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Scene-Conditioned PINN-GNN for Multipath RF Maps: Cross-Scene Generation and In-Scene Completion
Lizhou Liu, Xiaohui Chen, Zihan Tang, Mengyao Ma, Wenyi Zhang
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Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 98%
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Summary
The paper proposes a unified RF map construction framework called PINN-GNN that combines Physics-Informed Neural Networks (PINN) and Graph Neural Networks (GNN). The framework supports two tasks: cross-scene generation (using 2D/2.5D environmental geometry) and in-scene completion (using sparse receiver measurements). The PINN component embeds electromagnetic propagation constraints to ensure physical consistency of multipath parameters (path gain, ToA, AoA), while the GNN component models spatial correlations among receivers to ensure spatial consistency. A novel peak-weighted dynamic time warping (PW-DTW) metric is introduced to evaluate reconstruction quality.
Entities (9)
Relation Signals (6)
PINN-GNN → combines → Physics-Informed Neural Network
confidence 100% · This paper proposes a unified RF map construction framework based on a physics-informed neural network (PINN) and a graph neural network (GNN).
PINN → embeds → Electromagnetic Propagation Constraints
confidence 100% · The PINN embeds electromagnetic propagation constraints to establish a physically consistent mapping
GNN → enforces → Spatial Consistency
confidence 100% · the GNN enforces spatial consistency by modeling correlations among neighboring receivers.
Peak-weighted Dynamic Time Warping → evaluates → Multipath Reconstruction Quality
confidence 100% · To comprehensively evaluate multipath reconstruction quality, we propose a peak-weighted dynamic time warping metric
PINN-GNN → supports → Cross-Scene Generation
confidence 100% · supporting both cross-scene generation and in-scene completion
PINN-GNN → supports → In-Scene Completion
confidence 100% · supporting both cross-scene generation and in-scene completion
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Abstract
Abstract:Radio frequency (RF) maps provide a compact representation of multipath propagation characteristics and are fundamental to channel modeling, coverage analysis, and environment-aware wireless optimization. This paper proposes a unified RF map construction framework based on a physics-informed neural network (PINN) and a graph neural network (GNN), supporting both cross-scene generation and in-scene completion with 2D and 2.5D environmental representations. The PINN embeds electromagnetic propagation constraints to establish a physically consistent mapping from receiver locations to multipath parameters, including path gain, time of arrival, and angles, while the GNN enforces spatial consistency by modeling correlations among neighboring receivers. To comprehensively evaluate multipath reconstruction quality, we propose a peak-weighted dynamic time warping metric that jointly accounts for amplitude errors and peak delay misalignment in channel impulse responses. Extensive experiments demonstrate that the proposed method consistently outperforms image-based, diffusion-based, and interpolation baselines across both map-level and multipath-level metrics, achieving robust generalization and high-fidelity RF map construction under sparse observations.
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- Source: https://arxiv.org/abs/2607.01777v1
- Canonical: https://arxiv.org/abs/2607.01777v1
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Scene-Conditioned PINN-GNN for Multipath RF Maps: Cross-Scene Generation and In-Scene Completion Lizhou Liu, Xiaohui Chen, Zihan Tang, Mengyao Ma, and Wenyi Zhang Preliminary results of this work have been presented in part at 2025 International Conference on Future Communications and Networks [17]. Supported by the National Key Research and Development Program of China (2025YFF0514400) and the National Natural Science Foundation of China (62231022).Lizhou Liu, Xiaohui Chen, and Wenyi Zhang are with the Department of Electronic Engineering and Information Science, University of Science and Technology of China, Hefei, Anhui 230026, China (e-mail: liulizhou@mail.ustc.edu.cn; cxh@ustc.edu.cn; wenyizha@ustc.edu.cn). (Corresponding author: Wenyi Zhang)Zihan Tang and Mengyao Ma are with the Wireless Technology Lab, Huawei, Shenzhen 518129, China (e-mail: tangzihan1@huawei.com; ma.mengyao@huawei.com). Abstract Radio frequency (RF) maps provide a compact representation of multipath propagation characteristics and are fundamental to channel modeling, coverage analysis, and environment-aware wireless optimization. This paper proposes a unified RF map construction framework based on a physics-informed neural network (PINN) and a graph neural network (GNN), supporting both cross-scene generation and in-scene completion with 2D and 2.5D environmental representations. The PINN embeds electromagnetic propagation constraints to establish a physically consistent mapping from receiver locations to multipath parameters, including path gain, time of arrival, and angles, while the GNN enforces spatial consistency by modeling correlations among neighboring receivers. To comprehensively evaluate multipath reconstruction quality, we propose a peak-weighted dynamic time warping metric that jointly accounts for amplitude errors and peak delay misalignment in channel impulse responses. Extensive experiments demonstrate that the proposed method consistently outperforms image-based, diffusion-based, and interpolation baselines across both map-level and multipath-level metrics, achieving robust generalization and high-fidelity RF map construction under sparse observations. I Introduction To meet the requirements of 6G networks for immersive interaction, ultra-reliable low-latency communications (URLLC), and high-precision localization, prior knowledge of spatial radio frequency (RF) propagation is essential [21]. As systems evolve toward massive multiple-input multiple-output (MIMO) and ultra-wide bandwidths, conventional pilot-based active measurements incur substantial signaling overhead and latency [29]. Furthermore, deploying mobile access nodes like autonomous aerial vehicles (AAVs) requires proactive trajectory planning [18], necessitating the prediction of regional RF distributions before communication occurs. Consequently, wireless networks are shifting from measurement-driven reactive paradigms to environment-aware architectures [26]. As a key enabler of this transition, RF maps provide a spatial representation of wireless propagation characteristics. Specifically, multipath RF maps incorporate diverse parameters including path gain, time of arrival (ToA), and angle of arrival (AoA). These maps facilitate multi-dimensional channel state inference and intelligent resource allocation with reduced overhead, supporting applications like high-precision localization and integrated sensing and communication (ISAC) [31]. However, constructing high-fidelity multipath RF maps in urban environments featuring complex blockages and rich scattering remains fundamentally challenging. Existing RF map construction generally falls into two categories: physics-based models and data-driven interpolation. Empirical models like Okumura–Hata [20], [10] and 3GPP Urban Macro [1] primarily address large-scale path loss, failing to capture site-specific multipath details in complex geometries. While ray tracing (RT) explicitly models physical interactions like reflection and diffraction [22], [12], its prohibitive computational overhead limits large-scale or real-time applications. Alternatively, interpolation methods like Kriging [7], tensor completion [25], and matrix completion [6] exploit spatial correlation for computational efficiency but lack physical awareness. In non-line-of-sight (NLoS) urban environments, these methods struggle with strong non-stationarity and shadow boundaries, yielding physically inconsistent RF maps under sparse sampling. Recent artificial intelligence (AI) advances introduce new paradigms for RF map construction. RadioUNet [14] leverages deep neural networks to efficiently learn path loss distributions from simulation data. Building on this, cascaded U-Nets [15] integrate environmental and sparse RT data for refined spatial predictions, while graph neural networks (GNNs) [4] capture spatial topologies to enhance channel