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arxiv: 2606.08448 · v1 · pith:H24W6XIGnew · submitted 2026-06-07 · 🧮 math.NA · cs.NA

Multiscale Fourier Neural Operator for Inverse Wave Scattering in Highly Oscillatory Media

Zilin You , Zhenli Xu , Wei Cai This is my paper

Pith reviewed 2026-06-27 18:12 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords Fourier neural operatorHelmholtz equationinverse scatteringhighly oscillatory mediaoperator learningdiffusion modelwave propagationnumerical inversion
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The pith

Multiscale Fourier neural operator maps highly oscillatory media to scattered wavefields for inverse Helmholtz problems.

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

The paper develops the multiscale Fourier neural operator (MscaleFNO) to serve as a neural surrogate for the forward map in inverse medium scattering problems governed by the Helmholtz equation. Standard Fourier neural operators exhibit spectral bias that hinders their performance on highly oscillatory functions, but the multiscale version is designed to capture these high-frequency features more effectively. The method then employs a plug-and-play inversion scheme regularized by an elucidated diffusion model to solve for the medium properties from scattered wave data using least squares. Numerical experiments on two-dimensional oscillatory media with partial aperture data illustrate the method's ability to reconstruct medium properties accurately.

Core claim

The MscaleFNO provides a neural surrogate model with reduced spectral bias for the Helmholtz equations, mapping highly oscillatory medium profiles to scattered wavefields. A plug-and-play inversion using elucidated diffusion model is introduced to regularize the inverse solver based on least squares of data misfits. Numerical results for partial aperture inversion of oscillatory two-dimensional media demonstrate the advantage and effectiveness of MscaleFNO for accurate reconstruction of highly oscillatory medium properties.

What carries the argument

The multiscale Fourier neural operator (MscaleFNO), which processes inputs at multiple scales to overcome spectral bias in learning operators for highly oscillatory wave equations.

If this is right

  • Improved accuracy in reconstructing highly oscillatory medium properties from partial wave scattering data.
  • Effective regularization of inverse problems through integration with diffusion models.
  • Applicability to Helmholtz inverse scattering without requiring problem-specific retraining.
  • Advantage over standard FNOs in handling high-frequency oscillations in 2D media.

Where Pith is reading between the lines

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

  • Potential extension to three-dimensional or time-dependent wave problems if the multiscale structure generalizes.
  • Could enable faster surrogate-based inversions in applications like seismic imaging where oscillatory media are common.
  • Training on simulated data may need validation against noisy real-world measurements to confirm robustness.

Load-bearing premise

The multiscale architecture allows the neural operator to generalize across different highly oscillatory media without retraining or suffering from spectral bias.

What would settle it

If on a new set of highly oscillatory media the prediction error of MscaleFNO for the wavefields is not lower than that of a standard FNO, or if the inversion fails to reconstruct the media accurately.

Figures

Figures reproduced from arXiv: 2606.08448 by Wei Cai, Zhenli Xu, Zilin You.

Figure 1
Figure 1. Figure 1: As a result, standard FNOs may struggle to accurately represent [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 1
Figure 1. Figure 1: Architecture of the FNO network. The model consists of a lifting [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: MscaleFNO architecture for approximating the operator [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Training error of FNO and MscaleFNO under different wavenumbers [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Representative medium sample α(x, y) with source and receiver locations. 0.0 0.2 0.4 0.6 0.8 1.0 x 0.0 0.2 0.4 0.6 0.8 1.0 y total wave (k=10) 0.4 0.2 0.0 0.2 0.4 0.6 0.8 (a) k = 10 0.0 0.2 0.4 0.6 0.8 1.0 x 0.0 0.2 0.4 0.6 0.8 1.0 y total wave (k=20) 0.4 0.2 0.0 0.2 0.4 0.6 0.8 (b) k = 20 0.0 0.2 0.4 0.6 0.8 1.0 x 0.0 0.2 0.4 0.6 0.8 1.0 y total wave (k=30) 0.4 0.2 0.0 0.2 0.4 0.6 0.8 (c) k = 30 0.0 0.2 0… view at source ↗
Figure 5
Figure 5. Figure 5: Representative wavefields in the dataset for different wavenumbers. [PITH_FULL_IMAGE:figures/full_fig_p014_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: The Gaussian representation preserves the localized nature of the [PITH_FULL_IMAGE:figures/full_fig_p015_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Forward operator relative error versus wavenumber [PITH_FULL_IMAGE:figures/full_fig_p016_7.png] view at source ↗
Figure 4
Figure 4. Figure 4: The clean observation data are generated by solving the Helmholtz [PITH_FULL_IMAGE:figures/full_fig_p017_4.png] view at source ↗
Figure 8
Figure 8. Figure 8: Inversion error across different wavenumbers [PITH_FULL_IMAGE:figures/full_fig_p018_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Cross-sectional comparisons of reconstruction results by [PITH_FULL_IMAGE:figures/full_fig_p019_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Inversion error versus wavenumber k under different noise levels. Limited-aperture configuration In practical settings, data acquisition is often confined to a limited angular range by physical and experimental constraints. This incomplete coverage, where sources and receivers span only a subset of directions, introduces additional ill-posedness. We consider a two-dimensional setting where sources and rec… view at source ↗
Figure 11
Figure 11. Figure 11: Illustration of source (s) and receiver (r) configurations under [PITH_FULL_IMAGE:figures/full_fig_p021_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Reconstruction comparisons on the cross-section [PITH_FULL_IMAGE:figures/full_fig_p023_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Reconstruction results under different strategies at [PITH_FULL_IMAGE:figures/full_fig_p024_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Convergence behavior of different strategies at [PITH_FULL_IMAGE:figures/full_fig_p025_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Fourier spectra of the outputs from different branches in [PITH_FULL_IMAGE:figures/full_fig_p026_15.png] view at source ↗
read the original abstract

