Neural CKM: Channel Mapping Models

• Neural Channel Knowledge Map is a location-indexed representation that uses sparse measurements and environmental data to infer wireless channel properties.
• The approach leverages image super-resolution, inpainting, and conditional generative models to reconstruct diverse channel maps including gain, angle, and spatial correlations.
• Integration with communication, sensing, and beam management enables efficient AP placement, uncertainty-aware updates, and real-time decision support.

Neural Channel Knowledge Map (CKM) denotes a family of data-driven models for constructing, inferring, and exploiting location-specific channel knowledge as a site-specific prior for wireless communication and sensing. In the CKM literature, the mapped quantity can be a channel gain map (CGM), path loss, angle information, delay information, beam-conditioned equivalent gain, angle-delay statistics, or a spatial correlation matrix, and the map may be represented as a grid, a continuous coordinate-to-channel function, or a learned conditional distribution (Dai et al., 2024, Zhao et al., 8 May 2026, Wang et al., 22 May 2025, Chen et al., 22 Apr 2026). The common objective is to replace or reduce costly real-time channel acquisition with environment-aware prior knowledge learned from sparse measurements, environmental maps, historical observations, or the channel knowledge of other access points, which has led to the adoption of convolutional architectures, Transformers, super-resolution networks, diffusion models, flow matching, and coordinate-conditioned regressors (Wang et al., 2024, Huang et al., 6 Jan 2026).

1. Formal definitions and representational scope

A CKM is consistently defined as a location-indexed representation of channel knowledge. In the per-access-point formulation used for cross-AP inference, the map of AP $n$ is written as

$\mathcal{M}_n=\{\mathbf{c}_n;\mathbf{G}_n\},$

where $\mathbf{c}_n \in \mathbb{R}^{2\times 1}$ is the AP location and $\mathbf{G}_n \in \mathbb{R}^{W\times W}$ is a spatial map of a channel quantity over the region; in the differentiable formulation for low-altitude wireless networks, the learned map is a continuous function

$$\hat{\mathcal{H}}(\mathbf{q}) = \mathcal{F}(\mathbf{q};\mathcal{C}),$$

mapping continuous coordinates to expected channel power gain; in the beamforming-codebook-aware formulation for multi-antenna systems, the CKM becomes beam-conditioned,

$$\mathrm{CKM}(j) = \mathcal{F}_\Theta(\mathbf{p}_i \in \mathcal{M}; \mathbf{w}=\mathbf{W}[:,j]).$$

These formulations make explicit that neural CKM is not a single model class but a shared abstraction over location-conditioned channel priors (Dai et al., 2024, Zhao et al., 8 May 2026, Wang et al., 22 May 2025, Wu et al., 14 Apr 2025).

Most early neural CKM work concentrates on the CGM because it records a scalar field over space and is naturally compatible with image-based learning. Later work expands the target to channel angle maps, channel angle-delay maps (CADM), beam/codeword-conditioned maps, and channel spatial correlation maps (SCMs), thereby moving from first-order large-scale fading to richer angular, delay-domain, and second-order spatial statistics (Wang et al., 2024, Wu et al., 4 Jul 2025, Chen et al., 22 Apr 2026). This broadening is significant because it shifts CKM from coverage-oriented prediction toward covariance-aware beamforming, non-line-of-sight sensing, and multi-antenna inference.

Within the broader CKM literature, hybrid model-based construction predates fully neural approaches. EM-based map construction treated CKM estimation as a latent-variable maximum likelihood problem with piecewise statistical channel models, while interference-cancellation-based construction represented each spatial grid by effective path delays, angles, and powers and estimated them from BS received signals through Bayesian inference (Li et al., 2021, Jiang et al., 2024). Neural CKM inherits the environment-aware premise of these methods but replaces hand-crafted structure with learned multiscale, conditional, or generative mappings.