modeling. With the rapid development of generative models, RF map construction is increasingly formulated as a data generation problem. Generative adversarial networks (GANs) [32], [5], [23] and diffusion models [28], [27] are widely adopted for high-fidelity spatial generation and channel knowledge map (CKM) construction [8], [33]. Despite these advances, most methods frame RF map construction as a computer vision (CV) image-to-image translation problem [30]. These pixel-grid representations force continuous, floating-point physical measurements into quantized pixel intensities, inevitably losing fine-grained physical information during this image-centric transformation. Furthermore, they typically predict a single macroscopic quantity like path loss or received signal strength. While a few attempts incorporate spatial features like AoA [27], [19], they largely focus on the dominant path and overlook the multipath effects critical for practical wideband propagation [3]. Therefore, overcoming these image-centric quantization errors to construct high-fidelity, multi-parameter RF maps encompassing path gain, ToA, and AoA is essential for accurate wireless channel modeling. Practical multipath RF map construction involves two challenging tasks. First, cross-scene generation requires predicting the global multipath distribution in uncharted areas using only accessible two-dimensional (2D) or three-dimensional (3D) environmental geometry to support proactive network deployment. Second, in-scene completion demands reconstructing high-resolution global channel distributions using strictly sparse measurements from mobile nodes when prior environmental information is unavailable. Under these data-restricted conditions, purely data-driven pixel-grid methods exhibit significant limitations. Lacking physical constraints and the ability to model non-Euclidean spatial topologies, they struggle to infer highly coupled multipath characteristics. Therefore, these approaches often yield estimation errors and physically inconsistent predictions that violate fundamental electromagnetic laws. Accurately characterizing complex multipath features requires integrating electromagnetic propagation mechanisms with spatial topology modeling. Physics-informed neural networks (PINNs) address this by embedding physical constraints into the learning process. Widely recognized in fluid dynamics [13], PINNs are increasingly explored for wireless propagation [34]. By incorporating electromagnetic principles, like the geometric relationship among propagation distance, the speed of light, and ToA, as soft constraints, PINNs maintain physical consistency during data fitting and improve inference reliability. Furthermore, GNNs [24] expertly handle non-Euclidean topologies and model long-range node dependencies, effectively capturing spatial correlations induced by scattering and diffraction. Combining the physical constraint mechanisms of PINNs with the spatial reasoning capabilities of GNNs establishes a unified multipath RF map modeling framework. This integrated approach systematically overcomes the limitations of existing methods in complex urban environments under severely limited data conditions. To overcome the limitations of strictly pixel-grid representations and single-path predictions, we propose a unified PINN-GNN framework for RF map construction. This approach extends conventional modeling to joint multipath, multi-parameter characterization, simultaneously predicting path gain, ToA, and AoA. By directly utilizing original measurements and environmental semantics, the method extracts 2D or 2.5D geometric priors, including explicit height, from complex urban scenarios to assist LoS and NLoS identification. To decouple multipath parameters and prevent physical inconsistencies, PINNs embed electromagnetic propagation laws as soft constraints during data fitting. Subsequently, GNNs construct spatial graphs over sampling points and environmental features to model node correlations, ensuring the spatial consistency of the inferred parameters. We comprehensively evaluate this dual-consistency framework on cross-scene generation and in-scene completion tasks. Notably, to accurately characterize the temporal consistency of multipath structures, we propose a novel peak-weighted dynamic time warping (PW-DTW) metric. For cross-scene generation, comparing 2D and 2.5D maps verifies the critical role of explicit height information in recovering spatial multipath structures. Experimental results demonstrate that the proposed framework outperforms state-of-the-art learning and traditional interpolation methods in first-path prediction accuracy. Moreover, it achieves the stable, robust, and physically consistent inference of high-order multipath components in complex environments. Consequently, this work provides an effective solution for high-precision environment-aware communications and sensing in future 6G networks. The remaining part of this paper is organized as follows. Section I introduces the system model and formulates the multipath, multi-parameter RF map construction problem. Section I details the proposed scene-conditioned PINN-GNN framework. Section IV outlines the baseline methods and evaluation metrics, while Section V presents experimental results and analysis. Finally, Section VI concludes the paper. I System Model and Problem Formulation This section establishes a unified system model for multipath RF map construction based on two environment representations for scene conditioning. Grounded in wireless propagation theory, the multipath channel is parameterized by channel gain, delay, and angular information. Subsequently, RF map completion and generation are formulated within a common framework, which aims to map scene conditions to multipath parameters under varying observation settings. I-A System Scenario and Environment Representation We consider a wireless communication system consisting of a fixed transmitter (Tx) located at ∈ℝ3t ^3 and multiple receivers (Rx) distributed within a region of interest (RoI). The position of an arbitrary receiver is denoted by ∈ℝ3r ^3. To unify the description of signal propagation, the environmental geometry is modeled as a grid map, where each grid coordinate (u,v)(u,v) corresponds to a physical coordinate (x,y)(x,y). Depending on the available information granularity, the environment ℰE can be represented in two forms: • 2D environment map (ℰ2DE^2D): a binary occupancy map indicating buildings or accessible regions, which captures the topological structure of building boundaries and outdoor open spaces. • 2.5D environment map (ℰ2.5DE^2.5D): an extension of the 2D map that additionally records the building height at each grid location. This representation explicitly characterizes variations in blockage severity and provides more reliable geometric cues for LoS visibility and dominant path formation, while avoiding the high cost and complexity of full 3D modeling. To exclude non-target regions such as building interiors from prediction and evaluation, an effective region mask R(u,v)∈0,1R(u,v)∈\0,1\ is defined for each scene, where R(u,v)=1R(u,v)=1 indicates a valid outdoor location. RF maps are constructed exclusively over the effective region defined by R. I-B Multipath Signal Model and Tensor Representation In wireless signal propagation, information is encoded in both amplitude and phase of electromagnetic waves. Governed by Maxwell’s equations, transmitted signals reach receivers via LoS and NLoS paths induced by reflection, scattering, and diffraction. When waves interact with environmental scatterers having different dielectric properties, they produce multipath components with varying delay, direction, and fading characteristics. The superposition of these multipath components in both temporal and spatial domains creates the received signal, a phenomenon formally known as multipath propagation. Consider a wideband wireless communication system where a Tx sends a baseband equivalent signal s(t)s(t). For a Rx equipped with an M×NM× N uniform planar array (UPA), the received signal (t)r(t) under multipath propagation is given by (t)=∑l=1Lαl(θl,φl)s(t−τl)+(t), (t)= _l=1^L _la( _l, _l)s(t- _l)+n(t), (1) where L denotes the number of multipath components, αl=|αl|e−j2πfcτl _l=| _l|e^-j2π f_c _l is the complex gain of the l-th path with carrier frequency fcf_c and delay τl _l, θl∈[0,π] _l∈[0,π] and φl∈[−π,π] _l∈[-π,π] represent the elevation and azimuth