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Physics-Unrolled Neural Operator for Wireless Field Modeling

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

arXiv:2608.18495v1cs.LGcs.AI

TL;DR

High-fidelity indoor radio maps capture propagation details needed for wireless planning but are expensive to simulate, while affordable training labels remain noisy. PU-HNO progressively refines low-fidelity ray-tracing maps using propagation-aware neural-operator stages and outperforms noisy training labels and competing models across image-quality and wireless deployment metrics.

  • Problem

    High-fidelity radio maps are expensive to generate, motivating whether machine learning can convert low-cost, noisy simulations into accurate predictions.

  • Method

    PU-HNO uses geometry- and scene-conditioned neural-operator stages that progressively model reflection, diffraction, and scattering effects.

  • Results

    PU-HNO most accurately reconstructs held-out high-fidelity references across wireless and image-quality metrics, including 3.5 dB RMSE and 0.82 Outage F1.

  • Takeaways & Limitations

    Imperfect simulation labels can support high-fidelity wireless field prediction when propagation structure is learned under conditionally unbiased label noise.

  • Takeaways & Limitations

    PU-HNO depends on accurate scene priors and is evaluated only for indoor, single-transmitter received-signal-strength prediction using ray-tracing supervision.

Abstract

from arXiv · show

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.

1 Introduction

PU-HNO addresses noisy, costly supervision for high-fidelity wireless field modeling with a three-stage, propagation-aware neural-operator cascade. It progressively models reflection, diffraction, and scattering, while theory and deployment-oriented evaluation establish benefits beyond conventional image-quality assessment.

  • Motivation: Wireless fields exhibit sharp RSS variations because reflection, diffraction, and scattering interact coherently with scene geometry and materials.These variations matter for access-point placement, coverage planning, localization, and network reliability.
  • Motivation: High-fidelity labels are computationally expensive, so training typically uses richer intermediate-fidelity simulations that remain noisy and imperfect.The learning objective is to refine coarse fields rather than simply reproduce imperfect labels.
  • Method: PU-HNO is a three-stage neural-operator cascade that progressively refines low-fidelity ray-tracing outputs through reflection, diffraction, and scattering stages.The stages target broad reflected coverage, edge-driven diffraction, and fine local scattering fluctuations.
  • Method: Propagation-aware training objectives emphasize fine-scale wireless-field structure that standard pointwise losses tend to smooth out.The method learns from noisy intermediate-fidelity supervision and captures spatial structure in unseen environments.
  • Theory and evaluation: A deployment-oriented evaluation protocol uses wireless metrics to reveal coverage and planning differences that standard image-quality metrics miss.Experiments compare PU-HNO with standard vision architectures, general-purpose neural operators, and wireless-specific baselines across diverse indoor scenes.
  • Theory and evaluation: A zero-shot denoising theorem shows that, with enough training scenes, an operator trained on noisy labels can be closer to the high-fidelity reference than the labels themselves.The result extends classical single-image denoising to operator learning across many physical scenes.

2 Background and Related work

Radio maps discretize electromagnetic signal strength across environments and support wireless planning and localization. Because propagation combines mechanisms with distinct spatial signatures and Monte Carlo ray tracing trades fidelity against cost, related work spans physics-informed, neural-operator, denoising, and wireless surrogate methods.

  • Wireless Fields and Propagation Mechanisms: Radio maps assign received-signal-strength values to locations, supporting access-point placement, coverage planning, localization, and outage-region identification.
  • Wireless Fields and Propagation Mechanisms: Line-of-sight and specular paths form broad patterns, diffraction creates shadow-boundary changes, and scattering introduces fine fluctuations, requiring multiple mechanisms beyond higher spatial resolution.
  • Related Work: Related work includes neural operators, physics-informed networks, multi-fidelity learning, noisy-label denoising, and physics-prior wireless models for structured-field prediction.
  • Related Work: Wireless surrogates use scene geometry, sparse RF measurements, visual or geometric priors, and per-scene continuous representations such as NeRF-style fields and neural ray tracers.

3 Method

PU-HNO maps low-fidelity radio maps and scene priors to high-fidelity predictions through three cascaded residual operators for specular reflection, diffraction, and scattering. Its training objective supervises stages, preserves spatial transitions, and aligns residuals with propagation features, while conditionally unbiased label noise permits learning the ideal mapping.

