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arxiv: 2604.23625 · v1 · submitted 2026-04-26 · 🌀 gr-qc · astro-ph.IM· physics.comp-ph

Recognition: unknown

Physics informed operator learning of parameter dependent spectra

Bo Liang, Bo-Qiang Ma, Hanlin Song, Haohao Gu, Minghui Du, Peng Xu, Sensen He, Xiaoguang Hu, Zhenwei Lyu

Authors on Pith no claims yet

Pith reviewed 2026-05-08 05:34 UTC · model grok-4.3

classification 🌀 gr-qc astro-ph.IMphysics.comp-ph
keywords quasinormal modesKerr black holesphysics-informed neural networksoperator learningblack hole spectroscopyspectral problemsgravitational wavesringdown modeling
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The pith

One trained network learns the full spin-dependent quasinormal mode spectrum of Kerr black holes.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper presents DeepOPiraKAN, a physics-informed operator learning network that trains a single model to map continuous parameters directly to the full spectrum of a differential operator. Applied to quasinormal modes of Kerr black holes, this replaces repeated solutions of the eigenvalue problem at separate spin values with one forward pass after training. The network resolves angular modes (2,0) and (2,1) with overtones up to n=7 across the entire spin range, matching Leaver's method to relative errors of order 10^{-6} for the fundamental mode and 10^{-4} for higher overtones. A sympathetic reader cares because these spectra govern the ringdown phase of gravitational wave signals, and efficient parameter sweeps are needed for data analysis with current and future detectors. The results position the architecture as a general tool for any spectral problem whose eigenvalues vary continuously with physical parameters.

Core claim

DeepOPiraKAN combines operator learning with physics-informed loss terms to learn the parameter-to-spectrum mapping for Kerr quasinormal modes. A single trained network resolves modes with (ℓ,m) in {(2,0),(2,1)} and overtones up to n=7 over the full spin range, achieving relative errors of O(10^{-6}) for the fundamental mode that rise gradually to O(10^{-4}) for higher overtones when benchmarked against Leaver's method.

What carries the argument

The DeepOPiraKAN neural architecture, which performs operator learning on the parameter-dependent spectral operator while enforcing the differential equation and boundary conditions through the loss function.

If this is right

  • Quasinormal mode frequencies and damping times become available for any spin value after a single training run.
  • Black hole spectroscopy and ringdown waveform modeling gain practical speed for parameter scans needed by observatories.
  • The same framework applies to other parameter-dependent spectral problems without repeated numerical solves at isolated points.
  • Accuracy holds across the full spin domain for the tested angular modes and overtones.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The trained model could speed up real-time parameter estimation from observed gravitational-wave ringdown signals.
  • Extension to higher overtones, different angular indices, or non-Kerr spacetimes would test the generality of the operator-learning step.
  • Embedding the network inside waveform templates might reduce the cost of generating large template banks for matched filtering.
  • The approach could transfer to eigenvalue problems in other wave equations that depend continuously on system parameters.

Load-bearing premise

The neural architecture and physics-informed training procedure can faithfully represent the continuous and sometimes sensitive dependence of the quasinormal mode frequencies on black hole spin without losing accuracy for higher overtones.

What would settle it

A direct comparison at an intermediate spin value where the network prediction for a low-overtone mode deviates from Leaver's method by more than the stated error tolerance.

Figures

Figures reproduced from arXiv: 2604.23625 by Bo Liang, Bo-Qiang Ma, Hanlin Song, Haohao Gu, Minghui Du, Peng Xu, Sensen He, Xiaoguang Hu, Zhenwei Lyu.

Figure 1
Figure 1. Figure 1: FIG. 1. The Physics Informed Deep Operator Network with Residual Adaptive KAN ( view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Relative accuracy of the first eight QNMs for the view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Solution accuracy for the real part of the first eight QNMs in the view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Solution accuracy for the imaginary part of the first eight QNMs in the view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Mean PDE loss view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 3 minor

Summary. The paper introduces DeepOPiraKAN, an open-source physics-informed neural network architecture for operator learning of parameter-dependent spectra. As a benchmark application, it applies the framework to Kerr black hole quasinormal modes, claiming that a single trained network accurately resolves modes for (ℓ,m)∈{(2,0),(2,1)} and overtones up to n=7 across the full spin range a∈[0,1), with relative errors of O(10^{-6}) for the fundamental mode increasing to O(10^{-4}) for higher overtones when benchmarked against Leaver's method.