angles, respectively, and (t)n(t) is the additive white Gaussian noise (AWGN) vector. For a UPA placed on the x–y plane with antenna spacings dxd_x and dyd_y, the array response vector (θl,φl)∈ℂMN×1a( _l, _l) ^MN× 1 is (θl,φl)=[e−j2πλ(mdxsinθlcosφl+ndysinθlsinφl)]MN×1, ( _l, _l)= [e^-j 2πλ(md_x _l _l+nd_y _l _l) ]_MN× 1, (2) where λ=c/fcλ=c/f_c is the wavelength, c is the speed of light, and m∈0,1,…,M−1m∈\0,1,…,M-1\ and n∈0,1,…,N−1n∈\0,1,…,N-1\ index the antennas along the x- and y-axes, respectively. The corresponding time-domain channel impulse response (CIR) is expressed as (t)=∑l=1L|αl|e−j2πfcτl(θl,φl)δ(t−τl), (t)= _l=1^L| _l|e^-j2π f_c _la( _l, _l)δ(t- _l), (3) where δ(t)δ(t) denotes the Dirac delta function. As seen from (1)–(3), the multipath channel is fully characterized by the physical parameters αl,τl,θl,φl\ _l, _l, _l, _l\. To facilitate neural network processing while preserving physical interpretability, the complex-valued channel is represented as a fixed-size real-valued tensor. For each receiver location r, we define a multipath parameter tensor ()∈ℝL×4Y(r) ^L× 4 as ()=[1,2,…,L]T, (r)= [p_1,p_2,…,p_L ]^T, (4) where each row l=[gl,τl,θl,ϕl]p_l=[g_l, _l, _l, _l] contains four key physical parameters of the l-th path: the path gain (gl=|αl|g_l=| _l|), propagation delay (τl _l), and angles of arrival (AoA), including the elevation angle θl _l and azimuth angle ϕl _l. To accommodate spatially varying path counts, the tensor size is fixed to L (e.g., L=5L=5). Zero-padding is applied for fewer paths, whereas excess paths are truncated to retain the L components with the highest received power. This representation enables the model to jointly learn the energy distribution and spatial directionality of multipath propagation. I-C Unified Task Statement This work aims to generate or complete the multipath RF map across the target region. To unify different levels of information availability, all prior knowledge or observations are collectively referred to as conditioning information, and a common output space Y is adopted to describe the task. Depending on the available conditioning information, the problem encompasses two representative operating modes: • Geometry-conditioned cross-scene generation: In this setting, the environment map ℰE, represented as either ℰ2DE_2D or ℰ2.5DE_2.5D, and the Tx configuration t are available, with no observations provided (=∅S= ). The objective is to directly generate the complete tensor Y over the effective region. This approach enables transferable prediction in unseen environments by learning the geometry-to-propagation mapping from extensive training data. • Measurement-conditioned in-scene completion: When the environment map is unavailable (ℰ=∅E= ), the model relies on a small set of observed locations =(k,k)k=1KS=\(r_k,Y_k)\_k=1^K. The goal is to exploit these sparse observations alongside physical constraints to recover the spatial distribution Y across the entire effective region. To effectively accomplish these tasks and ensure physical consistency across both operating modes, the design and implementation details of the proposed scene-conditioned PINN–GNN framework are presented in Section I. I A Unified Scene-Conditioned PINN-GNN Framework This section presents a unified framework for constructing multipath RF maps. As illustrated in Fig. 1, the cross-scene task takes the environmental information of the target scene as input, whereas the in-scene completion task relies on sparse sampled points within the scene and does not require explicit environmental information. The adaptive scene condition encoder serves as a unified input interface that flexibly processes heterogeneous inputs according to the current operating mode. Using the encoded conditions, a physics-guided predictor generates an initial estimate of the multipath parameter tensor by incorporating electromagnetic physical constraints. Finally, a graph-consistent refiner constructs a spatial graph over the predicted points and aggregates neighborhood information to suppress local noise and sharpen signal boundaries. Figure 1: PINN-GNN architecture for RF map construction. Missing input modalities (ℰE or S) are automatically zero-padded to form the unified condition iz_i. I-A Overall Pipeline and Two Operating Modes Let validQ_valid denote the set of valid query locations determined by the environmental mask or validity rules. For any query location i∈validr_i _valid, the model outputs a predicted multipath tensor ^i∈ℝL×Q Y_i ^L× Q, where L denotes the number of paths and Q is the dimension of the physical parameters for each path. The overall mapping process is formulated as ^i=Φ(~i,k(i)),~i=ℱΘ(i,i), Y_i=G_ \! ( Y_i,\ N_k(i) ), Y_i=F_ (r_i,z_i), (5) where ℱΘF_ denotes the point-wise predictor, ΦG_ acts as a spatial refiner that aggregates local information from the k-nearest-neighbor set k(i)N_k(i) of node i, primarily responsible for mitigating local noise and enhancing spatial consistency. Specifically, ℱΘF_ produces a physically plausible initial estimate ~i Y_i conditioned on the feature vector iz_i. To provide a unified interface across different operating modes and govern model transferability, iz_i is decomposed into geometric, environmental condition, and observation-guided features, i=[i,i,i], _i=[g_i,\ e_i,\ s_i], (6) where ig_i characterizes the Tx-Rx geometric relationship, supplying essential spatial cues and references for physical regularization. All spatial inputs are normalized to eliminate cross-scene scale discrepancies and enhance training stability. ie_i encodes the environmental geometry, and is_i represents sparse observation guidance. To maintain a unified interface across operating modes, ie_i and is_i automatically default to zero vectors when unavailable. Through this task-aware encoding mechanism, the framework switches functionality by adjusting the conditional features. When environmental maps are available but sparse observations are absent, the framework performs cross-scene generation, whereas given only sparse observations, the model performs in-scene completion. I-B Environment Encoding and Feature Alignment To exploit environmental geometry and provide condition inputs with physical semantics for the subsequent PINN predictor, we design a unified U-Net-based environment encoder ℰψE_ψ. This module transforms environmental maps into high-dimensional feature representations spatially aligned with query locations. In the cross-scene generation mode, the input is a 2D or 2.5D environmental grid tensor ℰ∈ℝCin×H×WE ^C_in× H× W, where H and W denote the height and width of the grid map, respectively, and CinC_in is the number of input channels, which adapts to the environment representation. Specifically, Cin=1C_in=1 for a 2D binary occupancy mask, and Cin=2C_in=2 for a 2.5D map combining a normalized height field and an occupancy mask. The encoder extracts multi-scale features via successive downsampling, while the decoder upsamples and fuses shallow geometric features through skip connections. This preserves global context and local boundaries, yielding a dense feature field with matched spatial resolution as =ℰψ(ℰ)∈ℝDf×H×W, =E_ψ(E) ^D_f× H× W, (7) where DfD_f denotes the dimensionality of the encoded feature channels. The resulting feature field F maintains the same spatial resolution as the input environment, ensuring geometric alignment for subsequent feature sampling. Given small batch sizes in cross-scene training, convolutional blocks adopt group normalization rather than batch normalization, enhancing robustness against distribution shifts. To explicitly capture physical semantics like height and blockage, auxiliary supervision is applied during training, using lightweight prediction heads to reconstruct the input height field or occupancy mask. Since convolutions alone struggle with long-range LoS occlusions, we construct point-wise environmental features ie_i via a “local sampling + profile enhancement” strategy. For an arbitrary query location ir_i, local sampling is performed at the corresponding coordinates in the dense feature field F using s×s× s regional pooling, resulting in a location-aligned local feature vector i∈ℝDff_i ^D_f. To explicitly incorporate occlusions, we sample along the direct path between the Tx and Rx to form a profile feature ip_i, encoding statistics like the first obstruction, maximum blockage depth, and 2.5D clearance. The final vector is concatenated as i=[i,i]e_i=[f_i,p_i]. For in-scene