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Rethinking Channel Estimation for Fluid Antenna Systems: A Physics-Informed Operator Learning

Zixin Wang, Lizhong Zheng, Yuanming Shi, Kaibin Huang, Khaled B. Letaief

arXiv:2610.36787v1cs.ITmath.IT

TL;DR

Sparse observations make full-aperture FAS channel reconstruction an underdetermined problem. The paper formulates it as continuous-field operator learning and proposes UFNO; it reports improved reconstruction and lower latency than model-based baselines.

  • Problem

    FAS requires high-resolution channel information across the aperture, but hardware allows only sparse observations, leaving estimation severely underdetermined.

    Source: §I · p. 1

  • Method

    The paper formulates FAS estimation as a stochastic inverse field-reconstruction problem and proposes UFNO with soft-thresholding, dual spectral-spatial branches, and a physics-aware Sobolev loss.

    Source: §I · p. 2

  • Results

    The proposed framework achieves grid-agnostic reconstruction and orders-of-magnitude lower latency than model-based baselines.

    Source: §I · p. 2

  • Takeaways & Limitations

    An aperture-dependent EDoF bound provides a physics-based reference for choosing UFNO's spatial mode budget.

    Source: §I · p. 2

Abstract

from arXiv · show

Fluid antenna systems (FAS) leverage continuous reconfigurable apertures to transcend the spatial limits of fixed-position arrays, yet acquiring full-aperture channel state information (CSI) from sparse observations remains a formidable bottleneck. In this paper, we establish a continuous-field formulation for FAS by recasting channel estimation as a stochastic inverse field-reconstruction problem constrained by the projected Helmholtz model. Through electromagnetic information theory (EIT)-based spectral leakage analysis, we demonstrate that finite-aperture reconstruction induces an intrinsic convolutional relationship in the spectral domain, rendering the estimation task a fundamental frequency-domain inverse deconvolution problem. To solve this, we propose a U-shaped Fourier neural operator (UFNO) specifically tailored for noisy, underdetermined wireless environments. Furthermore, we leverage the Landau--Pollak theorem to derive an aperture-dependent EDoF bound that guides the spatial mode budget, and employ a physics-aware Sobolev loss to promote consistency in both channel values and spatial variations. Unlike conventional learning-based approaches, a single frozen UFNO can be directly evaluated at unseen spatial--frequency resolutions without resolution-specific retraining or explicit output interpolation. Extensive simulations show that our framework significantly outperforms state-of-the-art benchmarks in accuracy while achieving orders-of-magnitude lower inference latency than iterative solvers at high spatial--frequency resolutions.

I. Introduction

FAS can reduce the hardware burden of dense apertures, but estimating its full channel field from sparse measurements remains underdetermined. The paper motivates a continuous-field formulation and operator learning for this setting.

I. Introduction

  • FAS uses movable antennas within a continuous aperture to reduce hardware overhead, but sparse measurements leave full-domain channel estimation severely underdetermined.Dense CAP-MIMO also faces mutual coupling, near-field interactions, and high pilot and computational costs.Source: §I · p. 1
  • Compressed-sensing estimators face grid mismatch, high computational complexity, and reliance on sparsity assumptions that limit their applicability to shallow-scattering environments.Learning-based methods can require iterative sampling or site-specific optimization, while existing physical and kernel methods also face computational or resolution-generalization constraints.Source: §I · p. 2

A. Channel Propagation as a Physical Field

  • The paper models a wideband FAS link as a continuous complex channel field over a one-dimensional aperture and OFDM frequency band, observed through a limited number of RF chains.The physical-field formulation addresses limitations of discrete point-scatterer models, which can miss diffuse scattering and near-field interactions relevant to sub-wavelength positioning.Source: A. Channel Propagation as a Physical… · pp. 3–4

B. EIT-based SPDE Formulation

The channel field is modeled as a continuous integral over scattering density, with its spatial evolution constrained by a projected Helmholtz relation. Sparse RF measurements and a stochastic source together define the reconstruction problem.

