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

Abstract 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 ∼ 0.001% for the separated formulation with hard normalization, ∼ 0.1% for both the soft separated and soft joint formulations, and ∼ 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.