on large sets of arithmetic data. As part of our methodological review, we perform some rudimentary statistical learning algorithms by hand on simplified problems. We incorporate a self-contained number-theoretic background which places a significant emphasis on conjectures and examples relevant to the machine learning context.

We review recent work on calculating systematically the minimum volumes of large classes of Sasaki–Einstein manifolds in 2n + 1 dimensions corresponding to toric Calabi–Yau (n + 1)-folds. These non-compact Calabi–Yau (n + 1)-folds are characterized by n-dimensional convex lattice polytopes known as toric diagrams. The review summarizes the discovery of a universal lower bound on the minimum volume for toric Calabi–Yau (n + 1)-folds and a more restricted upper bound when the corresponding toric diagrams are reflexive polytopes. The review also covers the pioneering application of machine learning techniques in 2017, including a convolutional neural network that takes as its input the toric diagram, in order to identify new exact formulas for the minimum volume. We give a brief overview of how these discoveries on the minimum volume have implications for recent developments in identifying supersymmetric gauge theories in diverse dimensions that are realized by brane configurations based on the corresponding toric Calabi–Yau (n + 1)-folds. We put a particular emphasis on brane configurations known as brane tilings and brane brick models.

The following sections are included:

• Epilogue
• Index