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arxiv: 2508.00268 · v4 · submitted 2025-08-01 · 💻 cs.IT · math.IT

Channel Estimation for Flexible Intelligent Metasurfaces: From Model-Based Approaches to Neural Operators

Jian Xiao , Ji Wang , Qimei Cui , Yucang Yang , Xingwang Li , Dusit Niyato , Chau Yuen This is my paper

Pith reviewed 2026-05-19 01:57 UTC · model grok-4.3

classification 💻 cs.IT math.IT
keywords channel estimationflexible intelligent metasurfacesFourier neural operatormillimeter-wave communicationsdeep learningwireless channel modelingmorphing surfaces
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The pith

A hierarchical Fourier neural operator learns continuous mappings from flexible metasurface shapes to millimeter-wave channel responses.

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

Flexible intelligent metasurfaces can dynamically alter their 3D shape to make multipath signals add constructively, but this requires accurate channel estimates over a vast continuous space of possible deformations. Model-based methods frame the task as interpolation or sparse recovery that exploits angular sparsity in millimeter-wave channels. The paper introduces a hierarchical Fourier neural operator that directly learns the operator mapping any FIM shape to the corresponding channel response. This data-driven approach captures multi-scale spatial features and produces higher accuracy with fewer pilots than the explicit model-based baselines. The learned operator also reveals an anisotropic filter adapted to the physical geometry of the surface.

Core claim

By parameterizing a global convolution operator in the Fourier domain, the hierarchical Fourier neural operator (H-FNO) learns the continuous mapping from FIM deformation shapes to channel responses; it captures multi-scale features across a hierarchy of resolutions and reconstructs non-linear responses without relying on fixed physical assumptions, yielding higher estimation accuracy and better pilot efficiency than interpolation, kernel, or sparse-recovery baselines.

What carries the argument

The hierarchical Fourier neural operator (H-FNO), which learns a mesh-independent continuous operator from FIM shapes to channel responses by composing Fourier-domain convolutions across spatial scales.

If this is right

  • FIM-assisted systems can achieve their intended performance gains using fewer pilots than model-based estimators require.
  • The operator reconstructs non-linear channel behavior across the full continuous deformation space once trained.
  • Interpretability analysis indicates the network implements an anisotropic spatial filter matched to the metasurface geometry.
  • Model-based approaches remain limited by the validity of their sparsity or interpolation assumptions, while the learned operator sidesteps those restrictions.

Where Pith is reading between the lines

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

  • The same neural-operator structure could be applied to other wireless problems where the environment varies continuously over high-dimensional parameter spaces.
  • Real-world deployment would require verifying that simulation-trained operators transfer to measured channels on physical prototypes.
  • If the approach scales, it points toward replacing explicit physics models with learned operators for propagation environments that are too complex to describe analytically.

Load-bearing premise

The collected training shapes must cover the continuous high-dimensional deformation space densely enough for the learned operator to generalize to unseen FIM configurations.

What would settle it

Evaluate the H-FNO on a set of FIM shapes deliberately excluded from the training distribution and check whether its channel estimation error rises above that of the model-based benchmarks.

Figures

Figures reproduced from arXiv: 2508.00268 by Chau Yuen, Dusit Niyato, Jian Xiao, Ji Wang, Qimei Cui, Xingwang Li, Yucang Yang.

Figure 1
Figure 1. Figure 1: FIM assisted multi-user mmWave communications. [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Frame structure for FIM channel estimation. [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Fourier neural operator (FNO) architecture for FIM c [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Hierarchical Fourier neural operator (H-FNO) archi [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: NMSE performance against SNR for different channel e [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: NMSE performance against pilot overhead for different channel estimation algorithms.               !    !    !    !   !   !   ×    [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: NMSE performance of different number of antennas [PITH_FULL_IMAGE:figures/full_fig_p011_7.png] view at source ↗
Figure 9
Figure 9. Figure 9: Fourier layer weight visualization of the proposed H [PITH_FULL_IMAGE:figures/full_fig_p012_9.png] view at source ↗
Figure 12
Figure 12. Figure 12: 3D visualization of channel gain over the joint ante [PITH_FULL_IMAGE:figures/full_fig_p012_12.png] view at source ↗
read the original abstract