B. EIT-based SPDE Formulation

  • Electromagnetic information theory represents the channel field as a continuous integral over scattering density in the angular-delay domain.The integral's phase kernel represents plane-wave eigenfunctions of the Helmholtz equation, and discrete path models are sampled realizations of this continuous representation.Source: B. EIT-based SPDE Formulation · p. 4
  • Evaluating the source-free Helmholtz field on the one-dimensional aperture yields a projected equation governing spatial channel evolution.The wavenumber is κ(f) = 2π(fc + f)/c.Source: B. EIT-based SPDE Formulation · p. 4
  • Channel estimation couples the projected Helmholtz relation to sparse measurements of the field over the spatio-frequency domain.The transverse wave dynamics are modeled as a stationary, zero-mean complex Gaussian random field with covariance induced by the scattering environment.Source: B. EIT-based SPDE Formulation · p. 4

C. From Physical Principles to Operator Learning

Finite-aperture observation convolves the ideal channel spectrum, making reconstruction an inverse deconvolution problem. The paper also derives an aperture-dependent spatial-dimension bound and uses the physical operator structure to motivate continuous-field operator learning.

C. From Physical Principles to Operator Learning

  • Finite-aperture truncation convolves the ideal band-limited spectrum with the spatial-window spectrum, causing spectral broadening and making reconstruction an inverse deconvolution problem.The convolution creates nonzero components beyond the theoretical cutoff wavenumber.Source: C. From Physical Principles to… · p. 4
  • The Landau-Pollak characterization gives an aperture-dependent upper bound on the continuous field's spatial dimensionality and a necessary observation count for perfect linear reconstruction.The effective dimension is ⌈2W⌉ + 1; when M is smaller, the sampling map has a nontrivial null space.Source: C. From Physical Principles to… · p. 5
  • The continuous-field operator formulation learns channel-distribution priors and a Helmholtz-motivated non-local mapping, without enforcing an explicit PDE residual during training.Sparse pilot samples are measurement constraints, not prescribed boundary values; the unknown stochastic excitation makes a homogeneous residual constraint inappropriate.Source: C. From Physical Principles to… · p. 5
  • Meeting the EDoF observation count does not guarantee stable recovery, because finite-aperture inversion can amplify noise as singular values decay.The physical singular functions also differ from UFNO's retained discrete Fourier coefficients, so EDoF guides rather than fixes its mode budget.Source: C. From Physical Principles to… · p. 5

III. Proposed Operator Learning Framework for Continuous SFC Reconstruction

The framework casts channel estimation as operator learning between infinite-dimensional function spaces. It parameterizes the operator efficiently in the spectral domain.

III. Proposed Operator Learning Framework for Continuous SFC Reconstruction

  • The framework formulates channel estimation as operator learning between infinite-dimensional function spaces to address FAS's continuous aperture.Source: §III · pp. 5–6
  • The network architecture parameterizes this operator efficiently in the spectral domain.Source: §III · p. 6

A. Operator Learning Formulation

The section formulates FAS channel reconstruction as a resolution-invariant operator mapping sparse pilot observations to continuous spatial-frequency channel fields. It motivates Fourier operator learning and outlines the UFNO’s multiscale architecture.

A. Operator Learning Formulation

  • The proposed resolution-invariant operator uses an FNO to map sparse pilot observations to continuous SFC fields.It approximates the inverse mapping from the observation space to the SPDE solution manifold.Source: A. Operator Learning Formulation · p. 6
  • The formulation models noisy observation functions and ground-truth channel fields in separable Banach spaces.The input and output spaces are defined over a bounded continuous spatio-frequency domain.Source: A. Operator Learning Formulation · p. 6
  • The optimal operator parameters minimize an expectation over random channel fields with an added regularization term.The objective uses the functional-space L2 norm and a regularizer.Source: A. Operator Learning Formulation · p. 6
  • FNO spectral convolution provides a non-local reconstruction parameterization motivated by the Green’s-function representation.The section links this structure to the continuous nature of the SFC.Source: A. Operator Learning Formulation · p. 6
  • A learned operator can be evaluated on unseen spatial-frequency grids without resolution-specific retraining or explicit output interpolation.The passage describes this property as resolution invariance.Source: A. Operator Learning Formulation · p. 6
  • The UFNO uses 2P blocks, with P down-sampling blocks followed by P up-sampling blocks to reconstruct the high-resolution channel field.The architecture proceeds through state initialization, dual-branch representation learning, and multiscale field projection.Source: B. UFNO Architecture Design · p. 6

