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The authors apply a unified physics-informed neural network (PINN) to recover the time-time component of static, spherically symmetric black-hole exterior metrics across parameter sweeps of electric charge and cosmological constant. Unlike prior distributed PINN approaches that partition the domain and truncate the inner boundary to avoid steep gradients, this formulation resolves the strong $1/r$ and $1/r^2$ curvature region down to $r_{\min}=10^{-2}M$ without hardcoding analytic metric terms. Across all configurations spanning three decades in radius, the model achieves relative $\mathcal{L}_2$ errors below 6% against analytic baselines without requiring labeled supervision.
Domain decomposition is unnecessary even in steep gravitational fields: a single unified PINN accurately resolves black-hole metric singularities across three decades of scale down to $10^{-2}M$ with sub-6% error.
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 $\Lambda=0$ and sweep $\Lambda\in\{0,0.1,-0.1\}$ at $Q=0.5$. All configurations achieve relative $\mathcal{L}_2$ errors below $6\%$ against the analytic reference. For the representative case $Q=0.5$, $\Lambda=0.1$, 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.