Flexible intelligent metasurfaces (FIMs) offer a new solution for wireless communications by introducing morphological degrees of freedom, dynamically morphing their three-dimensional shape to ensure multipath signals interfere constructively. However, realizing the desired performance gains in FIM systems is critically dependent on acquiring accurate channel state information across a continuous and high-dimensional deformation space. Therefore, this paper investigates this fundamental channel estimation problem for FIM assisted millimeter-wave communication systems. First, we develop model-based frameworks that structure the problem as either function approximation using interpolation and kernel methods or as a sparse signal recovery problem that leverages the inherent angular sparsity of millimeter-wave channels. To further advance the estimation capability beyond explicit assumptions in model-based channel estimation frameworks, we propose a deep learning-based framework using a Fourier neural operator (FNO). By parameterizing a global convolution operator in the Fourier domain, we design an efficient FNO architecture to learn the continuous operator that maps FIM shapes to channel responses with mesh-independent properties. Furthermore, we exploit a hierarchical FNO (H-FNO) architecture to efficiently capture the multi-scale features across a hierarchy of spatial resolutions. Numerical results demonstrate that the proposed H-FNO significantly outperforms the model-based benchmarks in estimation accuracy and pilot efficiency. In particular, the interpretability analysis show that the proposed H-FNO learns an anisotropic spatial filter adapted to the physical geometry of FIM and is capable of accurately reconstructing the non-linear channel response across the continuous deformation space.

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 paper develops model-based channel estimation methods for flexible intelligent metasurfaces (FIMs) in mmWave systems, framing the task as function approximation via interpolation/kernel methods or as sparse recovery exploiting angular sparsity. It then proposes a hierarchical Fourier neural operator (H-FNO) that learns a mesh-independent global convolution operator in the Fourier domain to map arbitrary FIM morphologies to channel responses. Numerical results are reported to show that H-FNO significantly outperforms the model-based baselines in estimation accuracy and pilot overhead, with additional claims that the learned operator captures an anisotropic spatial filter adapted to FIM geometry and reconstructs nonlinear responses over the continuous deformation space.

Significance. If the generalization claims hold, the work would offer a practical route to channel estimation for dynamically morphing metasurfaces without requiring explicit physics-based modeling of every morphology. The mesh-independent property of the FNO architecture and the hierarchical multi-scale design constitute a clear technical advance over standard neural-network baselines for this high-dimensional continuous-parameter setting.

major comments (2)
  1. [Numerical Results] Numerical Results section: the central claim that H-FNO 'significantly outperforms' the model-based benchmarks and 'accurately reconstructs the non-linear channel response across the continuous deformation space' is presented without any reported dataset size, training/validation split, number of deformation samples, error bars, or explicit description of how the high-dimensional continuous deformation manifold (curvature modes, surface points, etc.) was sampled. This information is load-bearing for distinguishing operator learning from memorization of the training distribution.
  2. [Abstract / Interpretability analysis] Abstract and interpretability analysis: the post-hoc claim that H-FNO 'learns an anisotropic spatial filter adapted to the physical geometry of FIM' is stated without quantitative metrics, ablation studies, or comparison against physically expected filter shapes derived from the model-based baselines. This weakens the interpretability argument that is used to support the superiority of the learned operator.
minor comments (1)
  1. [Method] Notation for the hierarchical levels and Fourier modes in the H-FNO architecture should be defined more explicitly before the numerical experiments to improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed comments, which help improve the clarity and rigor of our work. We provide point-by-point responses to the major comments below and have revised the manuscript to address the concerns raised.

read point-by-point responses

  1. Referee: [Numerical Results] Numerical Results section: the central claim that H-FNO 'significantly outperforms' the model-based benchmarks and 'accurately reconstructs the non-linear channel response across the continuous deformation space' is presented without any reported dataset size, training/validation split, number of deformation samples, error bars, or explicit description of how the high-dimensional continuous deformation manifold (curvature modes, surface points, etc.) was sampled. This information is load-bearing for distinguishing operator learning from memorization of the training distribution.

    Authors: We agree that these details are essential for reproducibility and to substantiate the generalization claims. The original manuscript emphasized performance comparisons but omitted a full description of the experimental setup. In the revised version, we have expanded the Numerical Results section with the total number of deformation samples generated, the training/validation/test split, error bars over multiple runs, and an explicit account of how the continuous deformation manifold was sampled (via parameterization of curvature modes and surface points). revision: yes

  2. Referee: [Abstract / Interpretability analysis] Abstract and interpretability analysis: the post-hoc claim that H-FNO 'learns an anisotropic spatial filter adapted to the physical geometry of FIM' is stated without quantitative metrics, ablation studies, or comparison against physically expected filter shapes derived from the model-based baselines. This weakens the interpretability argument that is used to support the superiority of the learned operator.

