RF-NeRF: Neural Radio-Frequency Radiance Fields

• RF-NeRF is a neural representation that encodes both amplitude and phase of electromagnetic signals, enabling high-fidelity RF behavior modeling.
• It leverages MLPs with Fourier features and grid-based hybrid approaches for efficient volumetric ray tracing and advanced channel modeling.
• Applications span channel prediction, localization, and RIS optimization, achieving significant improvements in accuracy and computational efficiency.

Neural Radio-Frequency Radiance Field (RF-NeRF) models extend volumetric neural scene representations originally developed for computer vision into the radio-frequency (RF) domain. By encoding amplitude and phase variations of electromagnetic propagation into neural fields, RF-NeRF frameworks deliver high-fidelity predictions of complex signal behaviors in challenging wireless environments, capturing effects such as multipath, attenuation, scattering, and programmable surfaces such as RIS (Reconfigurable Intelligent Surfaces). Across several instantiations, RF-NeRF paradigms have become central to channel modeling, localization, RF-based environment understanding, and optimization for next-generation wireless systems.

1. Mathematical Foundations and Core Formalism

RF-NeRF generalizes the concept of neural radiance fields from optics—where each point in a scene emits or transmits a color along rays—to RF, where each point modulates both the amplitude and phase of an incident electromagnetic wave. Instead of RGB color, the signal at each spatial coordinate is treated as a complex phasor $S(x, d) = A(x, d) e^{i\alpha(x, d)}$ and attenuation is encoded as a complex transmission $T(x) = \delta(x) e^{i\beta(x)}$ (Yang et al., 19 May 2024, Zhao et al., 2023).

The volumetric rendering integral for the received RF field along a ray parameterized by spatial coordinate $x(t) = o + t\cdot d$ is:

$$E(r, d) = \int_{t_n}^{t_f} T(t)\, \sigma(x(t))\, c(x(t), d) \; dt$$

where $$T(t) = \exp\left(-\int_{t_n}^t \sigma(x(s))\, ds\right)$$, and $c(x, d)\in\mathbb{C}$ encodes the local emission (amplitude and phase) in direction $d$, and $\sigma(x)$ models attenuation or scattering. Discretized along sampled voxels with learned phasor and transmission responses enables practical differentiable ray tracing through RF fields.

The neural parameterization uses multilayer perceptrons (MLPs) or encoder-decoder networks, typically with high-frequency Fourier-feature encodings $\gamma(\cdot)$ for position and direction, to represent scene-dependent functional mappings (Shen et al., 10 Apr 2025, Wang et al., 8 Dec 2025).

2. Architecture Variants and Scene Representation

Different RF-NeRF frameworks adapt the neural functional representation in ways tailored to their modeling targets and computational constraints:

• MLP-based volumetric fields: Standard RF-NeRF (including NeRF$^2$) and NeRF-APT use deep MLPs to map from high-dimensional encoded positions, directions, and transmitter coordinates to complex attenuation and radiance (Zhao et al., 2023, Shen et al., 10 Apr 2025). NeRF-APT further introduces a U-Net style architecture with attention-gated skip connections and spatial-pyramid pooling for enhanced feature propagation.
• Grid-accelerated hybrid approaches: VoxelRF replaces the heavy MLP with an explicit voxel-grid storing per-voxel densities and features, combined with shallow MLPs that modulate local deformation and radiance, using trilinear interpolation to query the scene (Zeng et al., 14 Jul 2025). This hybrid explicit–implicit design is optimized for efficiency.
• Reflectance fields: Surface-centric neural reflectance fields model per-surface, incident-angle-dependent complex RF reflectance, where each surface’s reflection coefficient $$\delta(\theta, s) = \Delta A(\theta, s) \cdot e^{j\Delta\Theta(\theta, s)}$$ is produced by an MLP taking incident angle and surface ID (Jia et al., 5 Jan 2025). The scene geometry is assumed known and decomposed into labeled surfaces.

RIS-enabled environments require explicit two-stage modeling, with the RF-NeRF decomposing propagation into “TX→RIS” (incoming fields to the meta-surface) and “RIS→RX” (programmable re-radiation) with separate networks at each stage (Yang et al., 19 May 2024).

3. Ray Tracing and RF Rendering Pipeline

All RF-NeRF models rely on differentiable ray tracing, either in a volumetric continuous formulation or discretized over sampled voxels. The canonical discrete ray-marching equation is:

$$R(\omega) = \sum_{n=1}^N \exp\left(-\sum_{m=1}^{n-1} \delta(V_m)\right) S(V_n, -\omega)$$

where each sample $V_n$ along direction $\omega$ contributes (potentially complex) amplitude and phase modulation, and the cumulative attenuation $\delta(V_m)$ enforces exponential depth-dependent loss.

Extensions support multi-bounce reflections, material-dependent attenuation, multipath superposition, and programmable surfaces (Shen et al., 10 Apr 2025, Jia et al., 5 Jan 2025, Yang et al., 19 May 2024). VoxelRF introduces empty-space skipping (ignoring samples with density $\sigma(x)<\tau$) and progressive grid upsampling for computational speed (Zeng et al., 14 Jul 2025).

In RIS scenarios, the two-stage process computes the field received at the RIS from the transmitter, sums over spatial samples to model interception, and then models outgoing fields emitted toward the receiver, accurately capturing reconfigurable metasurface effects (Yang et al., 19 May 2024).

