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Physics-informed VAE-EVT for Tail Aware Radio Map Prediction

Amanda Sheron Gamage, Niloofar Mehrnia, James Gross

arXiv:2608.15314v1cs.AI

TL;DR

URLLC radio-map prediction needs to identify rare, low-SNR outage regions, which traditional models may represent poorly. Physics-informed VAE-EVT separates bulk and tail SNR distributions using scene-geometry features and achieves 4.83 dB outage RMSE at the 0.1% quantile, versus 21.90 dB for RME-GAN.

  • Problem

    Traditional radio-map models provide limited evidence for accurately predicting rare, severe low-SNR regions that determine stringent URLLC outage reliability.

  • Method

    Physics-informed VAE-EVT uses deterministic LoS/NLoS, shadowing, and distance features with dual latent variables modeling bulk SNR by GMM and tail outages by GPD.

  • Results

    4.83 dB outage RMSE at the 0.1% threshold, outperforming RME-GAN at 21.90 dB; EVT-based models show lower error distributions especially at this extreme threshold.

  • Takeaways & Limitations

    The results support combining physics-grounded features with tail modeling for URLLC-grade outage characterization across 600 test environments.

Abstract

from arXiv · show

Ultra-reliable low-latency communication (URLLC) requires precise identification of spatial regions where the signal-to-noise ratio (SNR) falls below an outage threshold. In this context, an outage refers to instances in which SNR falls below a specified threshold, which, for URLLC, can be as stringent as the 0.1% quantile of the SNR distribution. Traditional generative radio map models tend to focus on reconstructing average signal levels, often overlooking the low SNR that is crucial for accurate outage prediction. To address this limitation, we introduce a physics- and tail-informed VAE-EVT (variational autoencoder-extreme value theory) framework that distinctly models both the bulk and tail distribution of SNR. Our approach begins with a physics-informed preprocessing stage that extracts deterministic features, including line-of-sight, shadowing, and distance, from the scene geometry. A dual-latent encoder then captures the bulk SNR using a Gaussian mixture and the tail using a generalized Pareto distribution (GPD). By employing a modified variational objective, the model is trained to jointly supervise both regimes, ensuring focused attention on extreme fading events. Evaluated on the RadioMapSeer dataset, our method achieves an SNR RMSE of 4.83 dB in the outage region defined by the low threshold of 0.1% SNR quantile. This significantly outperforms the state-of-the-art GAN-based model, which records an SNR RMSE of 21.90 dB, with the performance gap widening as the outage threshold becomes more stringent.

I. INTRODUCTION

URLLC demands highly reliable, spatially precise radio-map predictions, while existing methods struggle to scale or capture localized outage tails. The proposed Physics-informed VAE-EVT framework addresses this gap by separately modeling bulk and extreme signal distributions under deterministic scene-geometry constraints.

  • Motivation: URLLC requires outage probabilities as low as 10^-9 to 10^-5, making precise spatial radio-map prediction essential for reliable network operation.Radio maps support network planning and proactive resource allocation.
  • Limitations of Existing Methods: Deterministic ray tracing is accurate but difficult to scale, while generative approaches often neglect spatially localized deep fades and outage tails.An EVT-GenAI method addresses temporal tails on a single link rather than spatial outage distributions across environments.
  • Proposed Framework: The proposed Physics-informed VAE-EVT framework separately learns bulk and extreme signal quantiles while using deterministic scene geometry to enforce physical consistency.The framework integrates generative AI, EVT, and physics-informed learning for radio-map prediction.
  • Key Contributions: A physics-informed feature pipeline extracts LoS/NLoS structure, shadow depth, and distance attenuation as propagation priors for model inputs.These features directly encode scene-geometry information into the prediction process.
  • Key Contributions: The Dual Latent Encoder combines a standard VAE for bulk-signal conditions with EVT, using a Gaussian distribution for bulk signals and a GPD for extreme outages.This explicitly separates bulk and extreme regimes in the latent space.

II. SYSTEM MODEL AND PROBLEM FORMULATION … C. Bulk and Tail of the Conditional SNR Distribution

The section formulates conditional SNR prediction over valid receiver locations by separating deterministic geometric structure from probabilistic variability. It models the bulk with a Gaussian mixture and rare low-SNR fading with a generalized Pareto distribution, jointly supervised through a specialized VAE objective.

