标题： 机器学习负曲率流形上的引力紧化 显示英文标题

标题： Machine Learning Gravity Compactifications on Negatively Curved Manifolds

摘要： 通过直接求解低能（半）经典运动方程来构建高维引力理论的真空景观是出了名的困难。 在这项工作中，我们研究了机器学习技术作为解决广义扭曲引力紧致化运动方程工具的可行性。 作为一个概念验证，我们使用神经网络来求解爱因斯坦偏微分方程组，这些方程组是在通过填充双曲流形的一个或多个尖点获得的非平凡三维流形上定义的。 虽然在三维情况下，爱因斯坦度规也是局部双曲的，但机器学习方法的通用性和可扩展性、高维空间中显式双曲流形族的存在性以及填充过程的普适性表明，本工作中开发的方法和代码具有更广泛的应用前景。 具体而言，它们可以用于解决几何问题，即数值构造新的高维负曲率爱因斯坦度规，也可以用于解决物理问题，即在同一流形上构造M理论的四维de Sitter紧致化。 显示英文摘要

摘要： 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.