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HankelKANet: Lightweight Path Loss Estimation via Physics-Inspired Hankel Convolution

Yue Ruan, Xiaoxiao Cheng, Yunan Zhu, Wenlong Huang, Meng Hua

arXiv:2610.19524v1cs.LGcs.NI

TL;DR

Path-loss estimation matters for wireless network planning, but existing data-driven methods depend on their training distributions and lack physical constraints. HankelKANet embeds Hankel-based propagation priors into a lightweight network; it reports strong results on simulated and real-world data, alongside smoothness and transferability findings.

  • Problem

    Path-loss estimation is important for wireless planning, while data-driven approaches rely on training-data distributions and lack physical constraints.

    Source: §I · p. 1

  • Method

    HankelKANet embeds Hankel-guided learnable basis functions in convolutional kernels and combines them with urban-layout features in a lightweight network.

    Source: §I · pp. 1–2

  • Results

    HankelKANet achieves state-of-the-art performance on simulated RadioMapSeer and real-world Huawei RSRPSet; on RadioMapSeer, it reduces RMSE by 16.96% versus PEFNet.

    Source: §I · p. 2, 1) Results on RadioMapSeer · p. 7

  • Takeaways & Limitations

    The paper links Hankel-based structural smoothness to reduced overfitting to simulator artifacts and robust cross-dataset generalization, including target-domain adaptation.

    Source: §VII · pp. 12–13

Abstract

from arXiv · show

Accurate path loss (PL) estimation is essential for wireless network planning and optimization. This paper proposes HankelKANet, a lightweight physics-inspired network for PL estimation that structurally embeds electromagnetic propagation characteristics into the neural architecture. Built on a single residual UNet backbone, the proposed model integrates urban layout features with a convolutional Kolmogorov-Arnold network(KAN). Specifically, the convolutional kernels are parameterized by the radial amplitudes of the zeroth- and first-order Hankel functions. This formulation acts as a physics-inspired nonlinear distance encoding, enabling the network to capture dominant attenuation characteristics directly without requiring additional physics-based loss terms. Experimental results demonstrate that HankelKANet achieves state-of-the-art accuracy and efficiency on both the simulated RadioMapSeer dataset and the real-world Huawei RSRPSet. Compared with the leading baseline, PEFNet, it reduces the parameter count by 78% and lowers the RMSE by 16.96% on the simulated data, while further exhibiting robust cross-dataset transferability and high data efficiency during target-domain adaptation. To interpret these empirical gains, we mathematically establish a parameter-conditional spatial gradient bound. Using discrete Dirichlet energy (DDE) and Jacobian sensitivity heatmaps, we empirically verify that this physical encoding intrinsically constrains the network's spatial derivatives. This structural smoothness effectively eliminates non-physical spatial ripples, preventing overfitting to simulator artifacts and serving as the fundamental driver for robust real world generalization.

I. INTRODUCTION

HankelKANet embeds electromagnetic propagation priors into a lightweight neural architecture, addressing limitations of existing path-loss estimation approaches. The paper also examines PEFNet’s efficiency and optimization bottlenecks and reports accuracy, efficiency, and generalization results.

II. METHODOLOGY BACKGROUND

  • The background motivates Hankel functions as learnable bases because they are nonlinear components of the electromagnetic Green’s function, and reviews PEFNet as a hybrid VIE–deep-learning approach.PEFNet uses a dual-UNet architecture to separate physics-driven and data-driven learning.Source: §I · p. 1

I. INTRODUCTION

  • HankelKANet combines Hankel-guided learnable functions with a streamlined architecture to embed propagation priors directly, without complex hybrid loss functions.It uses a two-channel input and a single residual UNet with a dual-convolution fusion module.Source: §I · pp. 1–2
  • HankelKANet achieves state-of-the-art results on simulated RadioMapSeer and real-world Huawei RSRPSet, while reducing PEFNet’s parameters by 78%.The introduction also reports 26% lower training time and 22.2% faster inference than PEFNet.Source: §I · p. 2
  • The paper derives a parameter-conditional spatial gradient bound and uses DDE and Jacobian heatmaps to connect structural smoothness with reduced simulator-artifact overfitting and robust real-world generalization.The paper presents this smoothness as the driver of cross-domain generalization.Source: §I · p. 2

A. PEFNet Analysis

  • PEFNet’s four-channel inputs, dual-UNet architecture, and manually tuned hybrid loss are identified as sources of preprocessing, computation, and optimization burdens.Its architecture has approximately 118.38M parameters; loss-weighting errors can cause gradient conflicts and unstable convergence.Source: A. PEFNet Analysis · p. 2

B. Wave Field Representation via Hankel Functions

The methodological background links Hankel wave-field structure to learnable convolutional bases and uses spatial sensitivity and DDE to frame smoothness and generalization. These concepts motivate a physics-aware spatial prior for path-loss modeling.