  • II. SYSTEM MODEL AND PROBLEM FORMULATION: The urban environment is discretized into an M×M binary building occupancy grid, with valid receiver locations defined as free-space pixels.The objective is to characterize the received SNR distribution across these valid locations.
  • A. Outage Definition: An outage occurs when the SNR γ(p) falls below a predefined critical threshold γ_th, defining the corresponding outage region.
  • B. Deterministic Geometric Structure and Residual Variability: The SNR combines a deterministic large-scale component governed by scene geometry with small-scale variability caused by diffraction, scattering, and building-edge interactions.
  • B. Deterministic Geometric Structure and Residual Variability: Distance, LoS/NLoS condition, and localized shadowing are computed from the building map and transmitter location, forming geometric descriptors X_geo for p(γ|X_geo).These descriptors are obtained through ray tracing.
  • C. Bulk and Tail of the Conditional SNR Distribution: Conditioned on X_geo, average-statistics SNR in the bulk regime is modeled with a Gaussian Mixture Model.The mixture represents distinct propagation states, including LoS, NLoS, and partial obstruction.
  • C. Bulk and Tail of the Conditional SNR Distribution: The GMM uses K Gaussian components with mixing weights, means, and variances that describe the conditional bulk distribution.The mixing weights are constrained to sum to one.
  • C. Bulk and Tail of the Conditional SNR Distribution: A finite Gaussian mixture underestimates severe rare fading, so SNR below γ_th is modeled with a generalized Pareto distribution for low-threshold exceedances.This tail model follows the Pickands-Balkema-de Haan theorem.
  • C. Bulk and Tail of the Conditional SNR Distribution: A specialized VAE objective jointly supervises the full conditional distribution, using Gaussian reconstruction for the bulk and GPD reconstruction for the tail.

III. PROPOSED VAE-EVT FRAMEWORK FOR TAIL-AWARE RADIO MAP PREDICTION

The Physics-informed VAE-EVT framework combines physics-informed preprocessing with a modified dual-latent VAE and tail-aware objective. Together, these components learn bulk and tail regimes to estimate p(γ|X_geo).

  • The framework comprises physics-informed preprocessing and a modified VAE architecture with a dual-latent encoder and tail-aware training objective.Preprocessing extracts the deterministic structural tensor X_geo from scene information.
  • The combined framework simultaneously learns bulk and tail regimes to accurately estimate p(γ|X_geo).

A. Physics-informed Preprocessing · 1) Distance and Transmitter Proximity: · 2) LoS Mask and NLoS Penetration Depth:

Physics-informed preprocessing converts transmitter and building geometry into deterministic, grid-aligned descriptors. These features encode distance, transmitter proximity, line-of-sight conditions, and non-line-of-sight obstruction severity, with building pixels masked during training and evaluation.

  • A. Physics-informed Preprocessing: The preprocessing stage converts transmitter position and building geometry into a tensor X_geo of deterministic geometric descriptors.All descriptor channels are defined on the full M×M grid.
  • A. Physics-informed Preprocessing: Building pixels with B(p) = 1 have all descriptor channels set to 0 and are masked during training and evaluation.This masking applies consistently across the full grid.
  • 1) Distance and Transmitter Proximity:: For each pixel, Euclidean transmitter distance d(p) = ∥p − p_tx∥_2 yields a min–max normalized log-distance channel with ε_d > 0 preventing log 0.The log-distance channel is defined as log(d(p) + ε_d).
  • 1) Distance and Transmitter Proximity:: A linearly normalized distance channel D_all(p) over Ω provides a stable conditioning input for the tail and outage branches.This channel complements the normalized log-distance representation.
  • 1) Distance and Transmitter Proximity:: A smooth transmitter-proximity map T_x(p) is formed as a truncated Gaussian centered at p_tx, with σ_T > 0 selected relative to grid scale.The map uses exponential decay from the transmitter location.
  • 2) LoS Mask and NLoS Penetration Depth:: The binary LoS mask M_LOS(p) equals 1 when the discretized transmitter-to-pixel ray avoids building pixels and 0 otherwise.The ray-intersection rule determines whether each pixel is line of sight.
  • 2) LoS Mask and NLoS Penetration Depth:: The obstruction severity for NLoS pixels is represented by a normalized penetration-depth feature based on the number of building intersections n_hits(p).The normalization uses a fixed d_max, such as the maximum n_hits observed in the training set.

