标题： 使用深度学习求解爱因斯坦方程 显示英文标题

标题： Solving Einstein equations using deep learning

摘要： 爱因斯坦场方程因其复杂的数学形式而以难以求解著称，在缺乏高度对称系统或理想物质分布的情况下，仅有少量解析解可用。然而，准确的解至关重要，尤其是在强引力场系统中，如黑洞或中子星。在本工作中，我们使用神经网络和自动微分来数值求解爱因斯坦场方程，这一方法受到物理信息神经网络（PINNs）理念的启发。通过利用这些技术，我们在给定物质能动张量的情况下成功获得了施瓦茨希尔德度规和带电施瓦茨希尔德度规。这种创新方法可能为求解时空耦合的爱因斯坦场方程开辟一条新途径，并成为数值相对论的重要组成部分。 显示英文摘要

摘要： Einstein field equations are notoriously challenging to solve due to their complex mathematical form, with few analytical solutions available in the absence of highly symmetric systems or ideal matter distribution. However, accurate solutions are crucial, particularly in systems with strong gravitational field such as black holes or neutron stars. In this work, we use neural networks and auto differentiation to solve the Einstein field equations numerically inspired by the idea of physics-informed neural networks (PINNs). By utilizing these techniques, we successfully obtain the Schwarzschild metric and the charged Schwarzschild metric given the energy-momentum tensor of matter. This innovative method could open up a different way for solving space-time coupled Einstein field equations and become an integral part of numerical relativity.