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

TL;DR

SpectralPINN presents a hybrid spectral-PINN solver for Kerr QNMs that solves the Teukolsky equation in both separated and joint 2D formulations by replacing standard activations with Chebyshev polynomials. It achieves high-precision QNM frequencies comparable to Leaver's method in the separable case (hard normalization at ~0.001% cumulative error) and robust results in the non-separable 2D regime (~0.1%–0.01% depending on normalization). The method supports soft and hard normalization, uses a complex-valued, block-coordinate training regime with a specialized optimizer (HAdamD), and demonstrates a proof-of-concept non-separable deformation that maps to ET-era ringdown constraints, illustrating practical applicability to beyond-Kerr spectroscopy. Overall, SpectralPINN offers a flexible, high-accuracy pathway for QNM spectra in complex settings where separability, asymptotics, or field content complicate traditional solvers, enabling precise tests of GR and alternative compact-object models with next-generation gravitational-wave detectors.

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.

Figures (5)

• Figure 1: Schematic representation of the SpectralPINN architecture. Each neuron corresponds to a Chebyshev basis function, and each "layer" is associated with one of the coordinate dependence. The scheme is loosely inspired by the PINN designs in luna2023solving.
• Figure 2: Cumulative ($\mathbb{R}+\mathbb{I}$) relative error of the perturbation frequency $\omega$ with respect to Leaver’s continued-fraction values, as a function of $a/M$ with $N_x = N_y = 100$. Results are shown for separated-ODE PINNs with soft (black circles) and hard (magenta inverted triangles) normalization; separated-ODE SpectralPINN runs with soft (red squares), dynamical (green crosses) and hard (orange diamonds) boundary conditions for $N=L=25$; and for the full 2D PDE SpectralPINN with soft (blue triangles) and hard (grey stars) normalization for $N=L=25$.
• Figure 3: Modulus of the 2D solution $|p(x,y)|$ (left) and training loss as a function of epoch for soft and hard NC (right).
• Figure 4: Dependence of the fundamental $s=-2,\,\ell=2,\, m=0$ QNM frequency on the quadrupolar deformation parameter $\omega(\varepsilon)$ for $a/M=\{0.3,\, 0.9\}$. The dashed lines correspond to the theoretical expected value: $\omega = 0.7540-i\, 0.1767$ for $a/M=0.3$ (red) and $\omega = 0.8240-i\, 0.1570$ for $a/M = 0.9$ (blue).
• Figure 5: Minimal per-mode ringdown SNR $\rho_{\min}(\epsilon)$ required to distinguish a deviation $\epsilon$ from Kerr at the $D=4$ level, using the ET-D noise curve. We show the single-mode thresholds for the $(2,0)$ and $(2,2)$ modes and the combined constraint when both modes are used jointly, for spins $a/M=0.3$ (left) and $a/M=0.9$ (right).