Teukolsky by Design: A Hybrid Spectral-PINN solver for Kerr Quasinormal Modes

TL;DR

This paper presents SpectralPINN, a hybrid spectral and physics-informed neural network solver for Kerr quasinormal modes, achieving high accuracy and enabling analysis of non-separable perturbations in the Teukolsky equation.

Contribution

The paper introduces a novel hybrid spectral-PINN approach for solving the Teukolsky equation, including methods for both separable and non-separable cases with high precision.

Findings

Achieves frequency errors of ~0.001% with hard normalization.

Demonstrates solving non-separable perturbed modes.

Provides a flexible framework for complex quasinormal mode calculations.

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 $\sim 0.001\%$ for the separated formulation with hard normalization, $\sim 0.1\%$ for both the soft separated and soft joint formulations, and $\sim 0.01\%$ 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.