    Authors: We acknowledge that the interpretability section would benefit from additional quantitative support. The manuscript already contains visualizations of the learned filters, but we agree these are primarily qualitative. In the revised manuscript, we have added quantitative alignment metrics between the learned filters and physically motivated anisotropic patterns, ablation studies on the hierarchical components, and explicit comparisons of the operator responses to those obtained from the model-based baselines. revision: yes

Circularity Check

0 steps flagged

No circularity: H-FNO performance is empirical validation of a data-driven operator, not a reduction to fitted inputs or self-citations

full rationale

The paper first presents explicit model-based estimators (interpolation/kernel and sparse recovery) that embed physical assumptions, then introduces a separate data-driven H-FNO that learns a mesh-independent operator directly from sampled training pairs of FIM shapes and channel responses. The reported outperformance is obtained by numerical comparison on test instances; no equation or claim equates the learned operator to its training inputs by construction, nor does any load-bearing step rest on a self-citation whose content is itself unverified. The derivation chain therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The approach rests on the assumption that mmWave channels exhibit angular sparsity and that a neural operator can learn a mesh-independent mapping from shape parameters to channel responses without explicit physics-based modeling.

axioms (2)
  • domain assumption Millimeter-wave channels are sparse in the angular domain
    Invoked to justify the sparse signal recovery model-based baseline.
  • domain assumption The mapping from FIM shape to channel response is a continuous operator that can be learned from discrete samples
    Central premise enabling the FNO and H-FNO frameworks.

pith-pipeline@v0.9.0 · 5810 in / 1281 out tokens · 17427 ms · 2026-05-19T01:57:35.884236+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

34 extracted references · 34 canonical work pages

  1. [1]

    Holographic MIM O communications: Theoretical foundations, enabling techn ologies, and future directions,

    T. Gong, P . Gavriilidis, R. Ji, C. Huang, G. C. Alexandrop oulos, L. Wei, Z. Zhang, M. Debbah, H. V . Poor, and C. Y uen, “Holographic MIM O communications: Theoretical foundations, enabling techn ologies, and future directions,” IEEE Commun. Surv. Tutorials , vol. 26, no. 1, pp. 196–257, 2024

    work page 2024

  2. [2]

    Stacked intelligent metasurface-aided MIMO transceiver design,

    J. An, C. Y uen, C. Xu, H. Li, D. W. K. Ng, M. Di Renzo, M. Debba h, and L. Hanzo, “Stacked intelligent metasurface-aided MIMO transceiver design,” IEEE Wireless Commun. , vol. 31, no. 4, pp. 123–131, Aug. 2024

    work page 2024

  3. [3]

    Covert communications with enhanced physic al layer security in RIS-assisted cooperative networks,

    X. Li, M. Liu, S. Dang, N. C. Luong, C. Y uen, A. Nallanathan , and D. Niyato, “Covert communications with enhanced physic al layer security in RIS-assisted cooperative networks,” IEEE Trans. Wireless Commun., vol. 24, no. 7, pp. 5605–5619, Jul. 2025

    work page 2025

  4. [4]

    Wideband beam forming for RIS assisted near-field communications,

    J. Wang, J. Xiao, Y . Zou, W. Xie, and Y . Liu, “Wideband beam forming for RIS assisted near-field communications,” IEEE Trans. Wireless Commun., vol. 23, no. 11, pp. 16 836–16 851, Dec. 2024

    work page 2024

  5. [5]

    STAR-RIS-assisted covert wireless co mmu- nications with randomly distributed blockages,

    X. Li, J. Zhao, G. Chen, W. Hao, D. B. d. Costa, A. Nallanath an, H. Shin, and C. Y uen, “STAR-RIS-assisted covert wireless co mmu- nications with randomly distributed blockages,” IEEE Trans. Wireless Commun., vol. 24, no. 6, pp. 4690–4705, Jun. 2025

    work page 2025

  6. [6]

    Flexible metasurfaces for mul tifunctional interfaces,

    Y . Zhou, S. Wang, J. Yin, J. Wang, F. Manshaii, X. Xiao, T. Z hang, H. Bao, S. Jiang, and J. Chen, “Flexible metasurfaces for mul tifunctional interfaces,” ACS Nano , vol. 18, no. 4, pp. 2685–2707, Jan. 2024

    work page 2024

  7. [7]

    Flexible intelligent microwave metasurface with shape- guided adaptive programming,

    F. Li, T. Pan, W. Li, Z. Peng, D. Guo, X. Jia, T. Hu, L. Wang, W . Wang, M. Gao et al. , “Flexible intelligent microwave metasurface with shape- guided adaptive programming,” Nat. Commun. , vol. 16, no. 1, p. 3161, Apr. 2025

    work page 2025

  8. [8]