B. Wave Field Representation via Hankel Functions

  • The wave-field representation separates primary radial propagation from secondary scattering, while explicitly computing the full scattering integral is intractable for large urban layouts.The primary field uses the zeroth-order Hankel function; its spatial gradient is characterized by the first-order mode.Source: B. Wave Field Representation via… · pp. 2–3

C. Kolmogorov-Arnold Network

  • Convolutional KANs embed learnable univariate functions in local kernels, allowing specialized physical priors to enter the convolution operation directly.KANs replace static linear weights with learnable functions, which can be parameterized by different basis families.Source: C. Kolmogorov-Arnold Network · p. 3

D. Geometric Complexity and Jacobian Sensitivity

  • The Jacobian norm measures local sensitivity, while DDE aggregates squared sensitivities to characterize global variation; irregular gradients in radio maps can indicate non-physical spatial noise.The paper relates spatial continuity in plausible radio maps to propagation coherence.Source: D. Geometric Complexity and Jacobian… · p. 3

III. SYSTEM MODEL

The system model frames path loss as a spatially averaged, large-scale quantity and simplifies propagation modeling by separating a Hankel-based direct-field prior from learned environmental effects. It retains the dominant zeroth- and first-order Hankel magnitudes while omitting phase and higher-order terms.

A. Large-Scale Channel and Path Loss Model

  • The large-scale path-loss target averages over local phase fluctuations, Doppler shifts, and small-scale variability, excluding fast fading by design.The path model describes each path’s complex amplitude as reflecting primary direct-field and secondary scattered-field contributions.Source: A. Large-Scale Channel and Path Loss… · p. 3
  • The model evaluates Hankel functions over transmitter-to-receiver distance to encode the primary field, while a parallel building-layout branch learns secondary scattering and diffraction.The system uses a 2D formulation based on fixed transmitter height, receiver-height points, and vertically extruded building approximations.Source: A. Large-Scale Channel and Path Loss… · p. 4
  • The spatial prior retains magnitudes of the zeroth- and first-order Hankel modes as dominant large-scale decay and gradient cues, omitting phase and higher-order scattering terms.The paper reports that omitting these terms does not degrade large-scale path-loss prediction accuracy.Source: A. Large-Scale Channel and Path Loss… · p. 4

B. Map-Based Scenario Representation

The scenario representation supplies building-layout and transmitter-distance maps as a two-channel input for path-loss map prediction. The proposed architecture combines environmental structure with propagation priors in a compact encoder–decoder framework, trained as a pixel-wise regression task.

B. Map-Based Scenario Representation

  • Each grid cell maps to a receiver location, and the target is a path-loss map paired with binary building-layout and normalized transmitter-distance maps.The two input maps are concatenated channel-wise to form a two-channel-to-one-channel image-translation task.Source: B. Map-Based Scenario Representation · p. 4

C. Learning Objective

  • The learning objective uses mean squared error because path-loss estimation is treated as pixel-wise continuous regression.The passage introduces the final learning objective but does not provide its full expression.Source: C. Learning Objective · p. 4

IV. PROPOSED METHOD

  • HankelKANet combines environmental structural information and propagation-related priors in a compact encoder–decoder framework.The architecture realizes the end-to-end mapping defined by the system model.Source: §IV · p. 4

A. Overall Architecture

HankelKANet combines an environment encoder with a physical branch that encodes distance-dependent propagation, within a single residual UNet. Its two branches extract complementary urban-structure and low-order Hankel features for path-loss prediction.

A. Overall Architecture

  • A single residual UNet fuses building-layout features from an environment branch with distance-based propagation features from a physical branch.The encoder-decoder uses skip connections to restore spatial detail; its maximum channel width is 512.Source: A. Overall Architecture · p. 4

1) Environment Branch:

  • The environment branch encodes local building boundaries and broader scene context, forwarding intermediate features to the decoder through skip connections.It progressively downsamples spatial resolution while increasing channels.Source: 1) Environment Branch · p. 5

2) Physical Branch:

  • The physical branch evaluates a distance-normalized primary-field envelope and processes the resulting map with Hankel-KAN convolutions.Its kernels use learnable functions parameterized by zeroth- and first-order Hankel responses, whose weights combine macroscopic energy decay and dominant spatial gradients.Source: 2) Physical Branch · pp. 5–6
  • The physical branch lower-bounds distance evaluation to avoid Hankel-function divergence and gradient explosion during backpropagation.The threshold restricts evaluation to a physically negligible sub-pixel region without distorting macroscopic features.Source: 2) Physical Branch · p. 5
  • The physical prior captures the unperturbed primary field, while fusion with environment features implicitly resolves complex secondary scattering interactions.The selected low-order modes represent macroscopic spreading and fundamental spatial gradients; higher-order oscillatory modes average out in large-scale path loss.Source: Figure 1 · p. 5, 2) Physical Branch · p. 6

C. Multi-Scale Feature Fusion and Decoder

The model fuses environment and propagation features across multiple scales, then decodes them into a path-loss map. Training uses an MSE objective, and evaluation covers simulated RadioMapSeer and measured Huawei RSRPSet data.

