Physics-Unrolled Neural Operator for Wireless Field Modeling

Rafid Umayer Murshed; Saif Ur Rahman; Mingyue Tang; Elahe Soltanaghai

cs.LGAug 19, 2026

Physics-Unrolled Neural Operator for Wireless Field Modeling

Authors: Rafid Umayer Murshed, Saif Ur Rahman, Mingyue Tang, Elahe Soltanaghai

Organizations: Department of Computer Science University of Illinois Urbana-Champaign Urbana, IL 61801, USA

Abstract

Radio maps are essential for wireless decision-making tasks such as access-point placement, coverage planning, and localization, but their fine spatial details are governed by complex propagation effects and are costly to simulate accurately. Machine learning offers a path to high-fidelity radio-map prediction without running expensive high-fidelity simulations for every scene. However, generating high-quality training labels at scale is also difficult: the affordable labels come from finite-ray simulations, which are richer than low-fidelity inputs but carry residual Monte Carlo noise. We address this challenge with Physics-Unrolled Hybrid Neural Operator (PU-HNO), a three-stage cascade that predicts high-fidelity indoor radio maps from low-fidelity ray-tracing outputs and scene priors by progressively capturing reflection, diffraction, and scattering effects, rather than treating radio maps as generic images. We prove that, under conditionally unbiased label noise, the model can learn stable propagation structure and outperform its own training labels. Experiments across diverse floorplans show that PU-HNO outperforms image-to-image baselines, wireless learning models, and monolithic neural operators across both image-quality and wireless deployment metrics.

Figures & tables

Figure 1: PU-HNO predicts high-fidelity radio maps from low-fidelity inputs and intermediate-fidelity supervision.

Figure 2: Individual propagation mechanisms jointly shaping the radio map.

Figure 3: PU-HNO estimates high-fidelity radio maps using a three-stage neural-operator architecture that captures different wireless propagation mechanisms.

		Wireless Performance				Computer Vision / Generic
Model Family	Model	Outage F1 $\uparrow$	Fading Ratio $\approx 1$	Tail Error (5%) $\downarrow$	MCESE $\downarrow$	RMSE (dB) $\downarrow$	MAE (dB) $\downarrow$	SSIM $\uparrow$	LPIPS $\downarrow$
SIONNA	Training Labels	0.172	1.782	1.661	1.311	14.012	5.486	0.725	0.476
Computer Vision	CNN	0.490	0.799	1.043	1.443	4.798	3.152	0.904	0.277
	UNET	0.178	0.815	0.448	1.076	3.851	2.326	0.919	0.266
	RESNET	0.107	0.761	0.531	1.152	4.228	2.692	0.905	0.291
	VIT	0.017	1.495	0.145	1.637	15.310	6.188	0.652	0.475

Table 1: Quantitative radio map prediction results showing that PU-HNO consistently outperforms all baselines across wireless and computer vision metrics.

Figure 4: PU-HNO recovers fine details from noisy labels and approaches the high-fidelity reference.

Category	Variant	Outage F1 $\uparrow$	Fading Ratio $\approx 1$	Tail Error $\downarrow$	RMSE (dB) $\downarrow$	SSIM $\uparrow$
Learning Paradigm	1A: Single Target (No Curr)	0.637	1.124	0.927	3.787	0.927
	1B: Progressive (No Curr)	0.784	0.956	0.217	3.407	0.926
Loss Formulation	2A: w/o Sobolev Gradient	0.111	0.814	0.381	3.584	0.923
	2B: w/o Transport (Ray)	0.740	0.910	1.273	3.512	0.916
	2C: Unweighted Masked Huber	0.128	0.818	0.197	3.655	0.916
Engineered Priors	3A: w/o Geometric Priors	0.780	0.967	0.110	3.660	0.923

Table 2: Ablation study showing that PU-HNO benefits from its staged architecture, geometry priors, and propagation-aware losses.

Panel of Figure 5 — Figure 5: Model sensitivity to structural distribution shifts and training label noise.

Appendix figures & tables10 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 6: Representative training and test floorplans. Environments are procedurally generated via recursive binary space partitioning (BSP) and randomized furniture spawning. The test set introduces structural distribution shifts by expanding the permitted range of room and obstacle counts beyond the training distribution.

Panel of Figure 7 — Figure 7: Qualitative evaluation of radio map reconstruction. PU-HNO accurately recovers the High-Fidelity Ground Truth (HF-GT) field from sparse LF-Inputs. Baselines such as GeNeRT (a) and SFNO (b) struggle to preserve these sharp, deployment-critical details and exhibit significant over-smoothing.

Component	Trainable parameters
Shared encoder, conditioning, and projection heads	$\sim 187$ K
Stage 1 (specular)	$\sim 1.56$ M
Stage 2 (diffraction)	$\sim 614$ K
Stage 3 (scattering)	$\sim 147$ K
Total	$\sim 2.50$ M

Appendix

Table 3: Parameter budget by component. Counts are obtained by instantiating the model used for the reported results and summing over the corresponding submodules. Totals are rounded to the nearest thousand.

Term	Symbol	Value
Stage-1 reconstruction	$\lambda_{1}$	$0.55$
Stage-2 reconstruction	$\lambda_{2}$	$1.00$
Stage-3 reconstruction	$\lambda_{3}$	$1.60$
Sobolev gradient	$\lambda_{\text{sob}}$	$0.15$
Stage-1 transport feature	$\mu_{1}$	$0.20$
Stage-2 transport feature	$\mu_{2}$	$0.28$

Appendix

Table 4: Loss hyperparameters used for all reported runs. “Distance scale” refers to

r_{\text{scale}}

in Eq. ( 14 ), in physical units relative to the

15\,\text{m}\times 15\,\text{m}

floorplan.

