Physics informed operator learning of parameter dependent spectra

TL;DR

This paper introduces DeepOPiraKAN, a physics-informed neural network that efficiently models parameter-dependent spectra, demonstrated on black hole quasinormal modes with high accuracy, reducing computational costs.

Contribution

The paper presents a novel neural network architecture that captures the parameter-to-spectrum mapping in a single model, avoiding repeated spectral computations across parameter space.

Findings

Accurately computes black hole quasinormal modes with relative errors of 10^{-6} for fundamental modes.

Achieves high accuracy across the full spin range for multiple modes and overtones.

Demonstrates potential for scalable spectral analysis in complex physical systems.

Abstract

Spectral problems governed by differential operators underpin a wide range of physical systems, yet remain computationally challenging because their spectra depend sensitively on continuous parameters and often demand repeated evaluations across parameter space. Here we present $\texttt{DeepOPiraKAN}$, an open source physics informed neural network architecture for spectral analysis. By combining operator learning with enhanced optimization stability, it captures the underlying parameter-to-spectrum mapping in a single model, avoiding repeated spectral solutions at isolated points in parameter space. As a representative and stringent benchmark, we apply this framework to the computation of quasinormal modes of Kerr black holes. A single trained network accurately resolves modes with $(\ell,m)\in \{(2,0),(2,1)\}$ and overtones up to $n=7$ across the full spin range, achieving relative errors of $\mathcal{O}(10^{-6})$ for the fundamental mode and gradually increasing to $\mathcal{O}(10^{-4})$ for higher overtones, benchmarked against the Leaver's method. This level of accuracy is already significant for black hole spectroscopy and practical ringdown modelling for current and future observatories. More broadly, these results highlight the potential of $\texttt{DeepOPiraKAN}$ as a general and scalable framework for parameter dependent spectral problems across complex physical systems.