2. Neural construction paradigms

One major paradigm reformulates CKM construction as image super-resolution or inpainting. In "Deep Learning-Based CKM Construction with Image Super-Resolution" (Wang et al., 2024), sparse observations $y^*=Hx^*$ are treated as a low-resolution version of a complete CKM $x^*$, and SRResNet is trained with mean squared error for numerical fidelity rather than perceptual realism. The network has 1,549,462 parameters, uses 5 residual blocks, two $3\times 3$ convolution layers with 64 feature channels per block, two Batch Normalization layers, PReLU activations, and two sub-pixel convolution layers for upsampling. On RadioMapSeer at $4\times$ super-resolution, SRResNet reports RMSE $\mathcal{M}_n=\{\mathbf{c}_n;\mathbf{G}_n\},$0 dB, PSNR $\mathcal{M}_n=\{\mathbf{c}_n;\mathbf{G}_n\},$1, SSIM $\mathcal{M}_n=\{\mathbf{c}_n;\mathbf{G}_n\},$2, LPIPS $\mathcal{M}_n=\{\mathbf{c}_n;\mathbf{G}_n\},$3, and MSE $\mathcal{M}_n=\{\mathbf{c}_n;\mathbf{G}_n\},$4, while on CKMImageNet it reports RMSE $\mathcal{M}_n=\{\mathbf{c}_n;\mathbf{G}_n\},$5 dB, PSNR $\mathcal{M}_n=\{\mathbf{c}_n;\mathbf{G}_n\},$6, SSIM $\mathcal{M}_n=\{\mathbf{c}_n;\mathbf{G}_n\},$7, LPIPS $\mathcal{M}_n=\{\mathbf{c}_n;\mathbf{G}_n\},$8, and MSE $\mathcal{M}_n=\{\mathbf{c}_n;\mathbf{G}_n\},$9; the same paper states that only $\mathbf{c}_n \in \mathbb{R}^{2\times 1}$0 of the locations need to be measured to achieve about $\mathbf{c}_n \in \mathbb{R}^{2\times 1}$1 dB RMSE in path loss (Wang et al., 2024).

A related but architecturally distinct line treats CKM as image-to-image inpainting over a geometric environment map. "An I2I Inpainting Approach for Efficient Channel Knowledge Map Construction" (Jin et al., 2024) discretizes the target area into a 2-D grid, normalizes channel gain to grayscale, and uses a Laplacian pyramid (LP) to obtain a reversible multiscale decomposition. Its LPCGMN architecture assigns more modeling capacity to low-frequency components, uses MHSA on low-frequency maps, and uses MHCCA on high-resolution residuals. On the DPM subset of RadioMapSeer, LPCGMN with $\mathbf{c}_n \in \mathbb{R}^{2\times 1}$2 reports NMSE $\mathbf{c}_n \in \mathbb{R}^{2\times 1}$3 and RMSE $\mathbf{c}_n \in \mathbb{R}^{2\times 1}$4, compared with U-Net NMSE $\mathbf{c}_n \in \mathbb{R}^{2\times 1}$5, RMSE $\mathbf{c}_n \in \mathbb{R}^{2\times 1}$6, and WNet NMSE $\mathbf{c}_n \in \mathbb{R}^{2\times 1}$7, RMSE $\mathbf{c}_n \in \mathbb{R}^{2\times 1}$8, while also reducing model size and FLOPs relative to WNet (Jin et al., 2024).