1) State initialization:

The UFNO initializes a feature tensor from sparse, noisy observations, then processes it with stacked Fourier blocks. Its spectral and local branches combine global propagation structure with local channel detail, with adaptive thresholding and mode selection shaping the spectral representation.

1) State initialization:

  • The input tensor is built from sparse, noisy observations and lifted into a higher-dimensional latent representation.Its channels include pilot components, a sampling mask, spatial-frequency coordinates, and a local SNR indicator.Source: 1) State initialization · p. 6
  • Removing dynamic zero-padding worsens overall, boundary-region, and interior NMSE by 1.45, 1.59, and 1.33 dB, respectively.The reported edge ringing is strongest at boundaries but also propagates into the interior through the global spectral operator.Source: 1) State initialization · p. 6

2) Dual-Branch Fourier Representation Learning:

  • Stacked Fourier blocks form the UFNO’s representation-learning engine, with parallel branches for global wave propagation and local scattering details.The global spectral branch is paired with a point-wise MLP that operates along the channel dimension without expanding the receptive field.Source: 2) Dual-Branch Fourier Representation… · pp. 6–7
  • The global branch transforms features into a mixed spectrum and retains a low-mode subset to reduce complexity and suppress out-of-band components.The spatial and frequency mode budgets are selected with reference to the aperture-dependent NDoF, but are not set equal to it because the bases differ and recovery also depends on SNR and representation capacity.Source: 2) Dual-Branch Fourier Representation… · pp. 6–7
  • Learnable spectral soft-thresholding suppresses weak, noise-dominated coefficients while retaining dominant global scattering structure.The threshold is shared across channels and modes; mode truncation supplies low-pass support, while thresholding performs amplitude-adaptive shrinkage.Source: 2) Dual-Branch Fourier Representation… · p. 7
  • The Fourier branch’s spectral-domain multiplication corresponds to global spatial convolution, allowing field points to aggregate information across the aperture.The figure also shows skip connections intended to preserve high-frequency details during reconstruction.Source: 2) Dual-Branch Fourier Representation… · p. 7, Figure 2 · p. 7

3) U-Shaped Multiscale Fusion:

The UFNO combines encoder-decoder multiscale features and skip connections to reconstruct the continuous field, then projects latent features to the physical SFC domain. Training uses a Sobolev loss on field values and derivatives, and the section considers evaluation cost across grid sizes.

3) U-Shaped Multiscale Fusion:

  • Encoder features are added to corresponding decoder inputs so multiscale fusion preserves spatial phase alignment and sub-wavelength fading details.The reconstructed tensor is cropped to physical dimensions and projected point-wise into the SFC domain.Source: 3) U-Shaped Multiscale Fusion · p. 8

C. Loss Design and Complexity Analysis

1) Sobolev Loss Design:

  • The Sobolev loss constrains both reconstructed channel values and their derivatives, addressing oversmoothing associated with pixel-wise MSE objectives.Spatial and frequency gradients correspond to local spatial phase progression and frequency-selective variation; the regularizer is not a Helmholtz-residual penalty.Source: 1) Sobolev Loss Design · p. 8

C. Loss Design and Complexity Analysis

  • The complexity analysis evaluates operator computation on a grid of N = Ns × Nf points to characterize scalability as discretization approaches the continuum limit.Source: 1) Sobolev Loss Design · p. 8

2) Parameter Complexity:

UFNO’s per-layer computational cost grows quasilinearly with grid size, while its learned parameter count is independent of spatial resolution. Its Fourier basis provides a global receptive field in one layer.