In this paper, we propose an operator learning method based on the multiscale Fourier neural operator (MscaleFNO) for inverse medium problems of Helmholtz equations. The MscaleFNO provides a neural surrogate model with reduced spectral bias for the Helmholtz equations, mapping highly oscillatory medium profiles to scattered wavefields. A plug-and-play inversion using elucidated diffusion model is introduced to regularize the inverse solver based on least squares of data misfits. Numerical results for partial aperture inversion of oscillatory two-dimensional media demonstrate the advantage and effectiveness of MscaleFNO for accurate reconstruction of highly oscillatory medium properties.

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 / 1 minor

Summary. The manuscript proposes the multiscale Fourier neural operator (MscaleFNO) as a neural surrogate for the forward Helmholtz map in inverse medium scattering, mapping highly oscillatory medium profiles to scattered wavefields with reduced spectral bias relative to standard FNOs. This surrogate is embedded in a plug-and-play least-squares inversion regularized by an elucidated diffusion model. Numerical results are presented for partial-aperture reconstruction of 2D oscillatory media, with the claim that these demonstrate the method's advantage and effectiveness.

Significance. If the MscaleFNO architecture demonstrably reduces spectral bias and generalizes to oscillation frequencies outside the training distribution, the work would supply a practical data-driven forward model for inverse scattering problems where conventional discretizations become prohibitive. The diffusion-regularized inversion framework is a modular contribution. The significance is currently limited by the lack of explicit scaling or ablation evidence tying performance to oscillation frequency.

major comments (2)
  1. [Numerical experiments] Numerical experiments section: no ablation study or scaling analysis is reported that measures reconstruction or forward-map error as a function of oscillation frequency relative to the training distribution. This verification is load-bearing for the central claim that MscaleFNO mitigates spectral bias sufficiently to enable reliable inversion in highly oscillatory regimes beyond the reported 2D cases.
  2. [Abstract] Abstract: the statement that 'numerical results ... demonstrate the advantage and effectiveness' is not supported by any quantitative metrics, baseline comparisons against standard FNO, error bars, or specification of the training frequency band; without these the strength of the evidence for reduced spectral bias cannot be assessed.
minor comments (1)
  1. [Method] The description of the multiscale Fourier layers would benefit from an explicit diagram or equation block showing how the scale-specific kernels are combined, to improve reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback, which highlights opportunities to strengthen the evidence supporting the reduced spectral bias of MscaleFNO. We address each major comment below and will incorporate revisions to improve clarity and rigor.

read point-by-point responses

  1. Referee: [Numerical experiments] Numerical experiments section: no ablation study or scaling analysis is reported that measures reconstruction or forward-map error as a function of oscillation frequency relative to the training distribution. This verification is load-bearing for the central claim that MscaleFNO mitigates spectral bias sufficiently to enable reliable inversion in highly oscillatory regimes beyond the reported 2D cases.

    Authors: We agree that an explicit scaling analysis with respect to oscillation frequency would provide stronger verification of the central claim. In the revised manuscript, we will add an ablation study in the Numerical experiments section that reports forward-map and reconstruction errors as functions of oscillation frequency, explicitly including cases both within and outside the training distribution. This will directly address the load-bearing aspect of the claim regarding mitigation of spectral bias in highly oscillatory regimes. revision: yes

  2. Referee: [Abstract] Abstract: the statement that 'numerical results ... demonstrate the advantage and effectiveness' is not supported by any quantitative metrics, baseline comparisons against standard FNO, error bars, or specification of the training frequency band; without these the strength of the evidence for reduced spectral bias cannot be assessed.

    Authors: We acknowledge that the abstract phrasing is general and would benefit from more specific support. The numerical experiments section contains quantitative metrics and baseline comparisons, but we will revise the abstract to explicitly reference the key quantitative improvements (including error reductions relative to standard FNO), note the presence of error bars, and specify the training frequency band. This revision will make the evidence for reduced spectral bias more assessable while remaining faithful to the reported results. revision: yes

Circularity Check

0 steps flagged

No circularity: standard supervised operator learning with no self-referential reductions

full rationale

The paper proposes MscaleFNO as a neural surrogate for the Helmholtz forward map and a diffusion-regularized least-squares inversion. No derivation chain, equations, or fitted parameters are shown that reduce a claimed prediction to its own inputs by construction. The approach follows standard supervised operator learning on simulated data without load-bearing self-citations, uniqueness theorems imported from the authors, or ansatzes smuggled via prior work. The central claims rest on empirical performance in 2D cases rather than any self-definitional or fitted-input structure.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities; all such quantities remain unknown.

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discussion (0)

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