  • Label-noise assumption: Under conditionally unbiased intermediate-fidelity labels, zero-mean errors can average across examples so training need not systematically move predictions away from the ideal mapping.High-fidelity references are held out for evaluation, while intermediate-fidelity labels are used during training.
  • Input representation: PU-HNO combines low-fidelity radio maps, geometry/material priors, and Fourier-expanded coordinate priors as inputs.The low-fidelity input includes valid-cell masks and dominant-direction features; priors encode floorplan, obstacles, materials, transmitter distance, and spatial context.
  • Physics-unrolled architecture: Three cascaded residual operators progressively refine specular structure, diffraction effects, and scattering corrections.Stage 1 captures broad reflection-driven coverage, Stage 2 targets edges, corners, and doorways, and Stage 3 models local geometry- and material-dependent fluctuations.
  • Diffraction refinement: Stage 2 restricts direction-selective and graph-based diffraction updates to unresolved regions identified after Stage 1.A learned mask s⋆ gates corrections involving geometry-guided filters and information exchange among candidate diffraction regions.
  • Training objective: The multi-term objective combines stage-wise Huber supervision, distance-modulated Sobolev regularization, and transport-feature alignment.These terms reduce sensitivity to occasional large label errors, preserve sharp transitions, and encourage residuals to follow amplitude, coherence, and orientation structure.

4 Evaluation

PU-HNO is evaluated on simulated indoor scenes against high-fidelity references using wireless and image-quality metrics, where it consistently outperforms image-to-image, wireless, and neural-operator baselines. Its staged physics-aware design improves propagation-structure recovery, robustness to scene shifts and noisy labels, and performance beyond intermediate-fidelity training labels.

  • Prediction accuracy relative to the high-fidelity reference: PU-HNO achieves the most accurate reconstruction, reducing tail error to 0.10, bringing fading ratio near 1.0, increasing Outage F1 to 0.82, and reducing RMSE to 3.5 dB.Predictions are compared against a held-out high-fidelity reference using wireless and image-quality metrics.
  • Impact of physics-based model decomposition: U-Net, TFNO, and WNO can appear competitive visually, but their wireless performance is weaker, with Outage F1 values of 0.18, 0.018, and almost no outage regions, respectively.The comparison shows that visual similarity and low average error do not ensure preservation of propagation structures.
  • Ablation study of model components: Removing semantic/material conditioning collapses Outage F1 from 0.82 to 0.001, showing that geometry alone is insufficient for predicting coverage failures.The ablation attributes the loss of wireless reliability to missing information about material and object effects on propagation.
  • Impact of out-of-distribution (OOD) scene shift: Across ID, Room-OOD, Clutter-OOD, and Both-OOD groups, PU-HNO obtains RMSE from 3.31 to 3.89 dB and Outage F1 between 0.74 and 0.87, while split-wise baselines reach only 0.34–0.53 Outage F1.The test groups vary whether room count and clutter count fall inside or outside the training range.
  • Impact of intermediate-fidelity label ray budget: With intermediate-fidelity labels generated from 50K to 5M rays, PU-HNO degrades more gracefully and maintains Outage F1 around 0.47 at 50K rays, while all baselines fall below 0.05.The evaluation keeps the high-fidelity reference fixed while varying the training-label ray budget.
  • Impact of training label noise structure: PU-HNO suppresses zero-mean IID, heteroscedastic, and spatially correlated label noise, but systematic bias remains problematic.Heteroscedastic noise improves RMSE from 3.84 to 3.62 dB, while the theorem explains that conditionally unbiased noise changes variance without shifting the optimal mapping.

5 Discussion

PU-HNO predicts high-fidelity radio maps by progressively refining propagation-specific effects rather than treating radio maps as generic images. Its effectiveness is accompanied by limitations involving scene priors, evaluation scope, and reliance on ray-tracing supervision.

  • Conclusion: PU-HNO decomposes radio-map prediction into propagation-aware stages that progressively refine specular structure, diffraction effects, and scattering-related details.The model uses low-fidelity simulation and scene priors to predict high-fidelity radio maps.
  • Limitations and Future Work: Accurate geometry, obstacle, and material priors are required, because missing floorplan details or unseen materials can introduce systematic prediction errors.Future work includes incorporating continuous material properties such as permittivity and conductivity.
  • Limitations and Future Work: The evaluation focuses on indoor, single-transmitter settings with fixed antenna assumptions, motivating studies of interference, outdoor environments, and antenna conditioning.Relevant antenna factors include orientation, polarization, and radiation patterns.
  • Limitations and Future Work: Ray tracing supplies all supervision and evaluation, while systematic bias in training labels can degrade performance and motivates measured-data calibration, sim-to-real adaptation, or hybrid training.These approaches are identified as requirements for real-world deployment.