completion where environmental maps are unavailable, ie_i either defaults to 0 or is generated by a lightweight implicit encoder as an embedding env,i∈ℝdef_env,i ^d_e, preserving the unified interface. I-C Physics-Guided Point-wise Predictor The point-wise predictor, implemented via a PINN, serves as the core physical engine of the proposed framework. It is essentially a physics-guided multi-task neural network that aims to learn a nonlinear mapping from the unified scene-conditioned feature vector iz_i to the multipath parameter tensor ~i Y_i, i.e., ~i=ℱΘ(i,i) Y_i=F_ (r_i,z_i). By embedding electromagnetic propagation principles into both the network architecture and the loss design, this module is able to produce physically plausible inferences even in regions with sparse observations. I-C1 PINN Architecture The point-wise predictor adopts a physics-guided multi-task multilayer perceptron (MLP) architecture to learn the mapping from point-level conditional features iz_i to the set of multipath parameters. The network input is the unified conditional feature vector iz_i constructed in Section I-B. Point-wise PINN Regressor: This backbone network extracts deep physical representations and regresses multipath parameters. Specifically, the geometric features ig_i and environmental features ie_i are fused through attention mechanisms to form a joint representation ih_i. This representation is then processed by a residual MLP to output preliminary estimates of the path gain, ToA, and angular information for each path. Propagation Type Classifier: As an auxiliary branch, the propagation type classifier takes the deep features extracted by the regressor backbone as input. It consists of a lightweight MLP and predicts the propagation mechanism probability vector ~i,l∈ℝ4 p_i,l ^4 for each path l, corresponding to LoS, reflection, scattering, and diffraction. These probabilities are utilized for both auxiliary supervision and computing adaptive entropy-based weights in the physical regularization terms. Physics-aware Predictions: To ensure regressed outputs strictly conform to electromagnetic propagation characteristics, raw network outputs undergo dedicated activations and normalizations. A scaled hyperbolic tangent activation constrains path gain to a physically meaningful negative dynamic range consistent with path loss definitions. Similarly, a sigmoid function restricts ToA to a non-negative interval bounded by the maximum admissible delay. To prevent numerical instability from direct angular regression, the model predicts sine and cosine components, enforcing unit-circle normalization via projection. These transformations yield physically consistent multipath parameter estimates. I-C2 Training Objective and Loss Design The network parameters are optimized by minimizing a composite loss function ℒPINNL_PINN. Let ℐsupI_sup denote the set of supervised samples, which depends on the operating mode. In the in-scene completion mode, ℐsupI_sup corresponds to the set of sparsely observed locations S; in the cross-scene generation mode, ℐsupI_sup consists of the sampled locations from the training scenes. The overall training objective is formulated as a weighted combination of a supervised loss term, a physics-based regularization term, and a propagation-type consistency term: ℒPINN=ℒsup+λphyℒphy+λtypeℒtype. _PINN=L_sup+ _phyL_phy+ _typeL_type. (8) Supervised Loss (ℒsupL_sup): On the supervising set ℐsupI_sup, the mean squared error (MSE) is adopted as ℒsup=1|ℐsup|∑i∈ℐsup‖~i−i‖22, _sup= 1|I_sup| _i _sup \| y_i-y_i \|_2^2, (9) where ~i=vec(~i) y_i=vec( Y_i) and i=vec(i)y_i=vec(Y_i) denote the vectorized predicted and ground-truth multipath parameter tensors, respectively. Propagation Type Consistency Loss (ℒtypeL_type): For each location i and each path l, the model outputs a probability vector over four propagation mechanisms, i,l=[pi,lLoS,pi,lRefl,pi,lScat,pi,lDiff],∑kpi,lk=1. _i,l=[p_i,l^LoS,\,p_i,l^Refl,\,p_i,l^Scat,\,p_i,l^Diff], _kp_i,l^k=1. (10) A cross-entropy loss is formulated as ℒtype=−1|ℐsup|∑i∈ℐsup∑l=1L∑k=14yi,l,klog(pi,lk+ϵ), _type=- 1|I_sup| _i _sup _l=1^L _k=1^4y_i,l,k (p_i,l^k+ε ), (11) where yi,l,ky_i,l,k denotes weak labels constructed based on electromagnetic propagation rules, and ϵε is a small constant introduced for numerical stability. Physics Loss (ℒphyL_phy) : To regularize predictions in unsupervised regions by enforcing fundamental electromagnetic propagation laws, the physics loss is introduced. It consists of four complementary components, jointly formulated as ℒphy=λ1ℒCG+λ2ℒToA+λ3ℒangle+λ4ℒconsist. _phy= _1L_CG+ _2L_ToA+ _3L_angle+ _4L_consist. (12) Specifically, channel gain, ToA, and angular terms constrain the physical range and geometric consistency of individual paths, while the last consistency term enforces self-consistency across the multipath set. Propagation-type priors are incorporated via probability expectations, with constraint strength adaptively tuned using entropy-based confidence weighting. Since diverse propagation mechanisms like LoS, reflection, scattering, and diffraction exhibit fundamentally different attenuation and delay characteristics, uniform physical bounds cannot apply to all paths. For example, LoS components strictly follow geometric ToA constraints, whereas scattered paths experience significant delay spreading. To enable type-dependent physical regularization, we define reference coefficient vectors for each propagation mechanism as -8.5359pt α =[αLoS,αRefl,αScat,αDiff], =[ _LoS, _Refl, _Scat, _Diff], (13) β =[βLoS,βRefl,βScat,βDiff]. =[ _LoS, _Refl, _Scat, _Diff]. Here, α specifies the allowable upper-margin tolerance per path type relative to free-space channel gain, with tighter margins for LoS paths and larger deviations for diffracted components. Similarly, β defines lower-bound scaling factors relative to the geometric LoS ToA, where βLoS _LoS approaches unity and larger values apply to non-LoS components. The predicted propagation type ~i,l p_i,l is a continuous probability distribution rather than a discrete label. To preserve differentiability during training, the discrete coefficients are transformed into continuous, path-level prior coefficients via probability-weighted expectations, which are expressed as α¯i,l=∑kpi,lkαk,β¯i,l=∑kpi,lkβk, α_i,l= _kp_i,l^k _k, β_i,l= _kp_i,l^k _k, -2.84544pt (14) where k∈LoS,Refl,Scat,Diffk∈\LoS,Refl,Scat,Diff\. Through this formulation, the boundaries of the physical constraints are dynamically adjusted according to the model’s confidence in the predicted propagation mechanism. To avoid imposing incorrect physical constraints during early training stages when the propagation-type predictions are highly uncertain, we introduce an entropy-based confidence gating mechanism. The entropy of the predicted probability distribution and the corresponding adaptive confidence weight are defined as H(i,l)=−∑kpi,lklog(pi,lk+ϵ), H(p_i,l)=- _kp_i,l^k (p_i,l^k+ε), (15) wi,l=(1−H(i,l)log4)η, w_i,l= (1- H(p_i,l) 4 )^η, (16) where η denotes a focusing factor. When the predicted distribution approaches a uniform one, the entropy increases and the weight wi,lw_i,l vanishes, effectively disabling physical regularization and preventing erroneous guidance. Conversely, when the predicted distribution becomes sharp, wi,lw_i,l approaches a zero-one degenerated distribution, and full-strength physical constraints are enforced. Out-of-Bound Penalties for Gain and ToA (ℒCG,ℒToAL_CG,L_ToA): Let the Tx–Rx distance be denoted by did_i. The theoretical LoS ToA is given by τiLoS=di/c _i^LoS=d_i/c, where c is the speed of light. The LoS reference power loss PiFSPLP_i^FSPL follows the free-space path loss model and is written as PiFSPL=20log10(di)+20log10(fc)+20log10(4πc), P_i^FSPL=20 _10(d_i)+20 _10(f_c)+20 _10\! ( 4πc ), (17) where fcf_c denotes the carrier frequency. For each path l, a smooth hinge penalty based on the softplus function softplus(x)=log(1+ex)softplus(x)= (1+e^x) is adopted to penalize boundary violations. Using the type-dependent coefficients α¯i,l α_i,l and β¯i,l β_i,l, the physical constraints on channel gain and ToA are formulated as (18), ℒCG=1|ℐsup|∑i∈ℐsup∑l=1Lwi,lsoftplus(g~i,l−(−PiFSPL+10log10α¯i,l))2. aligned L_CG= 1|I_sup| _i _sup _l=1^Lw_i,l\,softplus\! ( g_i,l- (-P_i^FSPL+10 _10 α_i,l ) )^2. aligned (18) and ℒToA=1|ℐsup|∑i∈ℐsup∑l=1Lwi,lsoftplus(β¯i,lτiLoS−τ~i,l)2, _ToA= 1|I_sup| _i _sup _l=1^Lw_i,l\,softplus\! ( β_i,l _i^LoS- τ_i,l )^2, (19) where g~i,l g_i,l and τ~i,l