Paper deep dive
Physics-informed VAE-EVT for Tail Aware Radio Map Prediction
Amanda Sheron Gamage, Niloofar Mehrnia, James Gross
Intelligence
Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 93%
Last extracted: 8/18/2026, 6:12:44 AM
Summary
The paper introduces a Physics-informed VAE-EVT framework for predicting radio maps with a focus on tail-aware signal-to-noise ratio (SNR) estimation for Ultra-reliable Low-Latency Communication (URLLC). The model uses a dual-latent encoder to separately model the bulk SNR distribution (using a Gaussian Mixture Model) and the tail distribution (using Extreme Value Theory with a Generalized Pareto Distribution). Physics-informed features derived from scene geometry, such as line-of-sight and shadowing, are used as input. The method significantly outperforms state-of-the-art GAN-based models in predicting low-SNR outage regions.
Entities (8)
Relation Signals (6)
Physics-informed VAE-EVT → evaluatedon → RadioMapSeer
confidence 95% · Evaluated on the RadioMapSeer dataset, our method achieves an SNR RMSE of 4.83 dB
Physics-informed VAE-EVT → outperforms → GAN
confidence 95% · This significantly outperforms the state-of-the-art GAN-based model, which records an SNR RMSE of 21.90 dB
Physics-informed VAE-EVT → uses → Generalized Pareto Distribution
confidence 95% · A dual-latent encoder then captures the bulk SNR using a Gaussian mixture and the tail using a generalized Pareto distribution (GPD).
Physics-informed VAE-EVT → uses → Extreme Value Theory
confidence 95% · we introduce a physics- and tail-informed VAE-EVT (variational autoencoder-extreme value theory) framework
Physics-informed VAE-EVT → extractsfeatures → Line-of-Sight
confidence 90% · Our approach begins with a physics-informed preprocessing stage that extracts deterministic features, including line-of-sight, shadowing, and distance, from the scene geometry.
Physics-informed VAE-EVT → targets → URLLC
confidence 90% · Ultra-reliable low-latency communication (URLLC) requires precise identification of spatial regions where the signal-to-noise ratio (SNR) falls below an outage threshold.
Cypher Suggestions (0)
No Cypher suggestions yet.
Abstract
Abstract: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.
Tags
Links
- Source: https://arxiv.org/abs/2608.15314v1
- Canonical: https://arxiv.org/abs/2608.15314v1
Trouble viewing inline? Open PDF directly →
Full Text
42,579 characters extracted from source content.
Expand or collapse full text
e-mails:asgamage,nilome,jamesgr@kth.se Physics-informed VAE-EVT for Tail Aware Radio Map PredictionNiloofar Mehrnia acknowledges the support of Vinnova (Sweden’s Innovation Agency) under Grant #2025-01333. Amanda Sheron Gamage, Niloofar Mehrnia, and James Gross Affiliation: Department of Electrical Engineering and Computer Science, KTH Royal Institute of Technology, Sweden Abstract 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%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 4.834.83 dB in the outage region defined by the low threshold of 0.1%0.1\% SNR-quantile. This significantly outperforms the state-of-the-art GAN-based model, which records an SNR RMSE of 21.9021.90 dB, with the performance gap widening as the outage threshold becomes more stringent. Index Terms: URLLC, Gen-AI, radio map, EVT, VAE, physics-informed learning. I Introduction The rapid expansion of mission-critical applications, such as autonomous driving and industrial automation, has placed unprecedented demands on wireless networks, establishing Ultra-reliable and low latency communication (URLLC) as an essential design paradigm for next-generation systems [1]. URLLC fundamentally requires outage probabilities as low as 10−910^-9 to 10−510^-5. Achieving such stringent reliability at the network level depends on precise radio map predictions, which provide detailed spatial insights into signal quality and serve as the geographic basis for guaranteeing reliability [17]. Identifying the spatial locations where signal strength drops below a threshold is vital, as these infrequent but severe fades largely determine overall reliability. Traditional model-based methods often fall short in capturing the intricate, site-specific fading characteristics anticipated in 6G environments. Consequently, data-driven approaches, particularly those leveraging Generative Artificial Intelligence (GenAI) models, have become indispensable for accurately modeling outage behavior. A radio map provides a spatially resolved representation of signal quality across a geographic area, serving as a foundational tool for network planning and proactive resource allocation [17]. Although deterministic methods like ray tracing can deliver highly accurate results, they depend on exhaustive, site-specific environmental data, which significantly limits their scalability in varied and rapidly changing settings. This limitation has driven sustained interest in learning-based