3) Localized Shadowing and Edge Map: · 4) Outage Prior:

The preprocessing augments binary LoS information with localized shadowing and an edge-guidance map, then supplies a coarse geometric outage prior and broadcast threshold encoding. These features target partial blockage, spatial transitions, and environment-specific outage thresholds.

  • 3) Localized Shadowing and Edge Map:: Localized shadowing scores S_NLOS(p) ∈ [0, 1] capture partial shadowing near corners and corridors that binary LoS masks miss.The score is defined from sampled points along the segment toward the transmitter.
  • 3) Localized Shadowing and Edge Map:: For each pixel p, S_NLOS(p) is computed as the fraction of N_s equally spaced segment samples falling inside buildings.LoS pixels are assigned S_NLOS(p) = 0.
  • 3) Localized Shadowing and Edge Map:: The edge map E(p) provides spatial-attention guidance by emphasizing LoS/NLoS boundaries and strong shadow transitions.Image gradients are estimated with Sobel operators and smoothed with a Gaussian kernel of standard deviation σ_E.
  • 3) Localized Shadowing and Edge Map:: E(p) combines weighted gradient terms, applies fixed normalization, and clips the result to [0, 1].The weighting coefficients α1 and α2 are nonnegative.
  • 4) Outage Prior:: The outage prior provides a coarse geometric baseline for outage risk.Its coefficients a1, b1, b2, and b3 are nonnegative, while ε ∈ (0, 0.5) avoids degenerate probabilities.
  • 4) Outage Prior:: For each radio map, γ_th is the q-th percentile of SNR values, with q ∈ (0, 100) fixed across the dataset.The normalized threshold encoding γ̂_th is broadcast to all pixels as an additional channel using training-set constants γ_min and γ_max.

5) Per-map Threshold and Input Tensor: · B. Dual-Latent Encoder

The method constructs a geometry-informed input tensor with per-map tail thresholds and fixed GPD anchors, then encodes bulk and tail information through separate latent pathways. The bulk latent is Gaussian, while the tail latent is transformed into a GPD-anchored representation.

  • 5) Per-map Threshold and Input Tensor:: The per-map threshold normalization includes ε_γ > 0 to prevent division by zero.The normalization uses the range γ_max − γ_min plus ε_γ.
  • 5) Per-map Threshold and Input Tensor:: Dataset-level GPD parameters (ξ̂, β̂) are maximum-likelihood estimates from training-set exceedances below γ_th and remain fixed during training.The exceedances are γ_th − γ for pixels satisfying γ < γ_th.
  • 5) Per-map Threshold and Input Tensor:: The final geometry-informed input tensor stacks B, T_x, M_LOS, d_L, S_NLOS, D_NLOS, E, P_outage, D_wall, and γ̂_th.This tensor corresponds to the Input Tensor block in Fig. 1.
  • B. Dual-Latent Encoder: The encoder approximates q_ϕ(z_bulk, z_tail | X_geo) with a convolutional backbone producing multi-resolution features for decoder U-Net skip connections.The encoder is shown as the Dual Latent Encoder block in Fig. 1.
  • B. Dual-Latent Encoder: Two bottleneck dense heads parameterize the bulk latent and an auxiliary tail pathway mapped to a GPD-anchored latent.The heads separately produce parameters for the two latent regimes.
  • B. Dual-Latent Encoder: The bulk latent uses a diagonal Gaussian with encoder outputs μ_g and log σ_g^2, sampled by reparameterization as z_bulk = μ_g + σ_g ⊙ ε.Here ε ∼ N(0, I), σ_g = exp(1/2 log σ_g^2), and d_z is the bulk latent dimension.
  • B. Dual-Latent Encoder: The tail pathway first samples an auxiliary diagonal-Gaussian variable z_aux from parameters μ_p and log σ_p^2.The auxiliary variable is reparameterized using ε ∼ N(0, I) and σ_p = exp(1/2 log σ_p^2).
  • B. Dual-Latent Encoder: A sigmoid transforms z_aux into u ∈ (0, 1)^d_t, which the fixed GPD quantile function maps into z_tail using (ξ̂, β̂).When |ξ̂| is very small, the stable limit z_tail ≈ −β̂ log(1 − u) is used.