2) Parameter Complexity:

  • UFNO’s parameter complexity is independent of spatial resolution N, allowing coarse-grid training and fine-grid deployment without parameter rescaling.The learned parameters do not need to be rescaled as the evaluation grid is refined.Source: 2) Parameter Complexity · p. 8

3) Computational Complexity:

  • UFNO achieves a global receptive field in one layer, whereas a single CNN layer captures only local features.The contrast reflects the global Fourier basis versus the CNN’s need to stack layers for global coverage.Source: 3) Computational Complexity · p. 8
  • UFNO scales as O(N log N), versus ViT’s O(N^2) and CNN’s effective O(N^1.5 K d_h^2) for global multipath correlations.The CNN’s effective complexity loses its linear advantage when enough layers are stacked to capture global correlations.Source: 3) Computational Complexity · pp. 8–9

IV. Simulation Results

The simulation section evaluates UFNO in a geometry-based multipath setting and specifies its evaluation protocol and comparison methods. The supplied passages describe the setup but do not report comparative outcomes.

A. Simulation Setup

  • The evaluation uses a 4λ aperture discretized into 48 spatial points and 48 OFDM subcarriers, with 20 paths and 4-spaced pilot sampling.The geometry-based setup uses a 3.5 GHz carrier, 15 kHz subcarrier spacing, and 300 ns delay spread.Source: A. Simulation Setup · p. 9
  • The reported UFNO configuration assigns 99.6% of trainable parameters to spectral convolution and 0.4% to point-wise MLP skips.The 640 learnable soft-threshold scalars add negligible overhead and enable adaptive spectral denoising.Source: A. Simulation Setup · p. 9
  • The comparison set includes signal-processing baselines, CNN and ViT, and UFNO ablations without SST or Sobolev loss.Signal-processing methods include LS interpolation, OMP/SBL recovery, and DC-GOMP.Source: A. Simulation Setup · p. 9
  • All learning methods use the same dataset and evaluation conditions, with results averaged over 5,120 independent channel realizations.CNN and ViT are representative fixed baselines, not products of exhaustive task-specific hyperparameter searches; the test set is not used for model selection.Source: A. Simulation Setup · p. 9
  • Table II frames physics-guided selection of the total two-sided spatial mode budget using complex NMSE across apertures and SNRs, with lower values preferred.The supplied table passage gives the metric and comparison scope but no cell values.Source: Table II · p. 10

B. Physics-Guided Spatial Mode Selection

The aperture-dependent EDoF provides a physical reference for selecting UFNO’s spatial mode budget, with the useful capacity depending on SNR. UFNO also maintains strong benchmark performance across SNR and delay-spread tests, with degradation at extreme delay spread.

B. Physics-Guided Spatial Mode Selection

  • Above the EDoF bound, moving from below it improves NMSE across W=2, 4, and 6 at 20 and 30 dB; at 0 dB, added modes give no systematic gain.The reported improvements are 0.56, 1.26, and 1.02 dB at 20 dB, and 0.69, 1.64, and 1.64 dB at 30 dB for W=2, 4, and 6, respectively. EDoF indicates the mode budget needed to avoid undersizing, while SNR governs how much capacity can be used.Source: B. Physics-Guided Spatial Mode Selection · p. 10

C. Performance Comparison

  • At 30 dB SNR, UFNO achieves NMSE below −35 dB and outperforms ViT and CNN by over 5 dB.At 30 dB, the full model also leads UFNO w/o SST and UFNO w/o S-Loss by approximately 3 dB; SST suppresses spectral leakage, while Sobolev loss preserves high-gradient structures.Source: Figure 3 · p. 10, C. Performance Comparison · pp. 10–11
  • UFNO maintains a dominant advantage over ViT and CNN from 100–500 ns, and multi-order-of-magnitude NMSE improvements over CS across the evaluated delay-spread domain.At the extreme 800 ns condition, UFNO degrades as low-pass truncation cuts off emerging high-frequency components.Source: Figure 4 · p. 10, C. Performance Comparison · p. 11

D. Sampling Ratio

As sampling density rises, UFNO achieves the strongest reported NMSE, while reconstruction progresses from coarse envelopes to structured multipath textures. The section compares methods over sampling ratios from 1% to 20% and includes uniform and random resource-block patterns.