A Broader Impact and Limitations … B.2 Propagation simulation: LF input, IF-GT labels, and HF-GT reference

PU-HNO is scoped as a learned indoor sub-6 GHz radio-map surrogate for network planning, with benefits from reducing repeated high-fidelity ray tracing but risks when applied outside its training regime. The experiments use procedurally generated indoor scenes and staged ray-traced fields: low-fidelity inputs, intermediate-fidelity training labels, and held-out high-fidelity references.

  • A Broader Impact and Limitations: PU-HNO targets indoor sub-6 GHz radio-map prediction for access-point placement, coverage planning, and what-if analysis, serving network designers or building-scale digital twins.Training and evaluation use synthetic scenes and synthetic ray-traced fields.
  • A Broader Impact and Limitations: Replacing repeated high-fidelity ray tracing reduces each candidate configuration’s marginal cost by several orders of magnitude, enabling more layouts and outer-loop optimization.The stated applications include sensor placement and environmental sensing.
  • A Broader Impact and Limitations: The main foreseeable risk is over-trust when the surrogate is treated as ground truth outside its training regime, including outdoor macrocells, dense metallic clutter, or other frequencies.The model has no decision-making role or direct path to harmful downstream uses beyond those enabled by ray tracing.
  • A Broader Impact and Limitations: The study is limited to 5.5 GHz, indoor 15 m × 15 m floorplans, one active transmitter, scalar RSS prediction, and ray-traced supervision without sim-to-real calibration.The model does not evaluate frequency transfer, outdoor propagation, multi-transmitter interference, phase-coherent quantities, or measured radio maps.
  • B Dataset Generation Pipeline: The dataset pipeline generates procedural indoor scenes, ray-tracing fields, and train/test splits to reproduce the experiments.This appendix specifies the construction of the three fidelity levels used by the study.
  • B.1 Scene generation: Each scene is a 15 m × 15 m floorplan with a 3 m ceiling, concrete outer structure, BSP-generated rooms, doorway gaps, and furniture-scale cuboid clutter.Rooms have a minimum dimension of 3 m; clutter uses rejection sampling with sampled obstacle sizes and indoor material assignments.
  • B.2 Propagation simulation: LF input, IF-GT labels, and HF-GT reference: Radio maps use Sionna RT at 5.5 GHz on a 128 × 128 RSS grid, with 10^4-ray LF inputs, approximately 10^6-ray staged IF labels, and approximately 10^8-ray HF-GT references.Y1 contains specular propagation, Y2 adds diffraction, Y3 adds diffuse scattering, and HF-GT is reserved for validation and testing.

B.3 Splits and out-of-distribution subgroups

The evaluation expands room and furniture-count ranges beyond training to measure both in-distribution performance and controlled scene-level distribution shift. The test set is partitioned into four disjoint subgroups based on whether room and furniture counts fall inside or outside training ranges.

  • Distribution ranges: Training scenes use 4–7 rooms and 2–6 furniture items, while tests span 3–8 rooms and 0–8 furniture items.This widens both ranges without changing the underlying floorplan generator.
  • OOD subgroups: The test set contains four disjoint groups: ID, Room-OOD, Clutter-OOD, and Both-OOD.These groups support the OOD sensitivity analysis in Section 4.3.
  • OOD subgroup definitions: ID scenes keep both room and furniture counts within training ranges, whereas Room-OOD shifts only room count outside that range.Clutter-OOD shifts only furniture count, and Both-OOD shifts both counts outside training ranges.

C Model Architecture … D.4 Composite objective

PU-HNO is a three-stage residual operator that progressively refines low-fidelity radio maps through specular, diffraction, and scattering modules, using geometry, material, and ray-evidence conditioning. Its curriculum and composite losses supervise stage-specific reconstruction, gradients, and transport features while accounting for spatially varying label reliability.