Symbol	Quantity	Value
$T_{\text{out}}$	Outage threshold	$-100$ dBm
$B$	Bandwidth	$20$ MHz
$F$	Receiver noise figure	$7$ dB
$N_{\text{dBm}}$	Thermal noise floor (per Eq. ( 25 ))	$\approx-94$ dBm
$Y_{\min}$	Simulator floor for valid mask	$-150$ dB
$R$	Dynamic range for PSNR	$170$ dB

Appendix

Table 5: Constants used by the wireless deployment metrics. These values are fixed across all reported experiments and are taken from standard 5–6 GHz indoor wireless practice.

Panel of Figure 8 — Figure 8: Performance comparison across generic and wireless-specific metrics with 95% bootstrap confidence interval.

Family	Model	F1 $\uparrow$	R $\uparrow$	Fad. $\approx 1$	SE5 $\downarrow$	MCESE $\downarrow$	RMSE $\downarrow$	MAE $\downarrow$	PSNR $\uparrow$	SSIM $\uparrow$	ESSIM $\uparrow$	LPIPS $\downarrow$	DISTS $\downarrow$	Grad $\approx 1.58$
Label	IF-GT	0.172	0.991	1.782	1.661	1.311	14.012	5.486	21.679	0.725	0.405	0.476	0.294	9.348
CV	CNN	0.490	0.348	0.799	1.043	1.443	4.798	3.152	30.989	0.904	0.548	0.277	0.202	1.380
CV	U-Net	0.178	0.100	0.815	0.448	1.076	3.851	2.326	32.897	0.919	0.551	0.266	0.199	1.388
CV	ResNet	0.107	0.058	0.761	0.531	1.152	4.228	2.692	32.085	0.905	0.509	0.291	0.211	1.368
CV	ViT	0.017	0.045	1.495	0.145	1.637	15.310	6.188	20.909	0.652	0.157	0.475	0.336	10.313
Wireless	NeRF 2	0.026	0.015	0.696	1.725	3.406	5.671	3.041	27.291	0.857	0.399	0.345	0.203	1.935

Appendix

Table 6: Extended quantitative comparison on the HF-GT indoor test set. R denotes outage recall. Fad. is the local fading ratio and should be close to

1

. SE5 is the absolute 5th-percentile spectral-efficiency tail error in high-variation regions. Grad is the mean spatial RSS-gradient magnitude. The HF-GT reference has GradMean

1.58

; hence Grad should be interpreted by closeness to

1.58

, not by monotone increase or decrease. Bold marks the best learned predictor.

Model	Outage F1 $\uparrow$	Recall $\uparrow$	Missed area $(\mathrm{m}^{2})$	Extra APs	Extra cost
CNN	0.490	0.348	$652$	$3$	$4,500
GeNeRT	0.000	0.000	$1{,}000$	$4$	$6,000
WNO	0.191	0.108	$892$	$4$	$6,000
PU-HNO	0.820	0.852	148	1	$1,500

Appendix

Table 7: Engineering interpretation of outage detection for AP placement on a

10{,}000~\mathrm{m}^{2}

enterprise floorplan. The assumed true hard-coverage area is

1{,}000~\mathrm{m}^{2}

Model	SE5 error $\downarrow$	Throughput loss	Achieved edge throughput $\uparrow$
CNN	1.043	$83.4$ Mbps	$16.6$ Mbps
GeNeRT	1.405	$112.4$ Mbps	$0.0$ Mbps
WNO	0.373	$29.8$ Mbps	$70.2$ Mbps
PU-HNO	0.100	8.0 Mbps	92.0 Mbps

Appendix

Table 8: Engineering interpretation of 5% tail spectral-efficiency error as achieved cell-edge throughput. The target edge throughput is

100

Mbps and the assumed bandwidth is

80

MHz.

Domains and spaces
Symbol	Meaning
$\Omega\subset\mathbb{R}^{2}$	2D floorplan/deployment domain
$\mathcal{Y}$	Separable Hilbert space of RSS fields on $\Omega$
$\langle\cdot,\cdot\rangle_{\mathcal{Y}}$ , $\\|\cdot\\|_{\mathcal{Y}}$	Inner product and norm on $\mathcal{Y}$
$\mathcal{G}$	Space of scene encodings (geometry, materials) on $\Omega$
$\mathcal{C}$	Space of coordinate priors (Rx grid, Tx/Rx elevations)

Appendix

Table 9: Comprehensive notation. Symbols are used consistently across this appendix.

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Radio maps describe how wireless signals propagate across space and are essential for wireless communication, sensing, and network planning. However, constructing accurate radio maps traditionally requires either dense measurements or computationally expensive physical simulations, which limits scalability and real-time deployment. Recent advances in generative artificial intelligence offer a promising alternative, but existing approaches lack fine-grained control and physical consistency when applied to real-world wireless environments. Here we present ControlRadio, a controllable generative framework that produces radio maps from natural-language descriptions and environmental layouts, including building structures and transmitter locations. Joint semantic and spatial conditioning enables interpretable, propagation-plausible generation, while a controlled latent prior and layout-aware conditioning improve stability and structural consistency. Extensive experiments demonstrate that ControlRadio achieves state-of-the-art accuracy and strong generalization across diverse urban scenarios, while reducing computation time by more than four orders of magnitude compared with conventional simulation-based methods. Such results suggest a new paradigm for scalable and controllable wireless environment modeling, with broad implications for next-generation communication systems and data-driven radio sensing.

Abstract

Radio Propagation Modeling

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