τ_i,l denote the predicted gain and ToA of the l-th path at location i, respectively. Angular Geometric Consistency Loss (ℒangleL_angle): This term enforces geometric self-consistency between the predicted spatial arrival directions and the inferred propagation types. A geometric consistency score si∈[0,1]s_i∈[0,1] is constructed based on the ToA and angular deviations of the first path relative to the geometric LoS reference, which is defined as si=exp(−|τ~i,1−τiLoS|στ)⋅exp(−d∠(θ~i,1,ϕ~i,1;θiLoS,ϕiLoS)σ∠). aligned s_i= \! (- | τ_i,1- _i^LoS| _τ )· \! (- d_ ( θ_i,1, φ_i,1; _i^LoS, _i^LoS) _ ). aligned -14.22636pt (20) The angular distance metric accounting for phase periodicity is expressed as d∠=|θ~i,1−θiLoS|+min(|ϕ~i,1−ϕiLoS|,360∘−|ϕ~i,1−ϕiLoS|). aligned d_ =| θ_i,1- _i^LoS|+ \! (| φ_i,1- _i^LoS|,360 -| φ_i,1- _i^LoS| ). aligned -14.22636pt (21) The score sis_i intuitively measures the similarity between the predicted first path and an ideal LoS component. Using sis_i as a soft label, a binary cross-entropy loss supervises the predicted LoS probability p~i,1LoS p_i,1^LoS, which is written as (22). ℒangle=−1|ℐsup|∑i∈ℐsup[silog(p~i,1LoS+ϵ)+(1−si)log(1−p~i,1LoS+ϵ)]. aligned L_angle=- 1|I_sup| _i _sup [s_i ( p_i,1^LoS+ε)+(1-s_i) (1- p_i,1^LoS+ε) ]. aligned (22) This loss establishes closed-loop feedback where ToA or angular deviations reduces predicted LoS probabilities, while consistent estimates reinforce geometric alignment. Multipath Consistency Loss (ℒconsistL_consist) Beyond single-path constraints, the multipath parameter set should also satisfy structural consistency at the set level. This term consists of a ranking consistency component and a gain–ToA coupling component, which are jointly defined as ℒconsist=λrankℒrank+λpairℒpair.L_consist= _rankL_rank+ _pairL_pair. -2.84544pt (23) To mitigate permutation ambiguity, paths are enforced into descending gain and ascending ToA orders. Slack variables mpm_p and mτm_τ permit minor ordering violations, yielding the ranking loss as (24). ℒrank=1|ℐsup|∑i∈ℐsup∑l=1L−1[softplus(g~i,l+1−g~i,l+mp)+softplus(τ~i,l−τ~i,l+1+mτ)]. _rank= 1|I_sup| _i _sup _l=1^L-1 [softplus( g_i,l+1- g_i,l+m_p)+softplus( τ_i,l- τ_i,l+1+m_τ) ]. (24) ℒpair=1|ℐsup|∑i∈ℐsup1||∑(l,m)∈softplus(−(g~i,l−g~i,m)(τ~i,m−τ~i,l)spsτ). aligned L_pair= 1|I_sup| _i _sup 1|P| _(l,m) softplus\! (- ( g_i,l- g_i,m)( τ_i,m- τ_i,l)s_ps_τ ). aligned -14.22636pt (25) Physically, longer ToA paths typically traverse greater distances or undergo more reflections, yielding higher attenuation. To penalize violations, we formulate a pairwise consistency loss (25), where =(l,m)∣1≤l<m≤LP=\(l,m) 1≤ l<m≤ L\ defines all path pairs, and training-set standard deviations sps_p and sτs_τ normalize scale differences. I-D Spatial Refinement via Graph Learning Although the PINN predictor incorporates physical constraints, it fundamentally operates in a point-wise manner, treating each query location separately and neglecting the strong spatial correlation induced by the continuous variation of wireless channels. To address this limitation and further suppress local prediction noise, a GNN is introduced to refine the PINN outputs by enforcing spatial consistency. Specifically, a k-nearest neighbor graph G=(V,E)G=(V,E) is constructed over the valid query set validQ_valid to explicitly model the topological relationship among receivers. Each node vi∈Vv_i∈ V corresponds to a valid receiver location, while edges in E connect the k closest neighbors according to the Euclidean distance in the coordinate space. In 2D scenarios, the distance metric is defined on the (x,y)(x,y) plane, whereas in 2.5D settings, the (x,y,h)(x,y,h) coordinates are used, where the height h is normalized to the same scale as the planar coordinates to ensure isotropic neighborhood construction. The initial node features are primarily given by the point-level predictions of the PINN, denoted as ~i∈ℝ4L y_i ^4L, optionally concatenated with a small set of geometric features to enhance discrimination in non-stationary regions such as building boundaries. For the network architecture, the GraphSAGE [9] framework is adopted to perform message passing, where local outliers are mitigated through neighborhood feature aggregation. To preserve the physical priors embedded in the PINN predictions as much as possible, the refinement network is formulated in a residual learning manner. Rather than directly predicting the final multipath parameters, the GNN learns a correction term Δi _i relative to the initial PINN estimates. The refined output is thus expressed as ^i=~i+Δi, y_i= y_i+ _i, -2.84544pt (26) where ^i y_i denotes the final multipath parameter estimation. This residual formulation encourages the graph network to focus on learning spatially consistent adjustments, thereby reducing the risk of violating the physical plausibility of the point-wise predictions. To ensure training stability, the objective function of the refinement stage, denoted by ℒgraphL_graph, is defined by minimizing the MSE between the refined predictions and the ground truth over the supervised sample set ℐsupI_sup, given as ℒgraph=1|ℐsup|∑i∈ℐsup‖^i−i‖22. _graph= 1|I_sup| _i _sup\| y_i-y_i\|_2^2. (27) I-E Joint Construction of RF Map via PINN and GNN As illustrated in the system architecture, the proposed framework constructs high-fidelity RF maps by integrating physical consistency with spatial correlation. First, the framework processes the raw environment map ℰE and geometric coordinates into unified condition vectors iz_i. A point-wise PINN then independently estimates the multipath parameters ~i Y_i for each location, enforcing physical validity via embedded constraints such as channel gain-ToA bounds and LoS alignment. To overcome the limitation of isolated point-wise predictions, a GNN is subsequently employed to refine these estimates. By constructing a k-nearest neighbor graph over the domain, the GNN models spatial dependencies and learns a residual correction Δi _i, effectively smoothing out local noise while preserving the physical structure. During training, the scene encoder is trained jointly with the PINN and GNN in an end-to-end manner. Its parameters are updated implicitly through the back-propagated physics-informed loss without introducing additional supervision. This design enables the encoder to learn task-specific environmental features, such as recognizing diffraction edges. The complete construction process, incorporating the two-stage training strategy to ensure stability, is summarized in Algorithm 1. Input: Scene information ℰE (2D / 2.5D environment map) or sparse samples m,mm=1M\x_m,y_m\_m=1^M Output: RF map or point-wise predictions Y 1 // Condition Initialization 2 i←e_i← Extract environment field =ℰψ(ℰ)F=E_ψ(E) (if available, else 0); 3 i←s_i← Encode sparse measurements (if available, else 0); 4 // Query Construction 5 Construct input queries i\q_i\ from either (i) grid locations implied by ℰE or (i) given sparse coordinates mx_m; 6 // Joint PINN-GNN Training 7 for each training epoch do 8 Point-wise PINN inference: 9 Obtain physics-aware multipath predictions ~i Y_i; 10 Graph construction: 11 Build spatial neighborhood graph G=(V,E)G=(V,E) according to query locations; 12 Graph-based refinement: 13 Refine point-wise predictions via message passing Y; 14 Loss computation: 15 Compute PINN loss ℒPINNL_PINN and GNN loss ℒGNNL_GNN ; 16 17 Parameter update: 18 Update all trainable parameters via backpropagation; 19 end for // RF Map Generation / Completion 20 Freeze model parameters; 21 Construct inference queries itest\q_i^test\ over the target scene; 22 Predict ~i Y_i; 23 Refine via GNN to obtain ^i Y_i; 24 Post-processing and inverse normalization; 25 return Y; Algorithm 1 RF Map Construction with Scene-Conditioned Joint PINN–GNN IV Comparison Schemes and Evaluation Metrics IV-A Baseline Methods and Ablation Studies Existing studies on RF map construction mainly focus on predicting the received power or path loss of the first-arriving path under cross-scene conditions, with the core objective of generating continuous radio coverage distributions given only environmental information. Since no prior work is capable of directly predicting the complete multipath parameter tensor in a cross-scene setting, a fair comparison with existing methods requires following their established evaluation protocol. Accordingly, the channel gain corresponding to the first path is extracted from the multipath predictions of the proposed