C. Multi-Scale Feature Fusion and Decoder

  • At four progressive scales, environment and physical features are concatenated and fused, with scale-matched Hankel-KAN convolutions extracting propagation-aware features.The physical kernels are 3 × 3 at scale two and 5 × 5 at scales three and four.Source: C. Multi-Scale Feature Fusion and… · p. 6
  • The symmetric decoder upsamples bottleneck features and uses encoder skip connections to reconstruct fine path-loss structure, especially near blockage boundaries and transitions.A final 1 × 1 convolution produces a single-channel continuous-valued path-loss map.Source: C. Multi-Scale Feature Fusion and… · p. 6

D. End-to-End Training Strategy

  • Hankel-KAN convolutions embed propagation priors in the architecture, allowing end-to-end training with MSE without additional physics-loss terms or manual loss-weight tuning.Source: D. End-to-End Training Strategy · p. 6

1) Dataset Description:

  • RadioMapSeer evaluation holds out 8,000 maps by scenario, while Huawei RSRPSet tests measured received power at valid locations using matched data budgets and a common held-out set.The experiments use within-dataset training and testing; cross-dataset transfer is evaluated separately.Source: 1) Dataset Description · p. 6, §V · p. 6, Figure 3 · p. 7

2) Evaluation Metrics:

On RadioMapSeer, HankelKANet has the lowest RMSE and highest R2 among the evaluated methods, with a 16.96% RMSE reduction versus PEFNet. The section also describes the evaluation metrics, training protocol, and robustness assessment across link and urban conditions.

2) Evaluation Metrics:

  • MAE, RMSE, and R2 are averaged across test maps with equal map weight; RMSE penalizes large errors more heavily, and higher R2 indicates better agreement.Both error metrics are reported in dB after inverse normalization.Source: 2) Evaluation Metrics · p. 7

3) Training Settings and Reproducibility:

B. Comparison with Baselines

  • On RadioMapSeer, empirical and classical machine-learning methods generally exceed 10 dB RMSE, while the independent ray-tracing solver reports 6.3250 dB.Convolutional neural networks RadioUNet and PEFNet outperform these approaches, according to the comparison passage.Source: 1) Results on RadioMapSeer · p. 7

1) Results on RadioMapSeer:

  • 16.96%: HankelKANet reduces RMSE from 2.2148 to 1.8392 dB versus PEFNet, while achieving the highest R2 (0.9849) and lowest RMSE (1.8392 dB).Its MAE is 0.7980 dB, marginally above PEFNet’s 0.7852 dB; the authors attribute the RMSE pattern to suppressing severe prediction errors.Source: 1) Results on RadioMapSeer · p. 7
  • HankelKANet remains robust in NLOS regions (RMSE 2.2506 dB) and high-density urban layouts (RMSE 2.2229 dB), while long-range links are most challenging (3.1182 dB).LOS and short-distance regions have RMSE < 1.4 dB.Source: 1) Results on RadioMapSeer · p. 8

2) Results on Huawei RSRPSet:

On Huawei RSRPSet, HankelKANet outperforms PEFNet in suppressing large prediction errors across target-data fractions. Its full-supervision evaluation reports the lowest RMSE and highest R2, while MAE is marginally higher than PEFNet’s.

2) Results on Huawei RSRPSet:

C. Ablation Studies on Architecture and Physical Priors

The ablations test the importance of Hankel-guided propagation and U-Net feature fusion, as well as choices within the physical branch. Removing these components or changing its parameterization worsens RadioMapSeer RMSE.

1) Architectural Components:

  • Removing the physics-guided branch raises RMSE from 1.8392 dB to 9.5872 dB, while removing U-Net skip connections raises it to 4.0146 dB.The authors associate the branch with a transmitter-dependent propagation prior and skip-connected feature fusion with recovery of localized details such as sharp shadowing transitions.Source: 1) Architectural Components · p. 8, Table IV · p. 9
  • Replacing the Hankel amplitude basis with B-splines raises RMSE to 2.7958 dB, and fixing k raises it to 2.1594 dB.The authors say a trainable k adapts the effective radial scale in the normalized feature domain.Source: 1) Architectural Components · p. 8, Table IV · p. 9

2) Hankel Orders and Angular Factors:

The paper favors a low-order radial Hankel representation without explicit phase modeling, alongside a compact implementation. Its analysis attributes smooth spatial behavior to the Hankel basis’s gradient envelope.