alternatives, such as RadioUNet [6], which demonstrated that generative-based encoder-decoder networks can efficiently reconstruct path loss maps from building geometry. However, a critical gap remains where most advanced models focus primarily on reconstructing average statistics and often overlook spatially deep fades in the signal tail. For URLLC applications, relying solely on average signal statistics is inadequate; instead, it is essential to explicitly model the rare but critical extremes that ultimately determine system reliability. Extreme Value Theory (EVT) provides a statistical foundation for characterizing rare fading events via the Generalized Pareto Distribution (GPD). The authors in [8, 7] have conducted foundational research on estimating the channel tail distribution for URLLC by fitting the GPD to the extreme quantiles of received signal power. This work is subsequently generalized in [9] to address channel prediction, where an EVT-based adaptive quantile loss function is introduced within a Deep Recurrent Neural Network (DRNN) framework. Additionally, the authors in [11] used EVT for constructing reliability-aware radio maps through Gaussian process interpolation of the GPD tail parameters at unobserved locations. However, these approaches lack a generative core for map synthesis and remain tied to the specific environments in which they were trained, requiring retraining when deployed in new settings, and underscoring the need for a more flexible, generative approach to EVT based radio map synthesis. GenAI has recently gained attention for large-scale radio map generation. For instance, [18] introduced a framework that uses a Generative Adversarial Networks (GAN) to estimate radio maps in complex urban environments, aiming to enhance the quality of the generated maps. GAN-based approaches are particularly efficient during inference but typically concentrate on modeling the central tendencies of the signal distribution, often neglecting the spatially localized deep fades that are crucial for URLLC. More recently, diffusion models have shown promise in capturing near-extreme quantiles of the signal distribution, as highlighted in [15]. Despite their effectiveness, diffusion models are computationally intensive and incur high inference latency. [13] integrates EVT with GenAI for channel estimation, employing a GAN to selectively enrich the scarce extreme region and to estimate the tail threshold and GPD parameters online. That framework, however, operates on a scalar received power time series at a single link, and therefore characterizes the temporal tail of one channel rather than the spatial distribution of outage across an environment. Extending tail-aware generation from a per-link time series to a spatially resolved radio map additionally requires conditioning on the scene geometry that determines where deep fades occur. Limited fidelity in the low-SNR tail and the high computational cost of tail-aware generation motivate incorporating domain structure beyond purely data-driven generative modeling. Physics-informed AI has recently emerged as a powerful approach in wireless communications to address the limitations of purely generative methods. By embedding propagation models, site-specific geometry, or analytical path loss priors into the learning framework, these methods infuse domain knowledge directly into the model training process [2, 4]. This physical conditioning enables the network to accurately resolve the transitions in signal power that define outage boundaries, thereby delivering a reliable and interpretable radio map prediction. In this paper, we propose a novel Physics-informed VAE-EVT framework that integrates GenAI, EVT, and physics-informed AI to predict radio maps. The generative model learns the bulk and extreme quantiles (tails) of the signal distribution separately, while deterministic scene-geometry conditions constrain the prediction to ensure physical consistency. The main contributions of the paper are as follows: • We develop a physics-informed feature extraction pipeline that computes spatial priors, including Line-of-Sight (LoS) and Non-Line-of-Sight (NLoS) segregation, shadow depth, and distance attenuation, to directly encode the physics of propagation into the model’s input. • We introduce a novel Dual Latent Encoder architecture that integrates a standard Variational Autoencoder (VAE) with EVT. This explicitly separates the latent space, modeling bulk-signal conditions with a standard Gaussian distribution and extreme-signal outages with a GPD. • We evaluate the performance of our framework compared to the state-of-the-art algorithms, using the RadioMapSeer dataset [16], a publicly available benchmark widely used for radio map generation. The remainder of this paper is structured as follows. Section I outlines the system model assumptions. In Section I, we detail the proposed Physics-informed