Organizations: School of Computer Science and Technology, Anhui University of Technology, Maanshan 243002, China · Department of Electrical and Electronic Engineering, Imperial College London, SW7 2AZ London, U.K.
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.
Figures & tables
Fig. 1: Electromagnetic interpretation of the structured Hankel prior for large-scale path loss. (a) A typical urban scenario exhibiting sharp PL contrasts between dense buildings and open areas. (b) The zeroth- and first-order Hankel magnitudes capture macroscopic cylindrical spreading and fundamental spatial gradients, whereas highly oscillatory higher-order modes ( n≥2 ) correspond to fine-grained scattering that averages out in large-scale PL. (c) Visualization of the combined effect of the zeroth- and first-order Hankel magnitudes extracted as the spatial prior.
Fig. 2: Overall architecture of the proposed HankelKANet. The network processes a streamlined two-channel input comprising a base station distance map (R_Map) and a building binary map. It is built upon a single residual UNet backbone with a maximum channel count of 512. The encoder-decoder structure utilizes skip connections to recover fine details in boundary and blockage regions, while the physical branches (PhyBlocks) embed distance-dependent propagation priors via 2D HankelKAN convolutions. The network finally outputs the predicted PL map.
Fig. 3: Examples of input and output maps from RadioMapSeer. From left to right: building layout, distance prior, and pathloss map for two representative scenarios.
Method
R2
MAE (dB)
RMSE (dB)
3GPP TR 36.873 [ 4 ]
0.2552
11.5416
14.7590
ray tracing [ 6 ]
0.8594
4.7591
6.3250
DTR [ 36 ]
0.4733 ( ± 0.0004)
8.7962 ( ± 0.0063)
12.1251 ( ± 0.0104)
KNN [ 37 ]
0.5808 ( ± 0.0002)
7.8994 ( ± 0.0095)
10.8649 ( ± 0.0053)
ANN [ 7 ]
0.6303 ( ± 0.0034)
7.4167 ( ± 0.0027)
10.1688 ( ± 0.0029)
RadioUNet [ 8 ]
0.9616 ( ± 0.0002)
1.4747 ( ± 0.0180)
2.8748 ( ± 0.0305)
TABLE I: Quantitative performance comparison of different methods on the RadioMapSeer test set.
Condition
Radio maps
Evaluated pixels
R2
MAE (dB)
RMSE (dB)
Overall outdoor test set
8,000
397,944,539
0.9849
0.7980
1.8392
LOS
8,000
125,925,669
0.9885
0.4614
1.3884
NLOS
8,000
272,018,870
0.9820
1.2583
2.2506
Short distance ( ≤72.80 m)
8,000
99,342,877
0.9910
0.3809
0.9827
Long distance ( ≥147.47 m)
8,000
99,980,657
0.9633
1.9792
3.1182
High building density ( ≥30.80% )
2,160
89,968,345
0.9715
1.1780
2.2229
TABLE II: Stratified performance of HankelKANet with Hankel orders {0,1} against RadioMapSeer reference maps.
Budget ( η )
Method
R2
MAE (dB)
RMSE (dB)
Training from Scratch
1%
PEFNet
0.6448
4.7843
7.8452
HankelKANet
0.6896
4.4375
7.4218
5%
PEFNet
0.7620
3.5207
6.4211
HankelKANet
0.7983
3.2884
5.9126
10%
PEFNet
0.8340
2.8129
5.3628
TABLE III: Within-dataset performance comparison on the Huawei RSRPSet (Training from scratch and full real supervision).
Ablation Variant
R2
MAE (dB)
RMSE (dB)
w/o Physical Branch
0.6783
6.7468
9.5872
Hankel replaced by B-spline
0.8734
1.7899
2.7958
w/o skip-connected module
0.9445
2.1316
4.0146
with non-trainable k
0.8833
1.1926
2.1594
HankelKANet (Proposed)
0.9849
0.7980
1.8392
TABLE IV: Architectural component ablation study on RadioMapSeer.
Representation Choice
Orders ( I ) / Mode n
R2
RMSE (dB)
FLOPs (G)
Latency (s)
Radial Envelope Only (Amplitude)
Zeroth-order only
I={0}
0.8559
3.1025
91.48
0.95
First-order only
I={1}
0.8352
3.6856
91.48
0.97
Proposed Variant
I={0,1}
0.9849
1.8392
91.58
1.11
Second-order added
I={0,1,2}
0.9851
1.8376
91.66
1.20
Third-order added
I={0,1,2,3}
0.9845
1.8662
91.78
1.29
TABLE V: Ablation of physical representation choices (mean over 5 runs). I denotes the set of utilized Hankel orders.
Fig. 4: Evolution of the Dimension-Normalized Geometric Complexity ( GCavg ) and prediction metrics over training epochs. Following [16], the shaded regions denote three optimization stages: fitting (capturing dominant large-scale structures), transition (refining finer details), and diffusion (stabilizing the learned mapping). Unlike PEFNet, which experiences a rapid initial surge in complexity before stabilizing at a highly elevated level, HankelKANet maintains a consistently low GCavg throughout training. Ultimately, HankelKANet settles at a value several orders of magnitude lower than PEFNet. This massive gap indicates that HankelKANet converges to an intrinsically smoother mapping that naturally aligns with large-scale propagation physics.
Fig. 5: Microscopic Jacobian Sensitivity visualization. (a) 2D spatial heatmap of the Jacobian gradient norm. (b) 1D sensitivity profile extracted along an iso-distance arc traversing both Line-of-Sight (LOS) and Non-Line-of-Sight (NLOS) regions. While both models react to diffraction boundaries, PEFNet suffers from pervasive high-frequency fluctuations across all regions. In contrast, HankelKANet imposes physics-inspired smoothness, yielding a stable sensitivity curve that aligns with Proposition 1.
Model
Pre-trained
Target Domain Data Budget ( η )
0% (Zero-shot)
1%
5%
10%
20%
Scratch Baseline
No
-
7.42
5.91
4.94
4.31
PEFNet
Yes
6.43
5.73
4.92
4.48
4.12
HankelKANet
Yes
5.82
5.14
4.57
4.23
3.98
TABLE VII: Cross-Dataset Generalization Validation: Transfer Performance (RMSE in dB) on the Real-World Huawei RSRPSet