  • C.1 Overall structure: PU-HNO applies three residual operators sequentially, producing the final prediction from the third-stage readout while using adaptive sizing to localize later corrections.The residual design supports stable localized updates, and the model has roughly 2.5M trainable parameters, concentrated mainly in Stage 1.
  • C.2 Encoder and conditioning: The shared encoder maps low-fidelity RSS, validity, coordinate, transmitter-distance, and ray-evidence inputs into a full-resolution latent field with width 128.Geometry and semantic streams are reused through lightweight FiLM and gating modules rather than repeated concatenation.
  • C.3 Stage operators: Stage 1 combines long-range Fourier coupling, localized wavelet-style processing, and reflection-aware oriented filtering to build the first usable field estimate.A learned sizing field then gates Stages 2 and 3 so they focus on unresolved regions rather than refining the entire map uniformly.
  • C.3 Stage operators: Stage 2 targets diffraction near geometric discontinuities with wedge-conditioned directional filtering and a sparse graph corrector, while Stage 3 models material-dependent local scattering corrections.Both stages are gated by the learned sizing field, and Stage 3 additionally uses a geometry-aware gate.
  • C.4 Prediction heads and curriculum: Small prediction heads and near-zero final-layer initialization keep early outputs close to the low-fidelity input, while progressive activation trains Stages 1, 2, and 3 with matching labels.Transport-feature auxiliary losses are linearly ramped after the main reconstruction objective is established, following the cascade’s physical ordering.
  • D.1 Confidence-modulated reconstruction loss: The reconstruction objective uses input-dependent confidence and geometry weights, masks invalid pixels, and applies weighted Huber supervision to each stage prediction.The weighting depends only on the input and is detached from optimization, while normalization by active weight keeps loss scales comparable across scenes.
  • D.2 Sobolev gradient loss: Stage 3 also receives a Sobel-based gradient-domain loss to discourage over-smoothing, because earlier stage targets are intentionally smoother than the final field.The supplied passage states that input-measurable weighted Sobolev losses preserve the same population target under conditionally unbiased label noise.
  • D.3 Transport-feature auxiliary loss / D.4 Composite objective: Fixed oriented-filter descriptors supervise residual updates through transport-feature losses, whose ramp τ(t) and coefficients are shared across reported runs in the composite objective.The descriptors summarize local amplitude, coherence, and dominant orientation and encourage directional structure without defining a second output target.

D.5 Optimizer, precision, and compute · E Baselines and Fair-Comparison Protocol

PU-HNO uses a reproducible AdamW training setup with mixed precision, fixed-seed controls, and single-GPU computation. Comparisons span image-to-image, wireless-specific, and monolithic neural-operator baselines under a documented protocol.

  • D.5 Optimizer, precision, and compute: PU-HNO is optimized with AdamW using β1 = 0.9, β2 = 0.95, weight decay 10−4, and base learning rate 8×10−4.The schedule applies three-epoch linear warm-up followed by cosine decay.
  • D.5 Optimizer, precision, and compute: Biases, normalization parameters, and one-dimensional parameters are excluded from weight decay.
  • D.5 Optimizer, precision, and compute: Training uses an effective batch size of 64 scenes, bfloat16 mixed precision, and gradient clipping at global norm 1.0.
  • D.5 Optimizer, precision, and compute: A fixed random seed is used for Python, NumPy, and PyTorch, while label-noise robustness is evaluated separately in Section 4.3.
  • D.5 Optimizer, precision, and compute: All reported PU-HNO runs use a single H200-class NVIDIA GPU and require approximately 1.5 GPU-hours for 35 epochs.No multi-GPU or multi-node training is required.
  • E Baselines and Fair-Comparison Protocol: The comparison protocol evaluates PU-HNO against image-to-image regressors, wireless-specific learning models, and monolithic neural operators.
  • E Baselines and Fair-Comparison Protocol: The appendix documents the comparison protocol and minimal implementation choices needed for reproducibility rather than reintroducing well-known architectures.

E.1 Shared protocol … F.4.2 Fading ratio (LocalStd9)

The evaluation compares PU-HNO with controlled baselines under shared data, capacity, tuning, optimization, and evaluation protocols. It assesses fidelity against HF-GT using standard image metrics and wireless deployment metrics designed to reveal coverage-hole and fading behavior.

