Physics-Informed Neural Networks for Static Black-Hole Exterior Metrics: Charge and Cosmological-Constant Sweeps
Abstract
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 [rmin,rmax], avoiding spurious jumps at subdomain interfaces. Moreover, whereas Ref.~\cite{Li2023} restricts training to r∈(10,300)---well outside the steep 1/r and 1/r2 curvature of the inner exterior---we begin at rmin=10−2M, 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∈{0,0.5,1.0,1.1} at Λ=0 and sweep Λ∈{0,0.1,−0.1} at Q=0.5. All configurations achieve relative ℒ2 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.
Related
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BibTeX
@misc{jin2026physicsinformed,
title = {{Physics-Informed Neural Networks for Static Black-Hole Exterior Metrics: Charge and Cosmological-Constant Sweeps}},
author = {Huan Jin and Fei Wu and Fei Xue},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.30332},
eprint = {2609.30332},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.30332},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Physics-Informed Neural Networks for Static Black-Hole Exterior Metrics: Charge and Cosmological-Constant Sweeps
AU - Huan Jin
AU - Fei Wu
AU - Fei Xue
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2609.30332
UR - https://arxiv.org/abs/2609.30332
AB - 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∈(10,300)---well outside the steep 1/r and 1/r^2 curvature of the inner exterior---we begin at r_min=10^−2M, 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∈{0,0.5,1.0,1.1} at Λ=0 and sweep Λ∈{0,0.1,−0.1} at Q=0.5. All configurations achieve relative 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.
ER -