VAE-EVT framework. Section IV discusses the measurement data and model assumptions, and provides a comprehensive evaluation. Finally, Section V concludes the paper.11 1 The implementation is available at https://github.com/AmandaGamage/physics-informed-vae-evt. I System Model and Problem Formulation We consider an urban radio propagation environment discretized into an M×M× M spatial grid. This is represented by a binary building occupancy map ℬ(p)∈0,1M×MB(p)∈\0,1\^M× M, where ℬ(p)=1B(p)=1 indicates a building at pixel p=(x,y)p=(x,y), and ℬ(p)=0B(p)=0 denotes free space. Given a single transmitter placed at a known location ptxp_tx, the set of valid receiver locations is defined as Ω=p:ℬ(p)=0 =\p:B(p)=0\. Our primary objective is to characterize the spatial distribution of the received Signal-to-Noise Ratio (SNR) across Ω . We place particular emphasis on accurately predicting the extremely low SNR s that trigger outage, as capturing these rare events is critical for evaluating the reliability of URLLC systems. I-A Outage Definition Let Prx(p)P_rx(p) denote the received power in dBm at pixel p. For a system with bandwidth B, noise spectral density N0N_0, and receiver noise figure NFNF, the thermal noise floor is calculated as Pnoise=10log10(B)+N0+NFP_noise=10 _10(B)+N_0+NF dBm. Then, the SNR in dB is therefore given by: γ(p)=Prx(p)−Pnoise.γ(p)=P_rx(p)-P_noise. (1) An outage is defined as the event where γ(p)<γth\γ(p)< _th\, for a predefined critical SNR threshold, γth _th. Consequently, the outage region for a given environment is defined as: =p∈Ω:γ(p)<γth.O=\p∈ :γ(p)< _th\. (2) I-B Deterministic Geometric Structure and Residual Variability The SNR, γ(p)γ(p), comprises two distinct components: The large-scale component is deterministic and entirely governed by the macroscopic scene geometry, and the small-scale component that captures the probabilistic variability arising from complex physical phenomena, such as diffraction, scattering, and signal interactions near building edges. The deterministic component can be analytically derived from physical propagation principles. Once ℬB and ptxp_tx are defined, essential geometric metrics, including distance from the transmitter, LoS/NLoS condition, and localized shadowing, can be explicitly computed via ray tracing [2]. These metrics are assembled into a comprehensive set of deterministic descriptors, denoted as geoX_geo. Then, small-scale statistical variability is modeled as the conditional distribution p(γ∣geo)p(γ _geo). I-C Bulk and Tail of the Conditional SNR Distribution Conditioned on geoX_geo, the SNR exhibits two inherently distinct statistical regimes. Under standard average-statistics channel conditions (the bulk), γ(p)γ(p) is captured by a Gaussian Mixture Model (GMM) [12]: fbulk(:γ(p)>γth∣geo)=∑k=1Kαk(;μk,σk2),f_bulk( γ\,:γ(p)> _th _geo)= _k=1^K _k\,N\! ( γ;\, _k,\, _k^2 ), (3) where K represents the total number of mixture components, N denotes the Gaussian probability density function, and αk _k, μk _k, and σk2 _k^2 denote the mixing weight, mean, and variance of the k-th component, respectively. The mixing weights are strictly constrained such that ∑k=1Kαk=1 _k=1^K _k=1. Each mixture component k corresponds to a specific propagation state, such as LoS, NLoS, or partial obstruction. A finite Gaussian mixture systematically underestimates the probability of severe, rare fading events. To overcome this limitation, under tail-statistics channel conditions (γ(p)<γthγ(p)< _th), γ(p)γ(p) is modeled using a GPD [8]. According to the Pickands-Balkema-de Haan theorem, the distribution of exceedances below a sufficiently low threshold γth _th [3] converges to: ftail(:γ(p)<γth∣geo)=1−(1+ξ(γth−γ(p))β)−1/ξ,f_tail( γ\,:γ(p)< _th _geo)=1- (1+ ξ( _th- _(p))β )^-1/ξ, (4) where β>0β>0 and ξ∈ℝξ represent the scale and shape parameters of the GPD fitted to the exceedances =γth−γ(p) γ= _th-γ(p), respectively. To robustly learn the full conditional distribution p(∣geo)p( γ _geo), we utilize a VAE [5] driven by a specialized training objective. The reconstruction loss is two-fold, ensuring the bulk is supervised under a Gaussian assumption while the tail is simultaneously supervised under a GPD assumption. I Proposed VAE-EVT Framework for Tail-Aware Radio Map Prediction The proposed Physics-informed VAE-EVT framework (shown in Fig. 1) comprises two core elements: (i) a physics-informed preprocessing pipeline that extracts the deterministic structural tensor geoX_geo, and (i) a modified VAE architecture featuring a dual-latent encoder and a tail-aware training objective. Together, these elements enable the simultaneous learning of the bulk and tail regimes to accurately estimate p(∣geo)p( γ _geo). Physics-informed preprocessing produces the multi-channel input tensor geoX_geo (Input Tensor). A Dual Latent Encoder maps geoX_geo to two latent variables: a Gaussian bulk latent bulkz_bulk and a GPD-anchored