Teukolsky by design: A hybrid spectral-PINN solver for Kerr quasinormal modes

We introduce SpectralPINN, a hybrid pseudo-spectral/physics-informed neural network (PINN) solver for Kerr quasinormal modes that targets the Teukolsky equation in both the separated (radial/angular) and joint two-dimensional formulations. The solver replaces standard neural activation functions with Chebyshev polynomials of the first kind and supports both soft via loss penalties and hard enforced by analytic masks implementations of Leaver's normalization. Benchmarking against Leaver's continued-fraction method shows cumulative (real+imaginary part) relative frequency errors of similar to 0.001% for the separated formulation with hard normalization, similar to 0.1% for both the soft separated and soft joint formulations, and similar to 0.01% for the hard joint case. Exploiting our ability to solve the joint equation, we add a small quadrupolar perturbation to the Teukolsky operator, effectively rendering the problem non-separable. The resulting perturbed quasinormal modes are compared against the expected precision of the Einstein Telescope, allowing us to constrain the magnitude of the perturbation. These proof-of-concept results demonstrate that hybrid spectral-PINN solvers can provide a flexible pathway to quasinormal spectra in settings where separability, asymptotics, or field content become more intricate and high accuracy is required.