4. Training Objectives, Losses, and Data Efficiency

RF-NeRF frameworks supervise training by minimizing discrepancies between predicted and measured RF observables:

• Signal magnitude losses: Mean-squared error (MSE) on magnitude or dB-scaled field/power values, either per spectrum direction or per sample (Zhao et al., 2023, Zeng et al., 14 Jul 2025, Yang et al., 19 May 2024).
• Structural similarity: SSIM loss on measured and synthesized spectrum “images” for dense sensor arrays (Shen et al., 10 Apr 2025, Zeng et al., 14 Jul 2025).
• Specialized loss terms: VoxelRF adds a background entropy loss on final ray transmittance to regularize empty/occluded regions (Zeng et al., 14 Jul 2025). NeRF-APT includes multi-scale losses at decoder outputs, and pseudo-supervised self-consistency terms for spectrum reconstruction appear in self-supervised pretraining (Wang et al., 8 Dec 2025).

The reliance on volumetric neural representations provides strong data efficiency, especially compared to surface parameterizations or traditional ray tracing, and grid-based hybrid methods (VoxelRF) further increase sample efficiency and accelerate convergence (Zeng et al., 14 Jul 2025, Shen et al., 10 Apr 2025).

5. Applications and Empirical Performance

RF-NeRF models achieve strong performance in a wide range of wireless channel modeling and RF sensing tasks:

• Channel prediction: NeRF-APT and NeRF$^2$ provide high-fidelity predictions of received RSSI, CSI, and spectrum, outperforming plain MLPs, VAEs, and classical interpolation (Shen et al., 10 Apr 2025, Zhao et al., 2023).
• Localization: RF-NeRF-based pretraining (e.g., RFRP) dramatically improves indoor localization accuracy. RFRP yields over 40% error reduction in few-shot localization benchmarks compared to standard transformers, and 21% versus supervised pretraining (Wang et al., 8 Dec 2025).
• RIS optimization: R-NeRF predicts full 3D field strengths for arbitrary RIS placements, facilitating automated site surveys and coverage optimization; over 94% of measured points achieve error within 5 dB, outperforming NeRF$^2$ and traditional interpolators (Yang et al., 19 May 2024).
• Efficiency and scalability: VoxelRF achieves ∼10× faster inference and ∼47× faster training than MLP-based RF-NeRF, with median SSIM ≈0.90 on realistic spatial spectrum tasks (Zeng et al., 14 Jul 2025).
• Synthetic dataset generation: NeRF$^2$’s “turbo-learning” strategy mixes a small amount of real measurements with vast amounts of NeRF-generated synthetic data, reducing error by ∼50% in localization and AoA estimation, and cutting the required real data by an order of magnitude (Zhao et al., 2023).

An overview of architectures and their comparative attributes:

Architecture	Scene Encoding	Training Time (RFID)	Med. SSIM/RSSI Error
NeRF$^2$ (Zhao et al., 2023)	Deep MLP ($\sim$8x512) + PE	15.5 hrs	0.80 / 2.85 dB
VoxelRF (Zeng et al., 14 Jul 2025)	160$^3$ grid + 2x shallow MLP (256)	20 min	0.90 / 2.66 dB
NeRF-APT (Shen et al., 10 Apr 2025)	U-Net+Attention APT, Conv	varies	0.866 / 3.52 dB

6. Limitations, Challenges, and Extensions

Several limitations remain in current RF-NeRF models:

• Scene assumptions: Most models assume static environments; handling of dynamic, time-varying, and frequency-selective channels is limited (Zeng et al., 14 Jul 2025, Shen et al., 10 Apr 2025).
• Compositional complexity: Modeling multiple transmitters, receivers, or RIS surfaces, or generalizing to large, variable-complexity scenes, is an open area (Yang et al., 19 May 2024).
• Physical priors: Diffraction, edge effects, polarization, and frequency-dependent reflection are not generally incorporated, though surface neural reflectance models suggest routes for integrating such priors (Jia et al., 5 Jan 2025).
• Computational footprint: High-resolution volumetric or grid-based encodings may impose significant memory costs for very large or complex venues. Adaptive or sparse data structures are proposed as future work (Zeng et al., 14 Jul 2025).
• Integration with geometry: Some methods require a priori geometric models; others learn volumetric properties directly from channel data, with tradeoffs in scalability and generalization (Jia et al., 5 Jan 2025, Zhao et al., 2023).

Extensions under investigation include hybrid representations, learned hashing for memory reduction, dynamic scene modeling, physics-informed priors, and real-time applications (e.g., scene-aware 6G deployment or automated site planning) (Yang et al., 19 May 2024, Zeng et al., 14 Jul 2025).

7. Impact and Future Directions

RF-NeRF methods now underpin a new class of physically-grounded, data-driven tools for modeling wireless propagation, channel estimation, localization, RIS placement optimization, and RF environmental understanding. Their differentiable nature enables integration with end-to-end learning paradigms, robust self-supervised pretraining, and generative synthetic data pipelines that bridge statistical priors with Maxwellian physics.

Research directions emerging from the RF-NeRF paradigm include: dynamic and wideband channel modeling, hierarchical scene decomposition, surface and edge diffraction integration, memory-efficient spatial representations, and cross-domain transfer for 6G and beyond. These models offer a foundation for real-time, sample-efficient, and physically-consistent wireless environment inference and design.

Key References:

• "Neural Radio-Frequency Radiance Fields" (Zhao et al., 2023)
• "NeRF-APT: A New NeRF Framework for Wireless Channel Prediction" (Shen et al., 10 Apr 2025)
• "R-NeRF: Neural Radiance Fields for Modeling RIS-enabled Wireless Environments" (Yang et al., 19 May 2024)
• "VoxelRF: Voxelized Radiance Field for Fast Wireless Channel Modeling" (Zeng et al., 14 Jul 2025)
• "Neural Reflectance Fields for Radio-Frequency Ray Tracing" (Jia et al., 5 Jan 2025)
• "Radiance-Field Reinforced Pretraining: Scaling Localization Models with Unlabeled Wireless Signals" (Wang et al., 8 Dec 2025)