C. Concatenation and U-Net Decoder with Spatial Attention … 2) Tail Branch:

The model concatenates bulk and tail latents to seed a spatial-attention U-Net decoder, whose masked output heads separately model bulk reconstruction and outage-focused tail SNR.

  • C. Concatenation and U-Net Decoder with Spatial Attention: The bulk and tail latents are concatenated as z = [z_bulk; z_tail], then transformed by a fully connected layer into a low-resolution decoder feature map.This feature map seeds subsequent spatial reconstruction.
  • C. Concatenation and U-Net Decoder with Spatial Attention: The decoder progressively upsamples features to M×M, combines matching-resolution encoder features through skip connections, and refines them with convolutional residual blocks.This follows a U-Net design for reconstructing the spatial SNR map.
  • C. Concatenation and U-Net Decoder with Spatial Attention: A spatial attention module is applied within the decoder to refine the reconstructed spatial feature representation.
  • D. Output Heads and Final Prediction: Three output heads receive the shared decoder feature map, while the free-space mask m(p) = 1−B(p) excludes building pixels from reconstruction losses.The mask is used to force building pixels to be excluded from reconstruction losses.
  • 1) Bulk Branch:: The bulk branch outputs μ(p) and log σ^2(p) per pixel, defining a Gaussian likelihood for non-outage SNR reconstruction.This branch models the non-outage regime.
  • 2) Tail Branch:: The tail branch outputs y_t(p), an outage-focused SNR estimate that may use a nonnegative shortfall parameterization tied to the outage threshold.With ŷ(p) ≥0, y_t(p) = γ_th−ŷ(p), so larger shortfalls produce smaller tail SNR values.

3) Outage Branch: · 4) Final Output: · E. VAE-EVT Training Loss

The outage branch predicts and sharpens per-pixel outage probabilities to route bulk or tail predictions into the final SNR estimate. VAE-EVT training combines reconstruction, outage-region emphasis, and latent regularization to prioritize rare low-SNR pixels and stabilize the tail pathway.

  • 3) Outage Branch:: The outage branch predicts a per-pixel outage probability π(p) ∈ [0, 1] and masks building pixels using m(p).Building pixels are assigned π(p) ← m(p)π(p).
  • 3) Outage Branch:: A steep sigmoid sharpens π(p) into π_s(p), with κ controlling steepness and t* selected on validation data by maximizing F1 score.The operation corresponds to π→π_s in Fig. 1.
  • 4) Final Output:: The final per-pixel SNR prediction blends the bulk mean μ(p) and tail prediction y_t(p) according to the sharpened outage mask π_s(p).The prediction is γ̂(p) = (1 − π_s(p))μ(p) + π_s(p)y_t(p).
  • E. VAE-EVT Training Loss: The model is trained end-to-end with reconstruction, latent regularization, and explicit supervision for outage routing.The objective uses scalar weights λ· ≥ 0 that are fine-tuned during training.
  • E. VAE-EVT Training Loss: The reconstruction loss routes learning by predicted outage probability, emphasizing bulk learning when π is small and tail learning when π is large.This routing makes the reconstruction objective adapt to the predicted regime.
  • E. VAE-EVT Training Loss: The outage loss increases penalties for rare outage pixels whose true SNR satisfies γ(p) ≤ γ_th.It averages squared prediction errors over O and uses ε_O > 0 to avoid division by zero when O is empty.
  • E. VAE-EVT Training Loss: Bulk and tail KL terms regularize their latent pathways, with the tail term acting on the auxiliary Gaussian because z_tail is a deterministic transform of z_aux.This stabilizes tail sampling while preserving the GPD mapping to z_tail.

4) Outage Supervision L휋:

The outage loss supervises pixelwise outage probabilities using masked labels and can combine classification, overlap, and calibration terms. Sharpening regularization promotes near-binary routing, while LoS emphasis addresses distance-driven outages that obstruction-based features may miss.