tail latent tailz_tail (Bulk Latent, Tail Latent). The two latents are concatenated and passed through a fully connected layer (Concatenation). A U-Net Decoder with spatial attention then reconstructs spatial feature maps and feeds three output branches: a Bulk Branch producing μ and logσ2 σ^2, a Tail Branch producing yty_t, and an Outage Branch producing π that is sharpened to πs _s, all used to estimate SNR, γ γ, as the Final Output. Input Tensor geoX_geo Dual Latent Encoder Bulk Latent bulk∼(μg,σg2)z_bulk ( _g, _g^2I) Tail Latent tail∼GPD(ξ^,β^)z_tail ( ξ, β) Concatenation [bulk;tail]→FC[z_bulk;z_tail] U-Net Decoder Spatial Attention Bulk Branch μ, logσ2 σ^2 Outage Branch π→πsπ→ _s Tail Branch yty_t Final Output γ^=(1−πs)μ+πsyt γ=(1- _s)μ+ _sy_t Fig. 1: Overview of the proposed Physics-informed VAE-EVT architecture. I-A Physics-informed Preprocessing The preprocessing stage converts (ℬ,ptx)(B,p_tx) into a tensor geoX_geo of deterministic geometric descriptors. All channels are defined on the full M×M× M grid; for building pixels (ℬ(p)=1B(p)=1), channels are set to 0 and masked during training and evaluation. I-A1 Distance and Transmitter Proximity For each pixel p∈Ωp∈ , the Euclidean distance to the transmitter is d(p)=‖p−ptx‖2d(p)=\|p-p_tx\|_2. We construct a min–max normalized log-distance channel Dnorm(p)D_norm(p) = log(d(p)+ϵd) (d(p)+ _d) with a small ϵd>0 _d>0 to avoid log0 0. We also construct a linearly normalized distance channel Dall(p)D_all(p) over Ω , used as a stable conditioning input for the tail and outage branches. A smooth proximity map Tx(p)T_x(p) is defined as a truncated Gaussian centered at ptxp_tx, Tx(p)=exp(−‖p−ptx‖222σT2)T_x(p)= \!(- \|p-p_tx\|_2^22 _T^2), with σT>0 _T>0 chosen relative to the grid scale. I-A2 LoS Mask and NLoS Penetration Depth The binary LoS mask MLOS(p)∈0,1M_LOS(p)∈\0,1\ is set to 1 when the discretized ray segment from ptxp_tx to p does not intersect any building pixel in ℬB, and 0 otherwise. Let nhits(p)∈ℕn_hits(p) be the number of building intersections along this ray under the chosen discretization rule. The LoS-masked log-distance feature is dL(p)=Dnorm(p)MLOS(p).d_L(p)=D_norm(p)\,M_LOS(p). (5) For NLoS pixels, the obstruction severity is captured by the normalized penetration depth DNLOS(p)=(1−MLOS(p))⋅min(nhits(p)dmax, 1),D_NLOS(p)= (1-M_LOS(p) )· \! ( n_hits(p)d_max,\,1 ), (6) where dmax∈ℕd_max is a fixed normalization constant (e.g., the maximum nhitsn_hits observed in the training set). I-A3 Localized Shadowing and Edge Map A binary LoS mask does not capture partial shadowing near corners and corridors. We define a localized shadowing score SNLOS(p)∈[0,1]S_NLOS(p)∈[0,1] by sampling NsN_s equally spaced points along the segment from p toward ptxp_tx, and computing the fraction of samples that fall inside buildings. We set SNLOS(p)=0S_NLOS(p)=0 for LoS pixels. To provide a guidance signal for spatial attention, we compute an edge map E(p)E(p) emphasizing LoS/NLoS boundaries and strong shadow transitions. Let ∇ denote the image gradient estimated with Sobel operators, and let GσEG_ _E denote a Gaussian smoothing kernel with standard deviation σE _E. We form E(p)=clip(norm(α1‖∇MLOS‖(p)+α2‖∇(GσE∗SNLOS)‖(p)))E(p)=clip\,(norm\!( _1\|∇ M_LOS\|(p)+ _2 \|∇\! (G_ _E*S_NLOS ) \|(p))), where α1,α2≥0 _1, _2≥ 0 are weighting coefficients, norm(⋅)norm(·) denotes a fixed normalization (e.g., min–max over Ω ), and clip(⋅)clip(·) truncates the result to [0,1][0,1]. I-A4 Outage Prior A coarse geometric baseline for outage risk is provided through an outage prior: Poutage(p)=clip(ϵ, 1−ϵ)(a1Dnorm(p)),MLOS(p)=1,clip(ϵ, 1−ϵ)(b1SNLOS(p)+b2DNLOS(p)+b3Dnorm(p)),MLOS(p)=0,P_outage(p)= casesclip_(ε,\,1-ε)\! (a_1D_norm(p) ),&M_LOS(p)=1,\\[4.0pt] clip_(ε,\,1-ε)\! (b_1S_NLOS(p)+b_2D_NLOS(p)+b_3D_norm(p) ),&M_LOS(p)=0, cases (7) where a1,b1,b2,b3≥0a_1,b_1,b_2,b_3≥ 0 are coefficients and ϵ∈(0,0.5)ε∈(0,0.5) avoids degenerate probabilities. I-A5 Per-map Threshold and Input Tensor For each environment (radio map), γth _th is set as the q-th percentile of the SNR values γ(p):p∈Ω\γ(p):p∈ \, where q∈(0,100)q∈(0,100) is fixed across the dataset. A normalized threshold encoding γ^th γ_th is then broadcast to all pixels as an additional channel. Concretely, γ^th γ_th can be formed using dataset-level normalization constants γmin _ and γmax _ (computed on the training set): γ^th=γth−γminγmax−γmin+ϵγ γ_th= _th- _ _ - _ + _γ, with ϵγ>0 _γ>0 preventing division by zero. To anchor the tail latent space, dataset-level GPD parameters (ξ^,β^)( ξ, β) are estimated by maximum likelihood using training-set exceedances, γth−γ _th-γ, restricted to pixels with γ<γthγ< _th. These anchors are treated as constants during network training. The final input tensor is geo=stack(ℬ,Tx,MLOS,dL,SNLOS,DNLOS,E,Poutage,Dall,γ^th)X_geo=stack(B,\,T_x,\,M_LOS,\,d_L,\,S_NLOS,\,D_NLOS,\,E,\,P_outage,\,D_all, γ_th), and corresponds to the Input Tensor block in