发表机构
School of Information Engineering, Jiangxi Institute of Technology（江西科技学院信息工程学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文用统一的物理信息神经网络从常微分方程恢复静态球对称黑洞外部度规，覆盖强曲率区域，扫描电荷和宇宙学常数，相对误差低于6%，验证了方法的鲁棒性。
AI 中文摘要
我们应用物理信息神经网络（PINNs）从约化的常微分方程（ODE）恢复静态、球对称黑洞外部度规的时间-时间分量。受近期利用深度学习求解爱因斯坦场方程工作的启发，我们通过残差损失和渐近边界约束来编码真空/带电外部解，而非将诸如$2M/r$之类的解析度规项直接嵌入网络输出。与参考文献中分布式PINN（DPINN）策略（将径向域划分为子域并使用独立网络）不同，我们采用一个统一的完全连接网络覆盖整个区间$[r_{\min},r_{\max}]$，避免了子域界面处的虚假跳跃。此外，参考文献将训练限制在$r\in(10,300)$，远在内部外部陡峭的$1/r$和$1/r^2$曲率区域之外，而我们则从$r_{\min}=10^{-2}M$开始，跨越三个数量级的半径，覆盖了DPINN通过域截断所避开的强变化区域。在固定质量$M=1$的情况下，我们在$\Lambda=0$时扫描电荷$Q\in\{0,0.5,1.0,1.1\}$，并在$Q=0.5$时扫描$\Lambda\in\{0,0.1,-0.1\}$。所有配置相对于解析参考解的相对$\mathcal{L}_2$误差均低于$6\\%$。对于代表性情况$Q=0.5$、$\Lambda=0.1$，三次使用固定种子的独立训练分别产生相对误差$3.89\\%$、$1.08\\%$和$2.76\\%$（平均值$2.58\\%$，标准差$1.41\\%$），证明了该无网格方法在无标记场数据情况下的鲁棒性。
英文摘要
We apply physics-informed neural networks (PINNs) to recover the time-time component of static, spherically symmetric black-hole exterior metrics from a reduced ordinary differential equation (ODE). Inspired by recent work on solving Einstein field equations with deep learning~\cite{Li2023}, we encode the vacuum/charged exterior through a residual loss and an asymptotic boundary constraint---not by embedding analytic metric terms such as $2M/r$ directly into the network output. Unlike the distributed PINN (DPINN) strategy of Ref.~\cite{Li2023}, which partitions the radial domain into subdomains with separate networks, we employ a \emph{unified} fully connected network over the entire interval $[r_{\min},r_{\max}]$, avoiding spurious jumps at subdomain interfaces. Moreover, whereas Ref.~\cite{Li2023} restricts training to $r\in(10,300)$---well outside the steep $1/r$ and $1/r^2$ curvature of the inner exterior---we begin at $r_{\min}=10^{-2}M$, spanning three decades in radius and covering the strongly varying region that DPINNs sidestep by domain truncation. Holding the mass fixed at $M=1$, we sweep electric charge $Q\in\{0,0.5,1.0,1.1\}$ at $Λ=0$ and sweep $Λ\in\{0,0.1,-0.1\}$ at $Q=0.5$. All configurations achieve relative $\mathcal{L}_2$ errors below $6\%$ against the analytic reference. For the representative case $Q=0.5$, $Λ=0.1$, three independent trainings with fixed seeds yield relative errors of $3.89\%$, $1.08\%$, and $2.76\%$ (mean $2.58\%$, standard deviation $1.41\%$), demonstrating robustness of the mesh-free approach without labeled field data.