framework, and the comparison is conducted at this scalar level. Under this alignment setting, RF map construction is formulated as a two-dimensional image generation or completion problem conditioned on environmental representations, where the goal is to predict first-path power maps or path loss maps. Three representative sampling-free, image-based methods are selected as cross-scene baselines. • RadioUNet [14] (Supervised Regression): RadioUNet is a classical end-to-end image regression approach that formulates RF map construction as a direct mapping from environmental layouts to path loss maps. It adopts a fully convolutional U-Net as the backbone and learns the nonlinear relationship between environmental structures and radio attenuation through pixel-wise supervision. Owing to its simplicity and effectiveness, RadioUNet has been widely used as a baseline in prior studies. • RME-GAN [32] (Adversarial Generation): RME-GAN represents RF map generation methods based on conditional GANs. Through adversarial training, it is able to better capture high-frequency details in path loss distributions. The original RME-GAN relies on sparse path loss samples as conditional inputs. To ensure fairness in the sampling-free cross-scene setting, only environmental information is retained as the conditioning input in our experiments. • RadioDiff [28] (Diffusion-Based Generation): RadioDiff is a recent sampling-free RF map construction method based on denoising diffusion probabilistic models. It formulates RF map generation as a conditional diffusion process and progressively recovers high-fidelity first-path power maps through multi-step reverse denoising. Its architecture integrates attention mechanisms and frequency-domain modeling modules, achieving strong performance in both structural similarity and prediction accuracy. It is worth noting that all the above cross-scene baseline methods are designed exclusively for modeling first-path power or path loss, and do not involve tensor-level prediction of multipath parameters. Therefore, they only participate in the aligned comparison at the first-path level. To ensure consistency and fairness across different methods, a unified alignment protocol is adopted for cross-scene first-path power evaluation. At the variable level, all comparisons are strictly restricted to the first-path power map G. For the proposed method, only the channel gain component G G corresponding to the first path is extracted from the predicted multipath parameter tensor Y for evaluation, ensuring consistency with methods that solely predict first-path power or path loss maps. At the spatial level, all image-based evaluation metrics are computed only within the receiver region ΩR=(u,v)∣R(u,v)=1 _R=\(u,v) R(u,v)=1\ to avoid interference from unreachable areas. At the normalization level, the structural similarity index (SSIM) is computed after min–max normalization within ΩR _R for each scene, thereby mitigating the impact of dynamic range variations across different environments. To systematically evaluate each component of the proposed unified framework, we report results under identical cross-scene settings using 2D and 2.5D representations. Accordingly, three ablation schemes analyze the distinct impacts of physical and spatial consistency modeling. • w/o PINN: The physical consistency constraint is removed, and the model relies solely on data-driven regression without incorporating physics-based loss terms. This setting is used to assess the impact of physical priors on the stability and generalization of multipath parameter learning. • w/o GNN: The graph-based spatial refinement module is removed, and RF maps are generated directly from point-wise predictions. This configuration evaluates the role of explicit spatial relationship modeling in continuous map construction. • Separate: A decoupled learning scheme is adopted as a baseline, where physics-aware regression and graph-based propagation are conducted in two independent stages. Specifically, a PINN is first trained to predict multipath parameters at each location under physical constraints, after which its outputs are fixed and a GNN is trained for neighborhood aggregation and spatial refinement. Unlike the proposed unified framework, this scheme does not employ end-to-end joint optimization and does not allow feature sharing across modules. In addition to cross-scene generation, the proposed method is further evaluated under a sparse-sample completion setting. In this scenario, besides the above ablation baselines, the classical spatial interpolation method Kriging is included for comparison. IV-B Evaluation Metrics We adopt a two-level evaluation protocol to comprehensively assess the performance of the proposed method under different task settings. The first level consists of map-level metrics for aligned cross-scenario comparison on first-path gain prediction, while the second level includes tensor-level metrics for evaluating the reconstruction accuracy of complete multipath parameters. Furthermore, the proposed PW-DTW metric is utilized to evaluate multipath temporal consistency. IV-B1 Map-Level Metrics (First-Path Gain) Let G(u,v)G(u,v) and G^(u,v) G(u,v) denote the ground-truth and predicted first-path gain maps, respectively. All map-level metrics are computed only within the receiver region ΩR=(u,v)∣R(u,v)=1 _R=\(u,v) R(u,v)=1\. The root mean square error (RMSE) is defined as RMSE(G,G^)=1|ΩR|∑(u,v)∈ΩR(G^(u,v)−G(u,v))2. (G, G)= 1| _R| _(u,v)∈ _R ( G(u,v)-G(u,v) )^2. (28) The normalized mean square error (NMSE) is given by NMSE(G,G^)=∑(u,v)∈ΩR(G^(u,v)−G(u,v))2∑(u,v)∈ΩRG(u,v)2. (G, G)= _(u,v)∈ _R ( G(u,v)-G(u,v) )^2 _(u,v)∈ _RG(u,v)^2. (29) To measure structural similarity, the structural similarity index (SSIM) is employed. To mitigate the impact of dynamic range variations across different scenarios, both the ground-truth and predicted maps are first min–max normalized within ΩR _R for each scenario as Gnorm=G−GminGmax−Gmin+ϵ,G^norm=G^−G^minG^max−G^min+ϵ. G^norm= G-G_ G_ -G_ +ε, G^norm= G- G_ G_ - G_ +ε. (30) Then, the SSIM metric is computed between the normalized maps x=Gnormx=G^norm and y=G^normy= G^norm. The standard SSIM formulation is applied as SSIM(x,y)=(2μxμy+C1)(2σxy+C2)(μx2+μy2+C1)(σx2+σy2+C2), (x,y)= (2 _x _y+C_1)(2 _xy+C_2)( _x^2+ _y^2+C_1)( _x^2+ _y^2+C_2), (31) where μx _x and μy _y denote the local means, σx2 _x^2 and σy2 _y^2 denote the local variances, and σxy _xy represents the local covariance of x and y. These local statistics are computed using an 11×1111× 11 Gaussian sliding window. The constants are set to C1=(0.01Lmax)2C_1=(0.01L_ )^2 and C2=(0.03Lmax)2C_2=(0.03L_ )^2 with LmaxL_ being the dynamic range of the normalized images. IV-B2 Tensor-Level Metrics (Multipath Parameters) Let Y, ^∈ℝH×W×L×Q Y ^H× W× L× Q denote the ground-truth and predicted multipath parameter tensors, respectively, where L is the number of paths and Q denotes the dimensionality of physical parameters per path. All tensor-level metrics are evaluated only within the effective propagation region ΩM _M, which is automatically inferred from the environmental map. Prior to calculating error metrics, all physical parameters are normalized using their respective statistical mean and standard deviation along each feature dimension to avoid bias caused by inconsistent units. The tensor-level RMSE is defined as RMSEten=1|ΩM|LQ∑(u,v)∈ΩM∑l=1L∑q=1Q(Δu,v,l,q)2, _ten= 1| _M|LQ _(u,v)∈ _M _l=1^L _q=1^Q( _u,v,l,q)^2, (32) where Δ denotes the prediction error of the corresponding parameter. For angular parameters with periodicity, such as the azimuth angle ϕφ, the circular distance is adopted as Δϕ=min(|ϕ^−ϕ|, 360∘−|ϕ^−ϕ|), φ= (| φ-φ|,\;360 -| φ-φ| ), (33) to ensure that the periodic nature of angular predictions is correctly handled. IV-B3 PW-DTW for CIR Evaluation Point-wise error metrics such as RMSE, NMSE, or SSIM are insufficient to capture the peak misalignment phenomenon commonly observed in multipath prediction. To jointly evaluate amplitude and delay alignment errors while assigning higher importance to the dominant or strong paths, PW-DTW is introduced to measure the similarity between predicted and ground-truth CIRs. CIR Synthesis and Normalization: For each receiver location, the Top-LpwL_pw dominant propagation paths are selected, with their gains and times of arrival represented as (gl,τl)\(g_l, _l)\. Here, glg_l denotes the path gain in dB and τl _l is measured in ns. The dB gains are first converted