A second paradigm is conditional generative modeling. "Channel Knowledge Map Construction via Guided Flow Matching" (Huang et al., 6 Jan 2026) formulates CKM generation as learning the conditional distribution $\mathbf{c}_n \in \mathbb{R}^{2\times 1}$9 and replaces diffusion-style stochastic denoising with a deterministic ODE

$\mathbf{G}_n \in \mathbb{R}^{W\times W}$0

under linear transport guided flow matching. For CGM construction it conditions on the observed low-resolution gain map, a building mask, and an edge map, and for SCM construction it conditions on neighboring SCM tensors. The reported CGM results are NMSE $\mathbf{G}_n \in \mathbb{R}^{W\times W}$1, PSNR $\mathbf{G}_n \in \mathbb{R}^{W\times W}$2, SSIM $\mathbf{G}_n \in \mathbb{R}^{W\times W}$3, FID $\mathbf{G}_n \in \mathbb{R}^{W\times W}$4, and time $\mathbf{G}_n \in \mathbb{R}^{W\times W}$5 ms, versus DDPM NMSE $\mathbf{G}_n \in \mathbb{R}^{W\times W}$6, PSNR $\mathbf{G}_n \in \mathbb{R}^{W\times W}$7, SSIM $\mathbf{G}_n \in \mathbb{R}^{W\times W}$8, FID $\mathbf{G}_n \in \mathbb{R}^{W\times W}$9, and time $$\hat{\mathcal{H}}(\mathbf{q}) = \mathcal{F}(\mathbf{q};\mathcal{C}),$$0 ms, corresponding to a $$\hat{\mathcal{H}}(\mathbf{q}) = \mathcal{F}(\mathbf{q};\mathcal{C}),$$1 inference-speed improvement (Huang et al., 6 Jan 2026).

A more explicitly physics-constrained generative formulation appears in "Channel Knowledge Map Construction via Physics-Inspired Diffusion Model Without Prior Observations" (Zhu et al., 2 Dec 2025). That work focuses on environment-only CGM generation and augments latent diffusion with three physically motivated regularizers: edge loss, regional propagation loss, and multi-scale feature loss. The reported static-CGM results are NMSE $$\hat{\mathcal{H}}(\mathbf{q}) = \mathcal{F}(\mathbf{q};\mathcal{C}),$$2, RMSE $$\hat{\mathcal{H}}(\mathbf{q}) = \mathcal{F}(\mathbf{q};\mathcal{C}),$$3, and PSNR $$\hat{\mathcal{H}}(\mathbf{q}) = \mathcal{F}(\mathbf{q};\mathcal{C}),$$4, while the dynamic-CGM results are NMSE $$\hat{\mathcal{H}}(\mathbf{q}) = \mathcal{F}(\mathbf{q};\mathcal{C}),$$5, RMSE $$\hat{\mathcal{H}}(\mathbf{q}) = \mathcal{F}(\mathbf{q};\mathcal{C}),$$6, and PSNR $$\hat{\mathcal{H}}(\mathbf{q}) = \mathcal{F}(\mathbf{q};\mathcal{C}),$$7; the paper states improvements over RadioDiff of $$\hat{\mathcal{H}}(\mathbf{q}) = \mathcal{F}(\mathbf{q};\mathcal{C}),$$8 in static CGM and $$\hat{\mathcal{H}}(\mathbf{q}) = \mathcal{F}(\mathbf{q};\mathcal{C}),$$9 in dynamic CGM (Zhu et al., 2 Dec 2025). This suggests that generative CKM research has moved from visual plausibility toward explicit propagation consistency.

3. Conditioning mechanisms, architectures, and physical priors

Cross-AP CKM inference makes the conditioning problem explicit. "Generating CKM Using Others' Data: Cross-AP CKM Inference with Deep Learning" (Dai et al., 2024) studies the mapping

$$\mathrm{CKM}(j) = \mathcal{F}_\Theta(\mathbf{p}_i \in \mathcal{M}; \mathbf{w}=\mathbf{W}[:,j]).$$0

where the input is the CKMs of existing APs together with the location of a potentially new AP, and the output is the inferred CKM of that AP. Each AP location is converted into a one-hot AP location map, fused with its CGM via

$$\mathrm{CKM}(j) = \mathcal{F}_\Theta(\mathbf{p}_i \in \mathcal{M}; \mathbf{w}=\mathbf{W}[:,j]).$$1

and stacked across APs. Because the target AP location map is sparse, the paper applies a pre-convolution with a $$\mathrm{CKM}(j) = \mathcal{F}_\Theta(\mathbf{p}_i \in \mathcal{M}; \mathbf{w}=\mathbf{W}[:,j]).$$2 all-ones kernel before overlaying it with the other AP feature maps. A UNet is trained with mean square error, and every AP is treated as the target AP in turn, so the learned rule is AP-role agnostic rather than tied to a fixed source-target pair (Dai et al., 2024).