Fig. 1. I-B Dual-Latent Encoder The encoder implements the approximate posterior qϕ(bulk,tail∣geo)q_φ(z_bulk,z_tail _geo) shown as the Dual Latent Encoder block in Fig. 1. It uses a convolutional backbone to produce multi-resolution feature maps, which are cached for the U-Net skip connections in the decoder. At the bottleneck, two dense heads produce parameters for the bulk latent and for an auxiliary tail pathway that is mapped to a GPD-anchored latent. I-B1 Bulk Latent Variable, bulkz_bulk The bulk latent is Gaussian with diagonal covariance. The encoder outputs μg∈ℝdz _g ^d_z and logσg2∈ℝdz _g^2 ^d_z, where dzd_z is the bulk latent dimension, and sampling is performed via reparameterization. bulk=μg+σg⊙ϵz_bulk= _g+ _g ε, where ϵ∼(,) ε (0,I), σg=exp(12logσg2) _g= \! ( 12 _g^2 ), and ⊙ denoting elementwise multiplication. This matches the Bulk Latent block bulk∼(μg,σg2)z_bulk ( _g, _g^2I) in the diagram. I-B2 Tail Latent Variable, tailz_tail The tail pathway begins with an auxiliary diagonal-Gaussian variable. The encoder outputs μp∈ℝdt _p ^d_t and logσp2∈ℝdt _p^2 ^d_t, where dtd_t is the tail latent dimension, and draws aux=μp+σp⊙ϵz_aux= _p+ _p ε, where, ϵ∼(,) ε (0,I), and σp=exp(12logσp2) _p= \!( 12 _p^2). A differentiable approximation of a uniform random variable is then obtained by =σ(0.5∗aux)∈(0,1)dtu=σ(0.5*z_aux)∈(0,1)^d_t, where σ(⋅)σ(·) is the logistic sigmoid. Finally, u is mapped through the GPD quantile function using the fixed anchors (ξ^,β^)( ξ, β). Accordingly, tail=β^ξ^((1−)−ξ^−)z_tail= β ξ((1-u)^- ξ-1), with the exponent applied elementwise. In practice, when |ξ^|| ξ| is very small, the numerically stable exponential-limit form can be used: tail≈−β^log(1−)z_tail≈- β (1-u). I-C Concatenation and U-Net Decoder with Spatial Attention The two latents are concatenated exactly as shown in the Concatenation block, =[bulk;tail]z=[z_bulk;\,z_tail], then passed through a fully connected (FC) layer and reshaped into a low-resolution feature map that seeds the decoder. The decoder follows a U-Net design: features are progressively upsampled back to M×M× M, concatenated with encoder features at matching resolutions through skip connections, and refined with convolutional residual blocks. A spatial attention module is applied within the decoder to emphasize spatially localized transitions highlighted by channels such as E and SNLOS_NLOS. I-D Output Heads and Final Prediction From the shared decoder feature map, three heads produce the outputs shown on the right of Fig. 1. Let m(p)=1−ℬ(p)m(p)=1-B(p) be the free-space mask, used to force building pixels to be excluded from reconstruction losses. I-D1 Bulk Branch The bulk branch outputs μ(p)μ(p) and logσ2(p) σ^2(p) for each pixel. These define a Gaussian likelihood for SNR reconstruction in the non-outage regime. I-D2 Tail Branch The tail branch outputs yt(p)y_t(p), a tail-focused SNR estimate intended for outage pixels. To keep the tail head aligned with the shortfall interpretation, the model may internally parameterize a nonnegative shortfall y^(p)≥0 y(p)≥ 0 and set yt(p)=γth−y^(p)y_t(p)= _th- y(p), so that larger shortfalls correspond to smaller tail SNR values. I-D3 Outage Branch The outage branch predicts a per-pixel outage probability π(p)∈[0,1]π(p)∈[0,1]. Building pixels are masked by setting π(p)←m(p)π(p)π(p)← m(p)\,π(p). A sharpened routing mask πs(p) _s(p) is obtained using a steep sigmoid, πs(p)=σ(κ(π(p)−t∗)) _s(p)=σ\! (κ(π(p)-t^*) ), where κ>0κ>0 controls the steepness and t∗∈(0,1)t^*∈(0,1) is selected on a validation set by maximizing the F1 score. This corresponds to π→πsπ→ _s operation in Fig. 1. I-D4 Final Output The final SNR prediction is computed per pixel as γ^(p)=(1−πs(p))μ(p)+πs(p)yt(p) γ(p)=(1- _s(p))\,μ(p)+ _s(p)\,y_t(p), matching the Final Output block in Fig. 1. I-E VAE-EVT Training Loss The model is trained end-to-end using a combination of reconstruction, latent regularization, and explicit supervision for outage routing. The total objective is ℒtotal=λrℒrecon+λKL,bℒKL,b+λKL,tℒKL,t+λπℒπ+λoutℒoutage+λsℒsharp+λLoSℒLoS, splitL_total&= _rL_recon+ _KL,bL_KL,b+ _KL,tL_KL,t\\ & + _πL_π+ _outL_outage+ _sL_sharp+ _LoSL_LoS, split (8) where λ⋅≥0 _·≥ 0 are scalar weights and fine-tuned during training. I-E1 Reconstruction Loss ℒreconL_recon Reconstruction is routed by the predicted outage probability so that bulk learning dominates when π is small and tail learning dominates when π is large, ℒrecon=1|Ω|∑p∈Ω[(1−π(p))ℒNLL(p)+π(p)(yt(p)−γ(p))2]L_recon= 1| | _p∈ [(1-π(p))\,L_NLL(p)+π(p)\, (y_t(p)-γ(p) )^2 ]. The bulk negative log-likelihood term is ℒNLL(p)=12(log(2πσ2(p))+(γ(p)−μ(p))2σ2(p))L_NLL(p)= 12 ( \! (2πσ^2(p) )+ (γ(p)-μ(p))^2σ^2(p) ), where, σ2(p)=exp(logσ2(p))σ^2(p)= \! ( σ^2(p) ). I-E2 Outage-Region Emphasis ℒoutageL_outage Because outage pixels are rare, ℒoutageL_outage increases the penalty inside =p∈Ω:γ(p)≤γthO=\p∈ :γ(p)≤ _th\. Accordingly, ℒoutage=1||+ϵO∑p∈Ωp∈(γ^(p)−γ(p))2L_outage= 