to linear amplitudes pl=10gl/10p_l=10^g_l/10. A discrete CIR is then synthesized on a unified delay grid tii=1Npw\t_i\_i=1^N_pw using Gaussian pulses as x(ti)=∑l=1Lpwplexp(−(ti−τl)22σg2), x(t_i)= _l=1^L_pwp_l \! (- (t_i- _l)^22 _g^2 ), (34) where σg _g controls the temporal spread of each path. The predicted and ground-truth CIRs are denoted by y(ti)y(t_i) and x(ti)x(t_i), respectively, and are jointly normalized using their global maximum amplitude as x~(ti)=x(ti)α,y~(ti)=y(ti)α, x(t_i)= x(t_i)α, y(t_i)= y(t_i)α, (35) with α=max(maxix(ti),maxiy(ti)). α= ( _ix(t_i), _iy(t_i) ). (36) Standard DTW Distance: Given normalized CIRs, the standard DTW local cost is c(i,j)=(x~i−y~j)2c(i,j)=( x_i- y_j)^2, and the cumulative cost matrix is computed via dynamic programming as G(i,j)=c(i,j)+minG(i−1,j),G(i,j−1),G(i−1,j−1), G(i,j)=c(i,j)+ \G(i-1,j),\,G(i,j-1),\,G(i-1,j-1)\, (37) yielding the DTW distance DDTW=G(Npw,Npw)D_DTW=G(N_pw,N_pw), where G(i,j)G(i,j) denotes the minimum cumulative alignment cost between prefixes x~1:i x_1:i and y~1:j y_1:j. However, treating dominant and weak paths equally fails to prioritize critical propagation alignments. Peak-Weighted Modeling: To highlight dominant and strong paths, a time-dependent weighting function is constructed based on peak detection. Peak sets xK_x and yK_y are identified on x~(t) x(t) and y~(t) y(t), respectively, and the Top-LpwL_pw peaks with the largest relative amplitude are retained. Taking x~(t) x(t) as an example, the peak-weighting function is defined as Wx(t)=maxk∈x(a~kexp(−(t−tk)22σ2)),a~k=x~(tk)maxix~(ti), aligned W_x(t)&= _k _x ( a_k \! (- (t-t_k)^22σ^2 ) ),\\ a_k&= x(t_k) _i x(t_i), aligned (38) where tkt_k denotes the peak location and σ controls the influence range of each peak. The function Wy(t)W_y(t) is constructed in the same manner, and both are linearly normalized to [0,1][0,1]. PW-DTW Weighted Cost and Constraints: By jointly considering amplitude discrepancies, delay misalignment, and peak importance, the weighted local cost in PW-DTW is expressed as (39), cw(i,j)=(w0+βpw2(Wx(ti)+Wy(tj)))⋅((x~i−y~j)2+λt(ti−tj)2). aligned c_w(i,j)= (w_0+ _pw2 (W_x(t_i)+W_y(t_j) ) )· (( x_i- y_j)^2+ _t(t_i-t_j)^2 ). aligned (39) where w0w_0 is the base weight, βpw _pw controls the strength of peak weighting, and λt _t balances the delay penalty. To constrain the maximum allowable peak displacement, a Sakoe–Chiba band constraint |i−j|≤r|i-j|≤ r is applied, where r≈Δtmax/Δtr≈ t_ / t, Δtmax t_ denotes the maximum permitted delay offset, and Δt t is the delay grid resolution. Under this constraint, dynamic programming is performed using cw(i,j)c_w(i,j) in place of c(i,j)c(i,j) in (37) to obtain the PW-DTW distance D. Since D increases with the length LDTWL_DTW of the optimal warping path, a normalized distance is defined as D¯=DLDTW. D= DL_DTW. (40) Averaging D¯ D across valid test-set receivers yields the final PW-DTW score, comprehensively evaluating CIR prediction quality regarding peak alignment and amplitude consistency. PW-DTW particularly suits CIR sequences featuring sparse peak structures dominated by a few strong paths. For scenarios involving dense multipath, indistinguishable peaks, or severe noise, increasing the detection threshold or pre-denoising the CIR sequences effectively improves stability. V Experiments and Analysis V-A Experimental Setup We comprehensively evaluate the proposed framework under cross-scene generation and in-scene completion settings. Given the prohibitive cost and complexity of acquiring dense real-world multipath measurements at scale, relying on RT-based datasets has become the standard practice. High-fidelity RT rigorously simulates electromagnetic propagation to yield accurate multipath parameters, serving as the widely accepted ground truth in related literature [14], [11]. V-A1 Cross-Scene Generation For the task of RF map generation in unseen environments, the large-scale OpenPathNet dataset [16] was adopted. Comprising tens of thousands of realistic urban scenes from multiple metropolitan cities, it provides diverse building layouts and environmental structures. Each 128 × 128 m2 scene includes environmental data and multipath RF ray-tracing measurements sampled at a 1 m resolution. The receiver height is fixed at 1 m, with the transmitter at (0,0,30) m operating at 3.6 GHz. Both 2D and 2.5D environmental features were extracted from the environment files provided by the dataset, as illustrated in Fig. 2. The model was trained on a subset consisting of 3,0003,000 distinct scenes. To rigorously assess generalization capability, the optimal model checkpoint was selected based on performance on an independent validation set, which was strictly excluded from gradient updates throughout training. Figure 2: Scene environment map: (a) 2D, (b) 2.5D. V-A2 In-Scene Completion For the task of reconstructing RF maps from sparse observations within a single scene, experiments were conducted on two representative scenarios: • Scenario 1 (Indoor): This scenario was constructed based on the DeepMIMO dataset [2] and consists of a 10 × 10 × 5 m3 indoor environment, as shown in Fig. 3. The access point was located at a height of 2.5 m and operated at 2.4 GHz. A total of 30,280 receivers were distributed over two grids with a spacing of 50 cm. • Scenario 2 (Outdoor): This scenario covers a large-scale area of 512×512m2512× 512~m^2 on the west campus of the University of Science and Technology of China (USTC), as illustrated in Fig. 3. The multipath data were generated using the Sionna ray-tracing platform [12], following the procedure described in [16]. Operating at 3.6 GHz, a base station sits 30 m high above the Third Electronics Building, serving receivers uniformly spaced at 1 m intervals at a 1 m height. Figure 3: Experimental scenarios: (a) DeepMIMO dataset, (b) USTC campus, (c) 3D model of USTC campus buildings. For both completion scenarios, model performance was evaluated under a fixed sparse sampling ratio of 10%. The randomly selected sparse observation points were divided into training, validation, and test sets with a ratio of 7:1.5:1.57:1.5:1.5. Since the fifth propagation path’s power typically drops to approximately 2%–5% of the dominant path, the number of multipath components is set to L=5L=5 to balance reconstruction fidelity and computational efficiency. The AdamW optimizer is adopted alongside a validation-driven adaptive decay strategy, using an initial learning rate of 1×10−41× 10^-4 globally and 5×10−55× 10^-5 specifically for the encoder. To mitigate overfitting, weight decay is set to 1×10−51× 10^-5, and a 0.2 dropout rate is applied to both the PINN and GNN modules. Finally, the key hyperparameters are summarized in Table I and remain fixed unless otherwise specified in the ablation studies. TABLE I: Key Hyperparameters and Default Settings Symbol Description Value Model Architecture P Number of predicted paths 5 Q Parameters per path 4 k Number of kNN neighbors 16 dgraphd_graph GNN hidden dimension 128 LgraphL_graph Number of GNN layers 4 Physics and Regularization α Power reference coefficients [1,0.7,0.3,0.1][1,0.7,0.3,0.1] β ToA reference coefficients [1,1.2,1.4,1.6][1,1.2,1.4,1.6] η Entropy exponent 2 ϵε Numerical stability constant 10−810^-8 σ∠ _ LoS angular tolerance 30∘30 λphymax _phy Maximum physics loss weight 0.5 λtypemax _type Maximum type loss weight 0.3 PW-DTW Evaluation KpwK_pw Number of dominant peaks 5 NpwN_pw Delay grid resolution 5000 σg,σ _g,σ Gaussian pulse width 3.5 ns βpw _pw Peak emphasis factor 3.0 λt _t Temporal penalty weight 1×10−41× 10^-4 Δtmax t_ Maximum delay offset 20 ns V-B Simulation Results V-B1 Performance on Cross-Scene Generation (Map-level) Table I reports the quantitative results for the cross-scene generation task. Our unified framework consistently outperforms all image-based baselines across all evaluation metrics. In particular, the proposed 2.5D model achieves the lowest RMSE and NMSE. In terms of structural fidelity, the highest SSIM is obtained by the proposed method, indicating that the physics-guided spatial refinement is more effective at preserving sharp signal discontinuities and shadow boundaries than purely generative models or the regression-based RadioUNet. Notably, even the proposed 2D setting yields lower prediction errors than the diffusion-based RadioDiff. This result suggests the performance advantage stems not solely from richer input representations, but largely from the robustness of the integrated PINN-GNN