Differentiable CKM for continuous trajectory optimization adopts a different conditioning strategy. "Towards Intelligent Low-Altitude Wireless Network Deployment: Differentiable Channel Knowledge Map Construction and Trajectory Design" (Zhao et al., 8 May 2026) uses a shared CNN encoder on a 5-channel tensor containing a BS position map, building height map, sampled channel gain map, LoS map, and KNN-augmented map. Location-specific features are then extracted by bilinear interpolation and concatenated with normalized coordinates to form the conditional input to either a c-MLP or a cKAN regressor. The key structural point is that the model is differentiable with respect to $$\mathrm{CKM}(j) = \mathcal{F}_\Theta(\mathbf{p}_i \in \mathcal{M}; \mathbf{w}=\mathbf{W}[:,j]).$$3, so the gradient $$\mathrm{CKM}(j) = \mathcal{F}_\Theta(\mathbf{p}_i \in \mathcal{M}; \mathbf{w}=\mathbf{W}[:,j]).$$4 can be embedded directly into alternating optimization and successive convex approximation for multi-UAV joint power, bandwidth, and trajectory optimization (Zhao et al., 8 May 2026).

Beam-aware and matrix-valued CKM require additional structure beyond scalar map regression. "Beamforming-Codebook-Aware Channel Knowledge Map Construction for Multi-Antenna Systems" (Wang et al., 22 May 2025) conditions CKM on DFT precoding vectors and uses a TransUNet architecture composed of a ResNet-based CNN encoder, a 12-layer Transformer, and a UNet-style decoder. The input is environmental data plus BS location in a $$\mathrm{CKM}(j) = \mathcal{F}_\Theta(\mathbf{p}_i \in \mathcal{M}; \mathbf{w}=\mathbf{W}[:,j]).$$5 tensor, and training uses a composite loss

$$\mathrm{CKM}(j) = \mathcal{F}_\Theta(\mathbf{p}_i \in \mathcal{M}; \mathbf{w}=\mathbf{W}[:,j]).$$6

with $$\mathrm{CKM}(j) = \mathcal{F}_\Theta(\mathbf{p}_i \in \mathcal{M}; \mathbf{w}=\mathbf{W}[:,j]).$$7. In parallel, "CKM Beyond Channel Gain: Spatial Correlation Map Construction with Deep Learning" (Chen et al., 22 Apr 2026) makes SCM tractable by decomposing it into a path gain map (PGM) and a path angle map (PAM), reconstructing those lower-dimensional maps with E-SRResNet, and then synthesizing the SCM from

$$\mathrm{CKM}(j) = \mathcal{F}_\Theta(\mathbf{p}_i \in \mathcal{M}; \mathbf{w}=\mathbf{W}[:,j]).$$8

That model also injects LoS, binary building, and BS priors, reflecting a broader trend toward physics-aware conditioning (Wang et al., 22 May 2025, Chen et al., 22 Apr 2026).

