Solving the Regge-Wheeler and Teukolsky equations: supervised versus unsupervised physics-informed neural networks

To expand on the burgeoning research on physics-informed neural networks (PINNs) and their ability to solve the eigenvalue problems in black hole (BH) perturbation theory, we implement a supervised learning approach to solve the Regge-Wheeler and Teukolsky equations, the equations of gravitational perturbations of Schwarzschild and Kerr BHs, respectively. To date, applications of PINNs using the data-free (unsupervised) learning approach have proven their ability to compute quasinormal mode frequencies of BHs, quantities with physical significance in gravitational wave astronomy. To investigate the potential use of PINNs to compute quasinormal mode overtones higher than the low-lying $n=0$ and $n=1$ modes (with $n$ indexing overtones), the present work has instead applied the supervised approach to simplify computations. Consistent with the universal approximation theory of neural networks, it is found that the PINN algorithm has the intrinsic ability to recover the complex frequencies for various spin sequences (i.e. $s=-2$, $a\in \{0.1,0.2,0.3,0.4\}$, $\mathrm{\ell }=2$, $m\in \{0,1,2\}$, $n\in \{0,1,2,3,4\}$), with approximation errors increasing with the rotation parameter $a$ and overtone number $n$ as a result of the residuals from the training data.