1|O|+ _O _p∈ I\p \\, ( γ(p)-γ(p) )^2, where ϵO>0 _O>0 prevents division by zero when O is empty. I-E3 Latent Regularization ℒKL,bL_KL,b and ℒKL,tL_KL,t The bulk KL term regularizes the bulk posterior toward a standard normal prior ℒKL,b=KL((μg,σg2)∥(,))L_KL,b=KL\! (N( _g, _g^2I)\,\|\,N(0,I) ). For the tail pathway, tailz_tail is a deterministic transform of auxz_aux. We therefore regularize the auxiliary Gaussian, ℒKL,t=KL((μp,σp2)∥(,))L_KL,t=KL\! (N( _p, _p^2I)\,\|\,N(0,I) ), which stabilizes tail sampling while preserving the GPD mapping to tailz_tail. I-E4 Outage Supervision ℒπL_π The outage loss supervises π(p)π(p) using labels ℓ(p)=γ(p)<γth,p∈Ω (p)=I\γ(p)< _th\, p∈ , after masking building pixels by m(p)m(p). In practice, ℒπL_π can combine a pixelwise classification loss (e.g., focal or weighted cross-entropy) with region-level overlap and calibration terms. All components operate on π (not πs _s) to adjust gradients. I-E5 Sharpening Regularization ℒsharpL_sharp To discourage indecisive routing, ℒsharpL_sharp penalizes high-entropy outage probabilities, ℒsharp=1|Ω|∑p∈Ω[−π(p)log(π(p)+ϵπ)−(1−π(p))log(1−π(p)+ϵπ)]L_sharp= 1| | _p∈ [-π(p) (π(p)+ _π)-(1-π(p)) (1-π(p)+ _π)], with a small ϵπ>0 _π>0 for numerical stability. This term encourages π (and therefore πs _s) to approach near-binary routing decisions. I-E6 LoS Outage Emphasis ℒLoSL_LoS NLoS outages are well explained by the obstruction descriptors SNLOS_NLOS and DNLOSD_NLOS, whereas a LoS pixel in outage has an unobstructed path and fails nonetheless, through distance attenuation rather than blockage. Since the outage prior of (7) and the shadowing channels all key on obstruction, a loss averaged uniformly over O is dominated by the NLoS majority, and the tail head converges to a shadowing-driven representation that systematically mispredicts the LoS minority. We therefore add an explicit term over the LoS outage subset LoS=p∈:MLOS(p)=1O_LoS=\p :M_LOS(p)=1\, ℒLoS=1|LoS|+ϵO∑p∈Ωp∈LoS[(γ^(p)−γ(p))2+(yt(p)−γ(p))2]L_LoS= 1|O_LoS|+ _O _p∈ I\p _LoS\ [ ( γ(p)-γ(p) )^2+ (y_t(p)-γ(p) )^2 ]. This term is the counterpart in the objective to the DallD_all channel in the input tensor. Both exist so that the tail and outage branches can express a distance-driven outage mechanism that the LoS masked geometric channels alone cannot represent. IV Numerical Results IV-A Experimental Setup IV-A1 Dataset We evaluate our framework on the RadioMapSeer dataset [16], which comprises 700 urban building layouts drawn from OpenStreetMap [10]. Each layout covers a 256×256256× 256 m2 grid at 11 m resolution, so that M=256M=256, with transmitter, receiver, and building heights of 1.51.5 m, 1.51.5 m, and 2525 m, respectively. We use the Dominant Path Model maps [14] simulated with WinProp, and restricted to Nmap=300N_map=300 layouts with Ntx=10N_tx=10 transmitter positions each, yielding 3000 environment transmitter pairs split 80/20 into training and test sets. The noise floor is computed with B=10B=10 MHz, N0=−174N_0=-174 dBm/Hz, and NF=0NF=0 dB, giving Pnoise=−104P_noise=-104 dBm. The SNR values are normalized to [0,1][0,1]. IV-A2 Implementation Details The LoS ray tracer uses 100100 samples per ray, sufficient to keep the inter-sample spacing below one pixel along the longest possible diagonal of a 256×256256× 256 map. The localized shadow feature uses 3030 samples, since it only needs to resolve obstruction in the immediate neighborhood of p rather than the full path to ptxp_tx. The NLoS penetration depth is normalized by dmax=10d_max=10, corresponding to the 99th percentile of observed wall crossings in the training set. The encoder uses four strided convolutional residual blocks with channel widths 32,64,128,256\32,64,128,256\, compressing the input to a 16×1616× 16 bottleneck. The post-hoc outage sharpening sigmoid uses a slope of 2020, which helps transitions πs _s from near-zero to near-one within a narrow window. Loss weights follow a KL warmup to prevent posterior collapse, with progressive ramping of the classification, outage reconstruction, and sharpening terms during training. IV-B Baselines Models To isolate the contribution of each architectural component, we compare four models, including: (i) Physics-informed VAE-EVT: the proposed model, with dual latent EVT encoder, physics preprocessing, and explicit LoS/NLoS feature routing; (i) VAE-EVT: dual-latent EVT encoder with no physics preprocessing; (i) Physics-informed VAE: a standard single-latent VAE with physics preprocessing but no EVT tail modeling; and (iv) RME-GAN [18]: a GAN-based radio map estimator with no explicit tail modeling. All models are trained using 2400 maps in the training set and are evaluated at two outage thresholds, q∈0.1%,10%q∈\0.1\%,10\%\. The evaluation metrics include outage region Root Mean Square Error (RMSE), F1-score, Precision, and Recall for outage pixel classification. The outage region RMSE directly