  • E.1 Shared protocol: All baselines share the same splits, IF-GT supervision, HF-GT reference, approximately 2.5M-parameter capacity, inputs, hyperparameter search, training recipe, and evaluation pipeline.The target is 2.5M trainable parameters, with realized counts within ±10%; six hyperparameter candidates are narrowed by successive halving.
  • E.2 Baseline families: Image-to-image, wireless deep-learning, and monolithic neural-operator families provide comparisons, while PU-HNO uniquely separates specular, diffraction, and scattering refinements.The image-to-image baselines are CNN, UNet, ResNet, and ViT; wireless baselines include NeRF2, RadioUNet, GeneRT, and WiGATr; operators include FNO, TFNO, UNO, SFNO, CodaNO, and WNO.
  • E.3 Reference: IF-GT labels; F.1 Reference field and aggregation: The IF-GT row measures residual training-label error against HF-GT, making improvement over it evidence that a learned prediction is closer to HF-GT than its labels.All metrics use the held-out HF-GT reference rather than IF-GT labels.
  • E.4 Why rankings differ across metric families; F Evaluation Metrics; F.4 Wireless deployment metrics: Image-quality scores can misrepresent wireless usefulness because smooth RSS structure dominates RMSE, SSIM, and LPIPS, whereas deployment depends on low-RSS, fading, and cell-edge behavior.This motivates reporting wireless-specific metrics alongside standard pixel and perceptual metrics.
  • F.1 Reference field and aggregation: Metrics are computed on valid pixels of the 128 × 128 receive-plane grid against HF-GT, with pixel averages over the test split and image-level averages across scenes.Validity requires finite HF-GT values above Ymin + ε, with Ymin = −150 dB; aggregation is identical for every model.
  • F.2 Pixel and perceptual metrics: MAE, RMSE, and PSNR provide standard pointwise reconstruction measures, with MAE and RMSE in dB and PSNR using the fixed dynamic range R = 170 dB.The PSNR range spans the simulator floor of −150 dB to the ceiling of 20 dB.
  • F.3 Perceptual metrics: SSIM, ESSIM, DISTS, LPIPS, and GradMean assess perceptual and structural similarity, with GradMean interpreted by closeness to HF-GT rather than monotone increase.dB fields are mapped to [0, 1] for SSIM, DISTS, and LPIPS; ESSIM focuses on HF-GT edges.
  • F.4.1 Outage F1; F.4.2 Fading ratio (LocalStd9): Outage F1 evaluates coverage-hole identification below Tout = −100 dBm, while LocalStd9 fading compares local variation to HF-GT, where 1 is ideal.Fading ratios below 1 indicate oversmoothing, and values above 1 indicate excessive local texture.

F.4.3 SE 5% tail error … F.6 Why we report all three families

The paper evaluates wireless predictions through spectral-efficiency tail and cell-edge errors, using fixed indoor-wireless constants and reporting pixel, perceptual, and wireless metric families. Wireless metrics receive the greatest emphasis for deployment-oriented assessment.

  • F.4.3 SE 5% tail error: Spectral efficiency is derived from RSS using a standard AWGN approximation with B = 20 MHz, F = 7 dB, and thermal noise −174 dBm/Hz.These parameters define the noise power used in the conversion.
  • F.4.3 SE 5% tail error: The SE 5% tail error reports absolute error in the 5th percentile over the high-variation mask H.It tests whether predictions preserve the weakest part of the spectral-efficiency distribution.
  • F.4.4 MCESE: mean cell-edge SE error: The cell-edge subset is defined as the bottom 10% of pixels by HF-GT RSS within H.This subset targets the weakest high-variation pixels.
  • F.4.4 MCESE: mean cell-edge SE error: MCESE measures mean absolute spectral-efficiency error on the cell-edge subset.The metric targets locations where coverage and access-point placement decisions are most sensitive.
  • F.5 Summary of constants: Wireless deployment metric constants are fixed across all reported experiments and follow standard 5–6 GHz indoor wireless practice.The constants are summarized in Table 5.
  • F.6 Why we report all three families: Pixel metrics measure average numerical reconstruction error, whereas perceptual metrics measure structural similarity when RSS fields are treated as images.These families evaluate reconstruction and image structure rather than deployment decisions.
  • F.6 Why we report all three families: Wireless metrics assess support for coverage planning, fading-margin estimation, and cell-edge performance assessment, so deployment discussions emphasize them most.Figure 8 compares generic and wireless-specific metrics with 95% bootstrap confidence intervals.

G Extended Results … H.5 Population decomposition

The extended results show that PU-HNO best preserves high-fidelity wireless structure and improves downstream AP-placement interpretations. The theoretical appendix explains that conditionally unbiased finite-ray noise permits population-risk minimization to recover the clean field or its projection onto the model class.