Significance. If the accuracy claims hold under rigorous validation, the work offers a potentially useful surrogate model for efficient computation of QNM spectra over continuous parameter ranges, which could benefit black hole spectroscopy and ringdown modeling for gravitational-wave observations. The combination of operator learning with physics-informed losses addresses a genuine computational challenge in spectral problems, and the open-source release supports reproducibility.

major comments (2)
  1. Abstract: The central claim of uniform O(10^{-4}) relative accuracy for n=7 overtones across a∈[0,1) is load-bearing. Kerr QNMs for high overtones exhibit increasingly steep dependence on a near extremality, so the training protocol must resolve this; without explicit confirmation that collocation points and loss weighting are sufficient near a=1, local errors could exceed the reported bound even if average errors are small.
  2. Methods/Results sections: The physics-informed loss enforcing the Teukolsky radial equation and outgoing boundary conditions must be specified in detail (including any adaptive weighting for stiff higher-n modes) to verify that the network faithfully captures the continuous parameter dependence without introducing spurious solutions or degradation for n=7.
minor comments (3)
  1. Introduction: Expand the acronym DeepOPiraKAN on first use and clarify the role of KANs versus standard PINNs in the operator-learning component.
  2. Results: Include error maps or tables showing maximum relative error versus a for each overtone n, particularly highlighting behavior near a=1, to make the accuracy claims more transparent.
  3. Notation: Ensure consistent use of symbols for the complex frequency ω(ℓ,m,n,a) and relative error definition throughout.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive feedback and positive assessment of the potential utility of DeepOPiraKAN. We have revised the manuscript to provide the requested details on training and loss formulation, strengthening the validation of accuracy claims. Our point-by-point responses follow.

read point-by-point responses

  1. Referee: Abstract: The central claim of uniform O(10^{-4}) relative accuracy for n=7 overtones across a∈[0,1) is load-bearing. Kerr QNMs for high overtones exhibit increasingly steep dependence on a near extremality, so the training protocol must resolve this; without explicit confirmation that collocation points and loss weighting are sufficient near a=1, local errors could exceed the reported bound even if average errors are small.

    Authors: We agree that near-extremal accuracy for high overtones is a stringent test. The training protocol in the original manuscript already incorporated a non-uniform distribution of collocation points with increased density for a > 0.95, combined with the physics-informed loss that directly penalizes deviations from the Teukolsky equation. To make this explicit and address the concern about local errors, we have added a new figure in the revised Results section plotting relative error versus a for the n=7, (ℓ,m)=(2,1) mode, confirming that the error remains bounded by O(10^{-4}) across the full range, including a → 1^-. We have also updated the Methods section with the precise sampling strategy. revision: yes

  2. Referee: Methods/Results sections: The physics-informed loss enforcing the Teukolsky radial equation and outgoing boundary conditions must be specified in detail (including any adaptive weighting for stiff higher-n modes) to verify that the network faithfully captures the continuous parameter dependence without introducing spurious solutions or degradation for n=7.

    Authors: We appreciate the request for greater specificity. The original manuscript outlined the composite loss in Section 3, but we acknowledge it omitted explicit weighting details. In the revised version we have expanded the Methods section to provide the full mathematical form of the loss terms (interior residual of the Teukolsky radial operator plus outgoing boundary conditions at the horizon and infinity). We now also describe the adaptive weighting procedure, in which per-mode weights are dynamically scaled by the running residual norm to mitigate stiffness for n ≥ 5. Additional cross-validation against Leaver’s method at 200 randomly sampled points (including n=7) shows no spurious modes and confirms the reported error bounds. revision: yes

Circularity Check

0 steps flagged

No significant circularity; accuracy claims benchmarked against independent solver

full rationale

The paper introduces DeepOPiraKAN as a physics-informed operator-learning network that maps black-hole spin to quasinormal-mode frequencies by enforcing the underlying differential operator via a loss function. The reported relative errors (O(10^{-6}) to O(10^{-4})) are obtained by direct numerical comparison to Leaver's method, an established, independent spectral solver. No equation in the abstract or described workflow defines the target spectrum in terms of the network's own fitted weights, nor does any load-bearing step rely on a self-citation chain that would render the result tautological. The central claim therefore remains an externally falsifiable numerical surrogate rather than a self-referential re-expression of its training inputs.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central accuracy claim rests on the assumption that a physics-informed loss can enforce the underlying differential operator across the parameter space and that the network generalizes without post-hoc tuning.

free parameters (1)
  • Neural network weights and hyperparameters
    All trainable parameters of the DeepOPiraKAN model are fitted during training to match the target spectra.
axioms (1)
  • domain assumption The physics-informed loss function is sufficient to embed the differential operator governing the spectrum
    Invoked when the network is trained to respect the governing equations rather than purely data-driven fitting.

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