measures tail prediction fidelity, while F1-score captures spatial outage localization accuracy. Results are reported across 600 test maps. (a) (b) Fig. 2: Distribution of outage RMSE across 600 test maps at (a) 0.1% threshold. (b) 10% threshold (Legends are common). IV-C Discussion Figure 3 shows predicted radio maps for three test maps. RME-GAN produces plausible large-scale propagation patterns but fails to reproduce the fine-grained shadow structure at building boundaries. VAE-EVT misses the high SNR near the transmitter region entirely. The proposed Physics-informed VAE-EVT most closely matches the ground truth, reproducing both the bright near-transmitter and the deep shadow regions. Fig. 3: Radio map for three test maps. From left to right: ground truth, Physics-informed VAE-EVT∗, Physics-informed VAE, VAE-EVT, and RME-GAN. TABLE I: Comparison of model performance across different thresholds for outage prediction. Model RMSE (dB) F1-Score Precision Recall 0.1% 10% 0.1% 10% 0.1% 10% 0.1% 10% Physics-informed VAE-EVT∗ 4.83 6.97 0.1120 0.2459 0.0608 0.8527 0.7088 0.1437 Physics-informed VAE 12.03 7.54 0.1015 0.2903 0.0559 0.2205 0.5504 0.6236 VAE-EVT 6.56 7.81 0.1222 0.2147 0.0673 0.9434 0.6638 0.1211 RME-GAN [18] 21.90 11.53 0.1179 0.5245 0.2524 0.5194 0.0769 0.5297 Table I summarizes the performance of different models for two different outage thresholds q∈0.1%,10%q∈\0.1\%,10\%\. At the 0.1% threshold, Physics-informed VAE-EVT achieves 4.83 dB outage RMSE, outperforming Physics-informed VAE (12.03 dB) by more than 7 dB and RME-GAN (21.90 dB) by roughly 17 dB. The standalone VAE-EVT also remains competitive at 6.56 dB, indicating that only the EVT-equipped models maintain predictive accuracy at these extreme outage levels. At the 10% threshold, however, the gap among all models narrows substantially: Physics-informed VAE-EVT records 6.97 dB, while Physics-informed VAE and VAE-EVT reach 7.54 dB and 7.81 dB, respectively, and even RME-GAN improves to 11.53 dB. This compression occurs because the evaluation at 10% includes a larger number of bulk-similar samples in the tail. Consequently, both the GPD fit and the RMSE metric become less indicative of extreme value prediction at this threshold, as they are increasingly influenced by non-extreme values. Although the F1 score at the 0.1% threshold appears low, the outage RMSE of γ γ remains low because the soft routing through πs _s combined with yty_t keeps misrouted pixels close to the true threshold SNR, so classification errors translate into only small RMSE penalties. Figure 2 illustrates the distribution of outage RMSE across all 600 test maps at the 0.1% and 10% thresholds. The distributions corresponding to the EVT-based models exhibit lower means and variances compared to the non-EVT models, including Physics-informed VAE or RME-GAN prediction, especially for the 0.1% outage threshold. This separation confirms that the EVT advantage is systematic across environments and not driven by a small number of favorable maps. At the 10% threshold, the distributions overlap, again consistent with the expectation that EVT provides diminishing returns outside the true extreme tail. Fig. 4: SNR distributions for a sample test map. Figure 4 compares the predicted and empirical SNR densities. The figure reveals a bimodal ground truth, with a dominant LoS peak and a secondary NLoS peak reflecting the separation between LoS-dominated pixels near the transmitter and deeply shadowed NLoS pixels behind buildings. In the bulk region, all physics-conditioned VAE models track the empirical curve closely, while RME-GAN shows a visible offset and VAE-EVT drifts above the empirical curve across the mid-SNR range. The crucial difference appears in the lower tail (inset). The proposed Physics-informed VAE-EVT tracks the empirical densities across both the bulk and the tail, underscoring the importance of physics conditioning. Moreover, models with physics preprocessing reproduce both peaks, while VAE-EVT collapses them and misses the LoS peak entirely. It is worth mentioning that the original RME-GAN evaluation reported in [18] uses all 700 RadioMapSeer regions, split into 500 training, 100 test, and 100 validation regions, whereas every model reported here is trained on the identical subset of Nmap=300N_map=300 layouts with Ntx=10N_tx=10 transmitter positions each. We further note that RME-GAN uses 1% of the true-SNR pixels as sparse observations, whereas the proposed model predicts from scene geometry alone. The results confirm that EVT is necessary to capture tail shape, and physics conditioning is necessary to position the distribution correctly along the SNR and preserve the bimodal bulk structure. It is also worth noting that, in contrast to diffusion-based radio map models that require iterative denoising at inference [15], the proposed framework generates a full 256×256256× 256 SNR map in a single forward