https://doi.org/10.1140/epjp/s13360-026-08019-3
Regular Article
A physics-informed neural network solver for a simplified scalar angular Teukolsky-like equation
Department of Informatics, Democritus University of Thrace, Kavala, Greece
a
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Received:
27
May
2026
Accepted:
26
June
2026
Published online:
6
July
2026
Abstract
The measurement of black-hole spin is considered one of the key problems in relativistic astrophysics. Existing methods, such as continuum fitting, X-ray reflection spectroscopy and quasi-periodic oscillation analysis, have systematic limitations related to model assumptions, calibration uncertainties and source-dependent parameters. In this work, a simplified physics-informed approach is proposed in which a scalar angular Teukolsky-like formulation is integrated with physics-informed neural networks (PINNs). The study is designed as a numerical proof of concept rather than as a complete observational black-hole spin-estimation pipeline. A PINN model is developed to approximate the scalar angular equation, with physical constraints directly embedded into the training process. Annotated observational data are not required; instead, the model is trained using the differential operator and boundary conditions as supervision. Experiments are performed under controlled synthetic conditions for several spin values, 
, with fixed 
, 
, and 
. An additional experiment is performed under controlled synthetic noise-contaminated conditions by perturbing the angular-mode samples with Gaussian noise of increasing amplitude. The results show that the relative reconstruction error remains low across the tested synthetic noise levels, with the largest single-run increase observed at 
. This non-monotonic behavior is interpreted as a consequence of stochastic noise realization and optimizer sensitivity during PINN training. The model reduces the training loss by several orders of magnitude across all tested configurations, reaching low near-final residual values. The results show that the proposed PINN can approximate the simplified scalar angular Teukolsky-like equation with low training loss under controlled synthetic conditions. The findings suggest that physics-informed machine learning may provide a useful numerical direction for exploratory black-hole spin-related modeling, although further validation with high-resolution numerical solvers, realistic Kerr perturbations and observationally motivated data remains necessary.
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© The Author(s), under exclusive licence to Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2026
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