2) Hankel Orders and Angular Factors:

  • Using Hankel orders {0, 1} avoids the RMSE degradation of single modes, while higher orders add negligible accuracy or worsen performance.Adding {0, 1, 2} changes RMSE by −0.0016 dB while increasing latency by 8.1%; {0, 1, 2, 3} yields 1.8662 dB RMSE. Explicit angular factors improve accuracy by at most 0.003 dB for |n| ≤1 but add up to 28% runtime; the authors describe phase reconstruction as largely redundant for scalar signal-power supervision.Source: 2) Hankel Orders and Angular Factors · pp. 9–10

D. Complexity and Efficiency Analysis

  • 78% fewer parameters and 81.9% fewer FLOPs than PEFNet, with inference memory reduced from 720 MB to 340 MB.Including online Hankel-function generation, latency is 1.110 s/map versus 1.426 s for PEFNet; the stated operating scope is macroscopic large-scale path loss in 2D outdoor urban scenarios below 6 GHz, with link distances up to about 350 m.Source: D. Complexity and Efficiency Analysis · p. 10

VI. IMPACT OF SPATIAL SMOOTHNESS ON EMPIRICAL GENERALIZATION

  • The paper frames physics-inspired spatial smoothness as a mechanism for reducing overfitting to simulator artifacts and improving simulated-to-real generalization.This is stated as the analysis objective, not a quantified result in these passages.Source: §VI · p. 10

A. Hankel Operators as an Inductive Bias for Smoothness

  • In the propagation regime κ∥r∥₂ ≫ 1, the spatial gradient is conditionally bounded by the modal parameters.The Hankel basis’s asymptotic envelope decays at O(∥r∥^−1/2) and is described as a distance encoding that prevents high-frequency spatial noise in the far field.Source: A. Hankel Operators as an Inductive… · p. 10

B. Empirical Verification of Spatial Smoothness

The empirical analysis compares geometric complexity during training and local Jacobian sensitivity to assess spatial smoothness. HankelKANet shows a flatter global complexity profile and smoother local sensitivity than PEFNet.

B. Empirical Verification of Spatial Smoothness

  • HankelKANet’s global geometric complexity remains consistently flat during training and orders of magnitude below PEFNet by the end.The authors interpret this as convergence to a smoother mapping rather than memorization of irregular functional landscapes.Source: B. Empirical Verification of Spatial… · p. 11
  • HankelKANet’s Jacobian sensitivity forms smooth, radially decaying contours, while PEFNet shows erratic flares and local hotspots.The analysis also extracts a constant-distance arc crossing line-of-sight and shadowed non-line-of-sight regions to isolate continuous-space behavior from radial path-loss decay.Source: B. Empirical Verification of Spatial… · p. 11
  • Along the constant-distance arc, both models capture diffraction-boundary step changes, but PEFNet fluctuates while HankelKANet avoids spurious oscillations.The authors attribute this smoothness to the embedded asymptotic envelope, which bounds spatial gradients and prevents high-frequency spatial noise.Source: B. Empirical Verification of Spatial… · p. 11

C. From Spatial Smoothness to Real-World Generalization

HankelKANet’s spatial smoothness is linked to stronger transfer from simulated training to real-world data, including zero-shot performance and efficient target-domain adaptation. The conclusion presents this smoothness as a driver of robust generalization and practical path-loss estimation.

C. From Spatial Smoothness to Real-World Generalization

  • Bounded spatial gradients make HankelKANet less prone to overfitting simulator artifacts and help it extract domain-invariant wave-propagation characteristics.The paper identifies structural spatial smoothness as a fundamental driver of robust real-world generalization and efficient target-domain adaptation.Source: C. From Spatial Smoothness to… · pp. 11–12
  • HankelKANet maintains low geometric complexity throughout training and settles several orders of magnitude below PEFNet, while its Jacobian sensitivity is stable across LOS and NLOS regions.PEFNet exhibits a rapid initial complexity surge and pervasive high-frequency sensitivity fluctuations; both models react to diffraction boundaries.Source: Figure 4 · p. 12, Figure 5 · p. 12

VII. CONCLUSION

  • The conclusion presents HankelKANet’s embedded Hankel prior as structural regularization that supports cross-dataset generalization and data-efficient target-domain adaptation.It also describes the model as a practical solution for coverage planning and interference management in emerging 5G/6G networks.Source: §VII · pp. 12–13
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