    Emergi ng technologies in intelligent metasurfaces: Shaping the fut ure of wireless communications,

    J. An, M. Debbah, T. J. Cui, Z. N. Chen, and C. Y uen, “Emergi ng technologies in intelligent metasurfaces: Shaping the fut ure of wireless communications,” IEEE Trans. Antennas Propag. , 2025

    work page 2025

  9. [9]

    Flexible intelligent metasurfaces for downlink multiuse r MISO com- munications,

    J. An, C. Y uen, M. D. Renzo, M. Debbah, H. V . Poor, and L. Han zo, “Flexible intelligent metasurfaces for downlink multiuse r MISO com- munications,” IEEE Trans. Wireless Commun. , vol. 24, no. 4, pp. 2940– 2955, Apr. 2025

    work page 2025

  10. [10]

    Flexible intelligent metasurfaces for enhancing MIMO communicatio ns,

    J. An, Z. Han, D. Niyato, M. Debbah, C. Y uen, and L. Hanzo, “Flexible intelligent metasurfaces for enhancing MIMO communicatio ns,” IEEE Trans. Commun., 2025

    work page 2025

  11. [11]

    Flexibl e intelligent metasurface for enhancing multi-target wireless sensing,

    Z. Teng, J. An, L. Gan, N. Al-Dhahir, and Z. Han, “Flexibl e intelligent metasurface for enhancing multi-target wireless sensing, ” IEEE Trans. V eh. Technol., pp. 1–6, 2025

    work page 2025

  12. [12]

    Decou- pling optical function and geometrical form using conforma l flexible dielectric metasurfaces,

    S. M. Kamali, A. Arbabi, E. Arbabi, Y . Horie, and A. Farao n, “Decou- pling optical function and geometrical form using conforma l flexible dielectric metasurfaces,” Nat. Commun. , vol. 7, no. 1, p. 11618, May 2016

    work page 2016

  13. [13]

    All-die lectric ultrathin conformal metasurfaces: lensing and cloaking application s at 532 nm wavelength,

    J. Cheng, S. Jafar-Zanjani, and H. Mosallaei, “All-die lectric ultrathin conformal metasurfaces: lensing and cloaking application s at 532 nm wavelength,” Sci. Rep. , vol. 6, no. 1, p. 38440, Dec. 2016

    work page 2016

  14. [14]

    Large-area and low-cost nanoslit-based fl exible metasurfaces for multispectral electromagnetic wave mani pulation,

    M. Zhang, X. Ma, M. Pu, K. Liu, Y . Guo, Y . Huang, X. Xie, X. L i, H. Y u, and X. Luo, “Large-area and low-cost nanoslit-based fl exible metasurfaces for multispectral electromagnetic wave mani pulation,” Adv. Opt. Mater ., vol. 7, no. 23, p. 1900657, Sep. 2019

    work page 2019

  15. [15]

    Flexible ac tive antenna arrays,

    M. Gal-Katziri, A. Fikes, and A. Hajimiri, “Flexible ac tive antenna arrays,” npj Flexible Electron. , vol. 6, no. 1, p. 85, Oct. 2022

    work page 2022

  16. [16]

    Soft shape-programmable surfaces b y fast electro- magnetic actuation of liquid metal networks,

    a. o. Ni, Xinchen, “Soft shape-programmable surfaces b y fast electro- magnetic actuation of liquid metal networks,” Nat. Commun. , vol. 13, no. 1, p. 5576, Sep. 2022

    work page 2022

  17. [17]

    A dynamically reprogrammable surface with self-evolvin g shape morphing,

    Y . Bai et al., “A dynamically reprogrammable surface with self-evolvin g shape morphing,” Nature, vol. 609, no. 7928, pp. 701–708, Sep. 2022

    work page 2022

  18. [18]

    Movable antenna-enhanced wireless communications: Gene ral archi- tectures and implementation methods,

    B. Ning, S. Y ang, Y . Wu, P . Wang, W. Mei, C. Y uen, and E. Bjo rnson, “Movable antenna-enhanced wireless communications: Gene ral archi- tectures and implementation methods,” IEEE Wireless Commun. , 2025

    work page 2025

  19. [19]

    MIMO evolution beyond 5G through reconfigurable intelligent surfaces and fluid antenna systems,

    Shojaeifard et al. , “MIMO evolution beyond 5G through reconfigurable intelligent surfaces and fluid antenna systems,” Proc. IEEE , vol. 110, no. 9, pp. 1244–1265, Sep. 2022

    work page 2022

  20. [20]

