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Numerical relativity (NR), solving Einstein equations numerically, plays an important role in source modeling for gravitational wave astronomy. Traditional methods for NR including the finite difference method, spectral method, and finite element method have been well developed. But newly developed neural network methods for partial differential equations (PDE) have not been well studied yet for NR. We present a physics-informed neural network (PINN) method to solve the Hamiltonian constraint equation for binary black hole (BBH) initial data in NR. This equation is a highly nonlinear elliptic PDE, posing significant challenges for conventional PINN approaches. To overcome these difficulties, we introduce a set of new techniques. We show that our PINN together with these techniques can successfully solve the Hamiltonian constraint equation for generic BBH systems. Validation against the traditional results demonstrates the high accuracy and robustness of our method, revealing the immense potential of constructing a PINN-based initial data solution to all BBH systems for NR.

We compute quasinormal modes of the Kazakov--Solodukhin quantum-corrected black hole using a high-precision Chebyshev spectral method. After factoring out the quasinormal mode asymptotics at the event horizon and at spatial infinity, the radial problem is reduced to a quadratic matrix pencil for the dimensionless frequency $Ω=Mω$. We apply this framework to minimally coupled scalar perturbations, electromagnetic perturbations, a non-minimally coupled scalar field, and an axial effective source Regge--Wheeler-type gravitational sector. The latter is treated as an effective axial model, rather than as the full gravitational perturbation problem, because the Kazakov--Solodukhin spacetime is not Ricci flat. Our results reproduce the available WKB, time-domain, Mashhoon, and asymptotic-iteration benchmarks in their common regimes of validity, after accounting for the different normalisations used in the literature. The spectral method also resolves additional overtones and candidate purely imaginary overdamped roots. In several sectors, these roots exhibit a spacing scale close to $Mκ=1/4$, with occasional multiple gaps in the retained numerical sequence. In the near-extremal, but still subextremal, regime, the detected purely imaginary branches approach an approximately equally spaced surface-gravity-scaled ladder, while the oscillatory spectra remain sector dependent. Within the parameter ranges and resolutions considered here, all retained modes have $\ImΩ<0$, and no stable growing mode is detected.