  Expanding on the research on physics-informed neural networks (PINNs) to
solve the eigenvalue problems in black hole (BH) perturbation theory, the
supervised learning approach was investigated to solve the Regge-Wheeler and
Teukolsky equations governing gravitational perturbations of Schwarzschild and
Kerr BHs. Previous works have applied unsupervised PINNs to compute quasinormal
mode frequencies, i.e. the discrete spectra of eigenvalues that satisfy the BH
perturbation equations; however, that was limited to only the zeroth and first
overtone (i.e. n = 0, 1). In this paper, supervised learning is used to compute
higher overtones with approximation errors accentuated primarily by the input
data rather than the PINNs themselves. Our results show the ability of PINNs to
approximate overtones up to the limits set by the input data, which concurs
with the universal approximation theory of neural networks.
