AI 中文总结
该研究将物理信息神经网络（PINNs）应用于静态时空爱因斯坦真空场方程求解，复现史瓦西解并扩展至轴对称q-度规，分析了边界条件等因素对训练的影响，验证了PINNs求解爱因斯坦方程的鲁棒性与灵活性。
AI 中文摘要
我们研究物理信息神经网络（PINNs）在静态时空爱因斯坦真空场方程数值求解中的应用。首先复现史瓦西解，再将该方法扩展至轴对称q-度规（一种以质量四极矩为特征的非平凡精确解）。我们分析边界条件、区域分解及方程冗余性对训练过程收敛性与稳定性的影响。所提框架可精确重构计算域内的度规函数，同时保持小残差误差。结果表明，PINNs为求解爱因斯坦方程提供了鲁棒且灵活的方法，也为研究未知精确解析解的引力构型提供了有前景的基础。
英文摘要
We investigate the application of Physics-Informed Neural Networks (PINNs) to the numerical solution of Einstein's vacuum field equations for static spacetimes. We first reproduce the Schwarzschild solution and then extend the method to the axisymmetric $q-$metric, a nontrivial exact solution characterized by a mass quadrupole moment. We analyze the influence of boundary conditions, domain decomposition, and equation redundancy on the convergence and stability of the training process. The proposed framework accurately reconstructs the metric functions in the computational domain while maintaining small residual errors. Our results demonstrate that PINNs provide a robust and flexible approach to solving Einstein's equations and offer a promising foundation for investigating gravitational configurations for which exact analytical solutions are unknown.