architecture. Finally, the additional performance gains observed in the 2.5D mode confirm that explicitly encoding height information ℰ2.5DE^2.5D provides critical geometric cues for modeling diffraction and blockage effects in complex urban environments. TABLE I: Map-level Comparison on Cross-Scene Generation Method RMSE ↓ NMSE ↓ SSIM ↑ RadioUNet [14] 0.0237 0.0069 0.9583 RME-GAN [32] 0.0296 0.0121 0.9299 RadioDiff [28] 0.0190 0.0049 0.9691 Proposed 2D 0.0188 0.0046 0.9700 2.5D 0.0171 0.0040 0.9725 V-B2 Performance on Cross-Scene Generation (Tensor-level) Fig. 4 presents the cumulative distribution function (CDF) curves of the prediction errors for four key physical parameters across 2D and 2.5D input configurations alongside ablation variants. Owing to the large dynamic range among path gains, path-gain error is evaluated as a relative error. The proposed 2.5D method consistently achieves the best first-path prediction performance across all parameters, significantly outperforming the 2D configuration and all ablated variants. Under the 2.5D setting, approximately 95% of first-path gain predictions exhibit relative errors below 20%, while 80% of ToA errors fall within 50 ns. Elevation and azimuth angle errors are largely constrained within 10∘10 and 20∘20 , respectively, demonstrating that explicit height information effectively improves modeling accuracy for complex geometric relationships. In contrast, the w/o PINN variant exhibits pronounced long-tail behavior in ToA and angular predictions, substantially increasing the proportion of large-error samples. This validates the critical role of physics-based constraints in preventing parameter divergence. Moreover, as path order increases, enhanced multipath randomness complicates the prediction task, slowing CDF convergence. Even in this regime, the proposed 2.5D method maintains competitive performance, highlighting its robustness in capturing secondary and weak multipath components. Figure 4: CDFs of multipath prediction errors for the cross-scene generation task: (a) channel gain, (b) ToA, (c) elevation angle, and (d) azimuth angle. TABLE I: Cross-Scene Ablation Results under 2D and 2.5D Environment Representations Method RMSE ↓ NMSE ↓ PW-DTW (×10−3) (× 10^-3) ↓ 2D 2.5D 2D 2.5D 2D 2.5D w/o PINN 2.0923 1.9879 0.0657 0.0597 3.102 2.351 w/o GNN 1.8986 1.7029 0.0519 0.0486 2.317 1.958 Separate 1.7982 1.5657 0.0497 0.0454 2.330 1.946 Proposed 1.3877 1.2714 0.0352 0.0324 1.642 1.353 Figure 5: CIR of cross-scene generation scheme: (a) 2D, and (b) 2.5D. Table I details the cross-scene generation ablation results, quantitatively evaluating how architectural components and input dimensionalities impact multipath parameter prediction. The experimental results indicate that the proposed 2.5D framework achieves the best performance across all evaluation metrics, thereby validating the effectiveness of jointly incorporating physics-based constraints and graph-based reasoning. Specifically, removing physical constraints (w/o PINN) causes the severest degradation, yielding the highest RMSE and NMSE. This underscores that purely data-driven regression struggles to model complex physical parameters across wide dynamic ranges accurately. Furthermore, the Separate scheme highlights the necessity of end-to-end joint optimization, as decoupled PINN and GNN training fails to fully exploit their complementary benefits. The PW-DTW metric and qualitative results in Fig. 5 further corroborate these conclusions. Under the 2D setting, the w/o PINN variant’s sharp PW-DTW error increase directly corresponds to the severe peak misalignment shown by the green dotted curve in Fig. 5. This confirms the physics loss’s decisive role in anchoring ToA predictions. Conversely, the proposed method minimizes numerical errors while generating power delay spectra whose peak locations and amplitudes closely track the ground truth. Furthermore, the 2.5D model consistently outperforms its 2D counterpart across all metrics, demonstrating that explicit height information provides critical diffraction-related geometric cues to enhance overall multipath prediction accuracy. V-B3 In-Scene Completion Results Figure 6: CDFs of multipath prediction errors for the in-scene completion task in S2: (a) channel gain, (b) ToA, (c) elevation angle, and (d) azimuth angle. Fig. 6 quantifies these performance gains through cumulative distribution function curves. The proposed method significantly outperforms all baselines in first-path prediction, whereas Kriging exhibits pronounced ToA and angle estimation deficiencies. Specifically, approximately 90% of first-path gain relative errors fall below 20%, 80% of ToA errors remain within 50 ns, and most azimuth errors are confined within 10∘10 . Furthermore, while enhanced multipath randomness complicates higher-order path predictions, the PINN-GNN framework successfully bounds these errors, validating its robustness in resolving multipath components across complex environments. TABLE IV: Performance Comparison on In-Scene Completion Method RMSE ↓ NMSE ↓ PW-DTW (×10−3) (× 10^-3) ↓ S1 S2 S1 S2 S1 S2 Kriging 2.3166 2.6835 0.0786 0.0914 2.673 3.445 w/o PINN 1.8952 2.2473 0.0566 0.0673 2.260 2.398 w/o GNN 1.6133 1.9072 0.0460 0.0542 1.803 2.356 Separate 1.5027 1.7492 0.0437 0.0501 1.818 2.232 Proposed 1.2106 1.3522 0.0317 0.0363 1.287 1.862 Figure 7: CIR of in-scene completion scheme: (a) S1, and (b) S2. Table IV details the in-scene completion performance across Scenarios 1 and 2. The PINN-GNN framework consistently outperforms the Kriging baseline and all ablation variants across all metrics. Particularly in the challenging Scenario 2, the proposed method reduces RMSE by approximately 50% against Kriging, highlighting the superiority of physics-aware propagation learning over purely statistical interpolation. Further insights can be obtained by jointly examining the PW-DTW metric and the power delay profiles shown in Fig. 7. Yielding the highest PW-DTW errors in both scenarios, Kriging produces severely misaligned spurious peaks, as visualized by the green dotted curve in Fig. 7. This failure stems from a lack of physical awareness regarding light-speed wave propagation and environmental geometry. Similarly, removing physical constraints (w/o PINN) severely degrades PW-DTW performance, confirming the physics loss’s critical role in anchoring ToA predictions within physically plausible distances. Conversely, the proposed method minimizes PW-DTW scores, generating curves that closely track the ground truth to capture dominant path delays and multipath structures accurately. Ultimately, these results validate the jointly optimized PINN-GNN framework’s effectiveness in reconstructing high-fidelity channel fingerprints. Note that the reported RMSE/NMSE values are not directly comparable to those obtained in our earlier work [17], as the current evaluation adopts per-path and per-feature normalization together with tensor-wise averaging. VI Conclusion In this paper, we propose a unified PINN-GNN framework for constructing high-fidelity multipath radio maps in complex urban environments, supporting both cross-scenario generation and sparse-sampling completion using 2D and 2.5D representations. By embedding physical constraints, the PINN learns a physically consistent mapping from receiver locations to multipath parameters, while the GNN captures spatial correlations via a k-nearest-neighbor graph to refine predictions. The PINN-GNN framework jointly models the physical relationships among multipath components and their spatial consistency, enabling accurate characterization of multipath propagation. Extensive experimental results demonstrate that the proposed method exhibits strong generalization performance in both cross-scenario generation and in-scenario completion tasks, achieving consistently superior map-level metrics compared to relevant baselines. Notably, under sparse sampling, the PINN-GNN framework maintains significantly higher robustness than Kriging interpolation. Through ablation studies and the PW-DTW metric, we further verify the decisive role of physical constraints in anchoring time-domain signal characteristics. These results validate the effectiveness of integrating physical knowledge with deep learning, and provide a reliable solution for high-precision digital twins and real-time channel prediction in future 6G networks. References [1] 3GPP (2023) 3GPP TR38.901. 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