  • G Extended Results: PU-HNO is the best learned predictor across the reported metrics, with GradMean 1.582 closely matching the HF-GT reference value 1.58.IF-GT has GradMean 9.348, indicating substantially greater high-frequency Monte Carlo roughness than the reference.
  • G.1 Downstream Case Study: Access-Point Placement: PU-HNO leaves 148 m2 of missed hard-coverage area, compared with 652 m2 for CNN, 1,000 m2 for GeNeRT, and 892 m2 for WNO.The calculation illustrates AP-placement implications on a 10,000 m2 enterprise floorplan and is not a separate benchmark.
  • G.1 Downstream Case Study: Access-Point Placement: PU-HNO achieves an estimated cell-edge throughput of 92.0 Mbps, compared with 16.6 Mbps for CNN, 0.0 Mbps for GeNeRT, and 70.2 Mbps for WNO.The interpretation uses an 80 MHz channel and a 100 Mbps cell-edge target.
  • G.1 Downstream Case Study: Access-Point Placement: PU-HNO has fading ratio 1.013, closer to the ideal value of 1 than CNN, GeNeRT, and WNO, whose ratios are 0.799, 0.608, and 0.809.The baselines underestimate fading by roughly 19–39%, whereas PU-HNO overestimates it by only 1.3%.
  • H Theoretical appendix: proofs for zero-shot denoising: The appendix formalizes the zero-shot denoising claim through assumptions, finite-ray unbiasedness, population decomposition, and Theorem 4, with extensions to realizable and Sobolev losses.The roadmap identifies these components as the supporting proof structure.
  • H.2 Setup and notation: PU-HNO predicts the scalar RSS field from indoor-scene inputs, while its physical motivation decomposes propagation into broad transport, sharp edge effects, and local fluctuations.The clean field is the simulator average, whereas training uses finite-budget noisy labels and evaluation uses a higher-budget reference.
  • H.3 Standing assumptions: The theoretical setup assumes Y = f⋆(X) + ε with E[ε | X] = 0 and uses bounded loss and model-class conditions to support the analysis.RSS values are clipped to [−150, 20] dB in practice for bounded-loss control.
  • H.4 MC RT as a Hilbert-valued unbiased estimator: Finite-budget Monte Carlo ray tracing is a conditionally unbiased Hilbert-valued estimator whose variance decays as O(m−1), supporting the noise model used in the proofs.The stated ray budgets are m = 10^6 for IF-GT and m = 10^8 for HF-GT.

H.6 Zero-shot denoising theorem · H.7 Realizable feature-regression analogue · H.8 Discrete Sobolev target preservation

Under conditionally unbiased label noise, the PU-HNO analysis shows that empirical risk minimization converges to the clean field and can eventually outperform noisy high-fidelity training labels. A realizable regression analogue explains the noise averaging, while discrete and weighted Sobolev losses preserve the clean population target.

  • H.6 Zero-shot denoising theorem: O(n^-1/2) convergence to the clean field holds under assumptions (A1)–(A4), with sufficiently many samples yielding predictions at least as close to the high-fidelity reference as noisy labels.The theorem identifies a finite-sample crossover n⋆ beyond which this denoising guarantee applies.
  • H.6 Zero-shot denoising theorem: The crossover condition 2ε2_MC > 0 captures an asymmetric-label-quality regime in which model approximation error is small relative to reference quality and training noise is sufficiently large.The left side is controlled through architecture and reference budget, while the right side is controlled through the IF-GT ray budget.
  • H.6 Zero-shot denoising theorem: The theorem’s proof combines a Rademacher oracle inequality, a clean-field bound, an HF-GT decomposition, and a conditional-zero-mean label identity whose cross-term vanishes.The label identity uses E[ε | X] = 0 to eliminate the cross-term.
  • H.7 Realizable feature-regression analogue: The exact feature-regression analogue assumes whitened features, a realizable linear target, conditionally zero-mean noise, and a well-conditioned empirical covariance, then derives an OLS noise-averaging guarantee.The proposition is presented as an independent finite-dimensional illustration of the abstract bound’s 1/√n scaling.
  • H.7 Realizable feature-regression analogue: m/(µn) estimation error arises with m effective parameters and per-sample variance σ2, while noise contributes through a σ2 prefactor dominated by 1/n once the sample-size threshold is exceeded.The stated threshold is n > mσ2/(µ · target).
  • H.8 Discrete Sobolev target preservation: The discrete Sobolev population minimizer remains f⋆ when a fixed finite-difference gradient term is added to the loss, because conditional zero-mean noise eliminates the transformed cross-term.The result applies to fields represented on a fixed grid and any α > 0.
  • H.8 Discrete Sobolev target preservation: X-measurable bounded geometry weights preserve consistency with f⋆ in the kernel-quotient sense, so the distance-modulated Sobolev loss does not bias the population target.The same argument covers confidence-modulated L2 stage losses when α = 0.
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