  • Outage Supervision Lπ: The outage loss supervises π(p) with masked labels ℓ(p) = I{γ(p) < γ_th} and applies all components to π rather than π_s.It may combine focal or weighted crossentropy with region-level overlap and calibration terms.
  • Sharpening Regularization L_sharp: Sharpening regularization penalizes high-entropy outage probabilities, encouraging π and π_s to approach near-binary routing decisions.A small ϵ_π > 0 provides numerical stability.
  • LoS Outage Emphasis L_LoS: Uniformly averaged losses can be dominated by NLoS outages, causing the tail head to learn shadowing-driven representations that mispredict the LoS outage minority.NLoS outages are explained by obstruction descriptors, whereas LoS outages can result from distance attenuation despite unobstructed paths.
  • LoS Outage Emphasis L_LoS: LoS outage emphasis complements the D_all channel, enabling tail and outage branches to represent distance-driven outages beyond LoS-masked geometric channels.Both mechanisms address outage behavior that obstruction-focused inputs alone cannot represent.

IV. NUMERICAL RESULTS … C. Discussion

The study evaluates Physics-informed VAE-EVT against ablations and RME-GAN on RadioMapSeer, showing improved extreme-outage prediction and more faithful spatial and bimodal SNR structure. EVT improves tail fidelity, while physics conditioning positions and preserves the distribution, with single-pass generation supporting URLLC deployment.

  • 1) Dataset:: 300 layouts with 10 transmitter positions each yield 3000 environment–transmitter pairs from RadioMapSeer’s 256 × 256 m^2 urban maps.The dataset uses Dominant Path Model maps simulated with WinProp and an 80/20 split.
  • 2) Implementation Details:: 100 LoS-ray samples, 30 localized-shadow samples, and d_max = 10 normalize geometry-derived features for the encoder.The sampling choices target pixel-scale path resolution and local obstruction detection; d_max corresponds to the 99th percentile of observed wall crossings.
  • B. Baselines Models: The comparison includes Physics-informed VAE-EVT, VAE-EVT, Physics-informed VAE, and RME-GAN, isolating dual-latent EVT modeling and physics preprocessing.The proposed model combines a dual latent EVT encoder, physics preprocessing, and explicit LoS/NLoS feature routing.
  • B. Baselines Models: 600 test maps are evaluated at q ∈ {0.1%, 10%} using outage RMSE, F1-score, Precision, and Recall.Outage RMSE measures tail prediction fidelity, while F1-score measures spatial outage localization accuracy.
  • C. Discussion: Physics-informed VAE-EVT most closely matches ground truth, reproducing bright near-transmitter regions, deep shadows, and both peaks of the bimodal SNR distribution.RME-GAN misses fine-grained building-boundary shadows, while VAE-EVT misses the high-SNR transmitter region and collapses the bimodal peaks.
  • C. Discussion: 4.83 dB outage RMSE is achieved by Physics-informed VAE-EVT at q = 0.1%, versus 12.03 dB for Physics-informed VAE and 21.90 dB for RME-GAN.VAE-EVT also achieves 6.56 dB, indicating that EVT-equipped models retain accuracy at extreme outage levels.
  • C. Discussion: EVT-based models have lower outage-RMSE means and variances across 600 maps, especially at 0.1%, while 10% evaluation is less indicative of extreme-value prediction.At 10%, bulk-similar samples increasingly influence the GPD fit and RMSE, compressing model differences.
  • C. Discussion: EVT captures tail shape, physics conditioning preserves bimodal bulk structure and positions SNR correctly, and the model generates a full 256 × 256 map in one forward pass.Unlike diffusion-based models requiring iterative denoising, the proposed framework predicts from scene geometry alone rather than sparse true-SNR observations.

V. CONCLUSIONS

The Physics-informed VAE-EVT framework improves tail-aware radio map prediction by decoupling bulk and tail SNR modeling and embedding deterministic LoS/NLoS features. It achieves 4.83 dB outage RMSE at the URLLC-critical 0.1% quantile, with systematic gains across 600 test environments.

  • Contributions and Results: 4.83 dB outage RMSE is achieved at the URLLC critical 0.1% quantile threshold.This compares with 12.03 dB for a Physics-informed VAE and 21.90 dB for the state-of-the-art RME-GAN.
  • Contributions and Results: The framework decouples bulk and tail modeling through a dual-latent encoder and embeds deterministic LoS/NLoS features.These design choices form the proposed Physics-informed VAE-EVT framework for tail-aware radio map prediction.
  • Contributions and Results: The RMSE distributions confirm that the performance advantage is systematic across all 600 test environments.The reported comparisons are against a Physics-informed VAE and the state-of-the-art RME-GAN.
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