用神经网络求解高维引力紧化方程，探索负曲率流形上的新解。
Machine Learning Gravity Compactifications on Negatively Curved Manifolds
- 用神经网络直接求解三维负曲率流形上的爱因斯坦方程
- 在双曲流形填补尖点后成功构造出新的爱因斯坦度量
- 方法可推广至高维负曲率几何与四维德西特紧化研究
通过直接求解低能（半）经典引力方程构建高维理论真空景观极为困难。本文探讨机器学习技术在求解一般翘曲引力紧化方程中的可行性。以三維非平凡流形为例，利用神经网络求解由双曲流形填补一个或多个尖点所得的爱因斯坦偏微分方程。尽管三维爱因斯坦度量局部为双曲，但机器学习方法的通用性、可扩展性，以及高维显式双曲流形的存在和填充过程的普适性，表明本工作开发的方法与代码具有更广泛的应用前景。具体而言，可用于数值构造新的高维负曲率爱因斯坦度量，也可用于在相同流形上构造四维德西特紧化模型。
原文摘要 · Abstract (English)
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.
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