hep-th

arXiv:2501.00093v1 Announce Type: cross Abstract: Constructing the landscape of vacua of higher-dimensional theories of gravity by directly solving the low-energy (semi-)classical equations of motion is notoriously difficult. In this work, we investigate the feasibility of Machine Learning techniques as tools for solving the equations of motion for general warped gravity compactifications. As a proof-of-concept we use Neural Networks to solve the Einstein PDEs on non-trivial three manifolds obtained by filling one or more cusps of hyperbolic manifolds. While in three dimensions an Einstein metric is also locally hyperbolic, the generality and scalability of Machine Learning methods, the availability of explicit families of hyperbolic manifolds in higher dimensions, and the universality of the filling procedure strongly suggest that the methods and code developed in this work can be of broader applicability. Specifically, they can be used to tackle both the geometric problem of numerically constructing novel higher-dimensional negatively curved Einstein metrics, as well as the physical problem of constructing four-dimensional de Sitter compactifications of M-theory on the same manifolds.

arXiv:2403.02237v2 Announce Type: replace-cross Abstract: We present our investigation of the study of two variable hypergeometric series, namely Appell $F_{1}$ and $F_{3}$ series, and obtain a comprehensive list of its analytic continuations enough to cover the whole real $(x,y)$ plane, except on their singular loci. We also derive analytic continuations of their 3-variable generalization, the Lauricella $F_{D}^{(3)}$ series and the Lauricella-Saran $F_{S}^{(3)}$ series, leveraging the analytic continuations of $F_{1}$ and $F_{3}$, which ensures that the whole real $(x,y,z)$ space is covered, except on the singular loci of these functions. While these studies are motivated by the frequent occurrence of these multivariable hypergeometric functions in Feynman integral evaluation, they can also be used whenever they appear in other branches of mathematical physics. To facilitate their practical use, we provide four packages: $\texttt{AppellF1.wl}$, $\texttt{AppellF3.wl}$, $\texttt{LauricellaFD.wl}$, and $\texttt{LauricellaSaranFS.wl}$ in $\textit{MATHEMATICA}$. These packages are applicable for generic as well as non-generic values of parameters, keeping in mind their utilities in the evaluation of the Feynman integrals. We explicitly present various physical applications of these packages in the context of Feynman integral evaluation and compare the results using other packages such as $\texttt{FIESTA}$. Upon applying the appropriate conventions for numerical evaluation, we find that the results obtained from our packages are consistent. Various $\textit{Mathematica}$ notebooks demonstrating different numerical results are also provided along with this paper.

arXiv:2404.18247v3 Announce Type: replace-cross Abstract: We study the integrability of two-dimensional theories that are obtained by a dimensional reduction of certain four-dimensional gravitational theories describing the coupling of Maxwell fields and neutral scalar fields to gravity in the presence of a potential for the neutral scalar fields. For a certain solution subspace, we demonstrate partial integrability by showing that a subset of the equations of motion in two dimensions are the compatibility conditions for a linear system. Subsequently, we study the integrability of these two-dimensional models from a complementary one-dimensional point of view, framed in terms of Liouville integrability. In this endeavour, we employ various machine learning techniques to systematise our search for numerical Lax pair matrices for these models, as well as conserved currents expressed as functions of phase space variables.

arXiv:2406.12916v3 Announce Type: replace Abstract: An important challenge in machine learning is to predict the initial conditions under which a given neural network will be trainable. We present a method for predicting the trainable regime in parameter space for deep feedforward neural networks (DNNs) based on reconstructing the input from subsequent activation layers via a cascade of single-layer auxiliary networks. We show that a single epoch of training of the shallow cascade networks is sufficient to predict the trainability of the deep feedforward network on a range of datasets (MNIST, CIFAR10, FashionMNIST, and white noise), thereby providing a significant reduction in overall training time. We achieve this by computing the relative entropy between reconstructed images and the original inputs, and show that this probe of information loss is sensitive to the phase behaviour of the network. We further demonstrate that this method generalizes to residual neural networks (ResNets) and convolutional neural networks (CNNs). Moreover, our method illustrates the network's decision making process by displaying the changes performed on the input data at each layer, which we demonstrate for both a DNN trained on MNIST and the vgg16 CNN trained on the ImageNet dataset. Our results provide a technique for significantly accelerating the training of large neural networks.