D. Sampling Ratio

  • The tested sampling ratios span 1%–20%, and both uniform and random resource-block occupation patterns are evaluated.Sampling ratio is defined using occupied spatio-frequency blocks relative to all available blocks.Source: D. Sampling Ratio · p. 11
  • At 20% sampling, UFNO achieves NMSE below −26 dB and significantly outperforms CNN and ViT.At that ratio, the full model outperforms UFNO w/o SST and UFNO w/o S-Loss by approximately 13–14 dB.Source: Figure 5 · p. 11, D. Sampling Ratio · p. 11
  • Across 1%–20% sampling, complex NMSE is averaged over 5,120 test channels; LS is nearly flat, while OMP and SBL gains are marginal and jagged.The passage attributes the CS behavior to rigid sparsity constraints and grid mismatch under discrete uniform masking.Source: D. Sampling Ratio · p. 11
  • From 1% to 20% sampling, the heatmaps show recovery shifting from coarse envelopes to structured multipath textures, with NMSE improving from approximately −1.9 dB to −29.5 dB.The caption specifies that panels show one randomly selected channel realization, so its annotated NMSE values are realization-specific.Source: D. Sampling Ratio · pp. 11–12, Figure 6 · p. 12

E. Resolution Invariance

A single frozen UFNO reconstructs channels across unseen spatial–frequency resolutions, outperforming the listed direct-inference and interpolation baselines. Its measured latency remains nearly resolution-independent, while the conclusion identifies measured-channel validation and time-varying channels as future work.

E. Resolution Invariance

  • The controlled comparison evaluates 512 channels at multiple resolutions using estimators trained only at 48 × 48, without parameter updates or resolution-specific fine-tuning.The study also tests sparse-pilot interpolation and resizing of dense CNN estimates as distinct controls.Source: E. Resolution Invariance · p. 12
  • Across five unseen resolutions, UFNO averages −29.39 dB NMSE, outperforming direct CNN, linear interpolation, cubic interpolation, and direct Standard FNO by 5.01, 7.86, 8.32, and 21.67 dB, respectively.It also exceeds CNN bilinear and bicubic upsampling by 4.90 and 4.93 dB.Source: E. Resolution Invariance · p. 12
  • Fig. 7 shows smooth reconstructions retaining dominant channel structures from 60 × 60 to 144 × 144, with visible mismatch emerging at 160 × 160.At 160 × 160, the retained Fourier-mode budget covers a smaller relative spectral portion and the test setting shifts further from training.Source: Figure 7 · p. 13, E. Resolution Invariance · p. 13
  • UFNO’s inter-layer skip connections and spectral suppression limit accumulated discretization mismatch, whereas Standard FNO degrades at every unseen resolution despite strong native-resolution accuracy.The paper attributes this difference to short paths for intermediate representations and suppression of low-energy coefficients sensitive to sampling changes and noise.Source: E. Resolution Invariance · p. 13

F. Computational Efficiency

V. Conclusion

  • The conclusion frames channel estimation as continuous physical-field reconstruction and presents UFNO, a learnable SST, a dual-branch architecture, and physics-aware Sobolev loss.The loss promotes consistency in channel values and spatial–frequency variations.Source: §V · p. 14
  • Future work will validate UFNO on measured channels, consider practical-receiver integration and planar or volumetric FAS, and develop spatio-temporal operators for time-varying channels.The fixed Fourier-basis representation may be insufficient for coupled non-stationarity across space, frequency, and time.Source: §V · p. 14

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