pass, making it directly deployable in URLLC scenarios. The proposed Physics-informed VAE-EVT, Physics-informed VAE, and VAE-EVT have comparable parameter counts of 38011962, 37882945, and 38000265, respectively. V Conclusions In this paper, we proposed a Physics-informed VAE-EVT framework for tail-aware radio map prediction. By decoupling bulk and tail via a dual-latent encoder and embedding deterministic LoS/NLoS features, the model achieves 4.83 dB outage RMSE at the URLLC critical 0.1% quantile threshold, compared to 12.03 dB for a Physics-informed VAE and 21.90 dB for the state-of-the-art RME-GAN. The RMSE distributions confirm that the advantage is systematic across all 600 test environments. These results show that URLLC grade outage characterization requires both physics-grounded features and principled tail modeling. Future work will extend the framework to multi transmitter scenarios and moving obstruction and receiver targets. References [1] M. Bennis, M. Debbah, and H. V. Poor (2018) Ultrareliable and low-latency wireless communication: tail, risk, and scale. Proceedings of the IEEE 106 (10), p. 1834–1853. External Links: Document Cited by: §I. [2] W. Chen and J. Chen (2024) Diffraction and scattering aware radio map and environment reconstruction using geometry model-assisted deep learning. IEEE Transactions on Wireless Communications 23 (12), p. 19804–19819. Cited by: §I, §I-B. [3] S. Coles (2001) An introduction to statistical modeling of extreme values. Springer Series in Statistics, Springer. Cited by: §I-C. [4] F. Jaensch, G. Caire, and B. Demir (2026) Radio map prediction from aerial images and application to coverage optimization. IEEE Transactions on Wireless Communications 25 (), p. 308–320. External Links: Document Cited by: §I. [5] D. P. Kingma and M. Welling (2014) Auto-encoding variational bayes. In International Conference on Learning Representations (ICLR), External Links: Link Cited by: §I-C. [6] R. Levie, Ç. Yapar, G. Kutyniok, and G. Caire (2021) RadioUNet: fast radio map estimation with convolutional neural networks. IEEE Transactions on Wireless Communications 20 (6), p. 4001–4015. Cited by: §I. [7] N. Mehrnia and S. Coleri (2022) Extreme value theory based rate selection for ultra-reliable communications. IEEE Transactions on Vehicular Technology 71 (6), p. 6727–6731. External Links: ISSN 1939-9359, Link, Document Cited by: §I. [8] N. Mehrnia and S. Coleri (2022) Wireless channel modeling based on extreme value theory for ultra-reliable communications. IEEE Transactions on Wireless Communications 21 (2), p. 1064–1076. External Links: Document Cited by: §I, §I-C. [9] N. Mehrnia, P. Valiahdi, S. Coleri, and J. Gross (2025) Channel prediction using deep recurrent neural network with evt-based adaptive quantile loss function. IEEE Communications Letters 29 (7), p. 1699–1703. External Links: Document Cited by: §I. [10] OpenStreetMap contributors (2017) Planet dump retrieved from https://planet.osm.org. Note: https://w.openstreetmap.org Cited by: §IV-A1. [11] D. E. Pérez, O. L. A. López, and H. Alves (2025) EVT-enriched radio maps for ultrareliable communication. IEEE Internet of Things Journal 12 (12), p. 22012–22022. External Links: Document Cited by: §I. [12] B. Selim, O. Alhussein, S. Muhaidat, G. K. Karagiannidis, and J. Liang (2016) Modeling and analysis of wireless channels via the mixture of gaussian distribution. IEEE Transactions on Vehicular Technology 65 (10), p. 8309–8321. External Links: Document Cited by: §I-C. [13] P. Valiahdi, N. Mehrnia, W. Saad, and S. Coleri (2026) EVT-based generative ai for tail-aware channel estimation. In 2026 32nd International Conference on Telecommunications (ICT), Vol. , p. 111–117. External Links: Document Cited by: §I. [14] R. Wahl, G. Wölfle, P. Wildbolz, and F. Landstorfer (2005) Dominant path prediction model for urban scenarios. In Proceedings of the IST Mobile and Wireless Communications Summit, Cited by: §IV-A1. [15] X. Wang, K. Tao, N. Cheng, Z. Yin, Z. Li, Y. Zhang, and X. Shen (2025) RadioDiff: an effective generative diffusion model for sampling-free dynamic radio map construction. IEEE Transactions on Cognitive Communications and Networking 11 (2), p. 738–750. External Links: Document Cited by: §I, §IV-C. [16] Ç. Yapar, R. Levie, G. Kutyniok, and G. Caire (2022) Dataset of pathloss and ToA radio maps with localization application. arXiv preprint:2212.11777. External Links: Link Cited by: 3rd item, §IV-A1. [17] Y. Zeng et al. (2024) A tutorial on environment-aware communications via channel knowledge map for 6G. IEEE Communications Surveys & Tutorials 26 (3), p. 1478–1519. Cited by: §I, §I. [18] S. Zhang, A. Wijesinghe, and Z. Ding (2023) RME-GAN: a learning framework for radio map estimation based on conditional generative adversarial network. IEEE Internet of Things Journal 10 (20), p. 18016–18027. External Links: Document Cited by: §I, §IV-B, §IV-C, TABLE I.