用神经网络求解弯曲时空中的极小曲面，能处理奇点和动边界。
Physics-informed neural network solves minimal surfaces in curved spacetime
- 将物理定律嵌入损失函数，设计带奇点特性的网络结构
- 在反德西特时空成功求解威尔逊环与胶子散射等典型问题
- 支持可训练边界的动态建模，适用于多领域复杂边界问题
我们提出一种基于物理信息神经网络（PINNs）的灵活框架，用于求解弯曲时空中的极小曲面边值问题，尤其关注奇点和移动边界。通过将基本物理定律编码进损失函数，并设计包含奇点行为与动态边界的网络结构，该方法能够鲁棒且精确地求解具有复杂边界条件的常微分方程与偏微分方程。我们在反德西特（AdS）时空中验证了该框架的通用性，包括与AdS/CFT对应相关的典型问题（如威尔逊环和胶子散射振幅），这些广泛应用于弦论研究。方法能高效处理边界处的奇点，同时支持“软”（基于损失）与“硬”（基于公式）两种边界条件施加方式，甚至可将边界位置设为可训练参数。所提技术不仅适用于高能理论物理，还可广泛应用于数学、工程与自然科学中的各类含奇点和移动边界的边值问题。
原文摘要 · Abstract (English)
We develop a flexible framework based on physics-informed neural networks (PINNs) for solving boundary value problems involving minimal surfaces in curved spacetimes, with a particular emphasis on singularities and moving boundaries. By encoding the underlying physical laws into the loss function and designing network architectures that incorporate the singular behavior and dynamic boundaries, our approach enables robust and accurate solutions to both ordinary and partial differential equations with complex boundary conditions. We demonstrate the versatility of this framework through applications to minimal surface problems in anti-de Sitter (AdS) spacetime, including examples relevant to the AdS/CFT correspondence (e.g. Wilson loops and gluon scattering amplitudes) popularly used in the context of string theory in theoretical physics. Our methods efficiently handle singularities at boundaries, and also support both "soft" (loss-based) and "hard" (formulation-based) imposition of boundary conditions, including cases where the position of a boundary is promoted to a trainable parameter. The techniques developed here are not limited to high-energy theoretical physics but are broadly applicable to boundary value problems encountered in mathematics, engineering, and the natural sciences, wherever singularities and moving boundaries play a critical role.
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