    Dynamic channel modeling of fluid antenna systems in UA V com - munications,

    H. Jiang, W. Shi, Z. Chen, Z. Zhang, K.-K. Wong, and H. Shi n, “Dynamic channel modeling of fluid antenna systems in UA V com - munications,” IEEE Wireless Commun. Lett. , 2025

    work page 2025

  21. [21]

    Flexible-ant enna systems: A pinching-antenna perspective,

    Z. Ding, R. Schober, and H. Vincent Poor, “Flexible-ant enna systems: A pinching-antenna perspective,” IEEE Trans. Commun. , 2025

    work page 2025

  22. [22]

    Channel estimation for pin ching-antenna systems (PASS),

    J. Xiao, J. Wang, and Y . Liu, “Channel estimation for pin ching-antenna systems (PASS),” IEEE Communications Letters , 2025

    work page 2025

  23. [23]

    Flexible antenna arrays for wireless communic ations: Modeling and performance evaluation,

    S. Y ang, J. An, Y . Xiu, W. Lyu, B. Ning, Z. Zhang, M. Debbah , and C. Y uen, “Flexible antenna arrays for wireless communic ations: Modeling and performance evaluation,” IEEE Trans. Wireless Commun., vol. 24, no. 6, pp. 4937–4951, Jun. 2025

    work page 2025

  24. [24]

    3GPP TR 38.901 V16.1.0 - Study on channel model for freq uencies from 0.5 to 100 GHz,

    “3GPP TR 38.901 V16.1.0 - Study on channel model for freq uencies from 0.5 to 100 GHz,” Dec. 2019

    work page 2019

  25. [25]

    Unified array man ifold de- composition based on spherical harmonics and 2-d fourier ba sis,

    M. Costa, A. Richter, and V . Koivunen, “Unified array man ifold de- composition based on spherical harmonics and 2-d fourier ba sis,” IEEE Transactions on Signal Processing , vol. 58, no. 9, pp. 4634–4645, Sep. 2010

    work page 2010

  26. [26]

    De Berg, O

    M. De Berg, O. Cheong, M. V an Kreveld, and M. Overmars, Computa- tional geometry: algorithms and applications . Springer, 2008

    work page 2008

  27. [27]

    Hastie, R

    T. Hastie, R. Tibshirani, J. H. Friedman, and J. H. Fried man, The elements of statistical learning: data mining, inference, and prediction. Springer, 2009, vol. 2

    work page 2009

  28. [28]

    C. M. Bishop and N. M. Nasrabadi, Pattern recognition and machine learning. Springer, 2006, vol. 4, no. 4

    work page 2006

  29. [29]

    Multi-user precoding a nd channel estimation for hybrid millimeter wave systems,

    L. Zhao, D. W. K. Ng, and J. Y uan, “Multi-user precoding a nd channel estimation for hybrid millimeter wave systems,” IEEE J. Sel. Areas Commun., vol. 35, no. 7, pp. 1576–1590, Feb. 2017

    work page 2017

  30. [30]

    Signal recovery from rand om mea- surements via orthogonal matching pursuit,

    J. A. Tropp and A. C. Gilbert, “Signal recovery from rand om mea- surements via orthogonal matching pursuit,” IEEE Trans. Inf. Theory , vol. 53, no. 12, pp. 4655–4666, Dec. 2007

    work page 2007

  31. [31]

    H. V . Poor, An introduction to signal detection and estimation . Springer Science & Business Media, 2013

    work page 2013

  32. [32]

    Fourier ne ural operator with learned deformations for pdes on general geometries,

    Z. Li, D. Z. Huang, B. Liu, and A. Anandkumar, “Fourier ne ural operator with learned deformations for pdes on general geometries,” J. Mach. Learn. Res. , vol. 24, no. 388, pp. 1–26, 2023

    work page 2023

  33. [33]

    Overview of AI and communication for 6G network: fundamentals, challenges, and future research opportunit ies,

    Q. Cui et al. , “Overview of AI and communication for 6G network: fundamentals, challenges, and future research opportunit ies,” Sci. China Inf. Sci. , vol. 68, no. 7, p. 171301, Apr. 2025

    work page 2025

  34. [34]

    U-net: Convol utional networks for biomedical image segmentation,

    O. Ronneberger, P . Fischer, and T. Brox, “U-net: Convol utional networks for biomedical image segmentation,” in International Conference on Medical image computing and computer-assisted intervention. Springer, 2015, pp. 234–241

    work page 2015