4. Datasets and empirical evaluation

Neural CKM research is heavily dataset-driven. Cross-AP inference uses RadioMapSeer with 700 environment maps, 500 training maps, 100 validation maps, and 80 AP location maps plus corresponding simulated CGMs for each physical map (Dai et al., 2024). SRResNet-based sparse CKM construction uses 21,000 RadioMapSeer path loss maps with 20,000 training and 1,000 test maps, as well as CKMImageNet with 11,064 path loss maps for training, 1,000 path loss test maps, 11,866 AoA training images, and 1,000 AoA test images, all at $$\mathrm{CKM}(j) = \mathcal{F}_\Theta(\mathbf{p}_i \in \mathcal{M}; \mathbf{w}=\mathbf{W}[:,j]).$$9 resolution (Wang et al., 2024). CKMImageNet itself was introduced as a large-scale paired numerical-and-visual propagation dataset built with Wireless InSite; the paper reports more than 10,000 different environment maps, over 25 million channel data entries, and about 40,000 channel knowledge images, with each image corresponding to a $y^*=Hx^*$0 physical area (Wu et al., 2024).

The reported metrics are heterogeneous because the tasks differ: some papers evaluate scalar path-loss fidelity, others evaluate beam-conditioned map quality, SCM structural consistency, or distributional realism. Direct comparison across papers is therefore limited by differences in datasets, target quantities, and conditioning information. Within each task family, however, the numerical record shows a consistent progression from interpolation-style baselines toward conditional neural and generative models.

Task	Representative result	Source
Cross-AP CKM inference	MSE $y^*=Hx^*$1, RMSE $y^*=Hx^*$2 dB; weighted inference RMSE $y^*=Hx^*$3 dB; model-based 3GPP inference RMSE $y^*=Hx^*$4 dB	(Dai et al., 2024)
Sparse CKM via SRResNet	RadioMapSeer RMSE $y^*=Hx^*$5 dB at $y^*=Hx^*$6 SR; CKMImageNet RMSE $y^*=Hx^*$7 dB; about $y^*=Hx^*$8 dB RMSE with $y^*=Hx^*$9 measurements	(Wang et al., 2024)
Beam-aware CKM via TransUNet	RMSE $x^*$0 vs RadioWNet $x^*$1; inference time $x^*$2 s vs accelerated Sionna ray tracing $x^*$3 s	(Wang et al., 22 May 2025)
LT-GFM generative CKM	NMSE $x^*$4, SSIM $x^*$5, FID $x^*$6, time $x^*$7 ms; DDPM time $x^*$8 ms	(Huang et al., 6 Jan 2026)
Physics-inspired diffusion	Static CGM NMSE $x^*$9; dynamic CGM NMSE $3\times 3$0	(Zhu et al., 2 Dec 2025)
SCM construction	Cosine similarity between constructed SCM and ground truth exceeds $3\times 3$1 in most regions	(Chen et al., 22 Apr 2026)

5. Communication, sensing, beam management, and optimization

In communication-oriented settings, neural CKM is used both for construction and for decision support. Cross-AP inference is motivated by initial CKM generation, environment-aware AP placement, and cost-effective CKM updates when APs are added or changed (Dai et al., 2024). In low-altitude wireless networks, differentiable CKM is embedded directly into joint power, bandwidth, and trajectory optimization, where the paper reports that CKM-JPBTO achieves a significantly higher minimum throughput than conventional statistical channel model-based JPBTO and that cKAN is slightly better than cMLP while being more parameter-efficient (Zhao et al., 8 May 2026). For active IRS systems, a Transformer-based neural CKM composed of LPS-Net and SE-Net predicts link power statistics and ergodic spectral efficiency from historical measurements tagged with user positions, after which the SM-IB scheduler reaches near-optimal max-min throughput with runtime from $3\times 3$2 s to $3\times 3$3 s compared with Gurobi from $3\times 3$4 s to $3\times 3$5 s in the 6-AIRS case (Chen et al., 9 Aug 2025).

Sensing and localization applications extend CKM beyond communications-only priors. "You May Use the Same Channel Knowledge Map for Environment-Aware NLoS Sensing and Communication" (Wu et al., 4 Jul 2025) treats the sensing target as a virtual UE and transforms communication channel priors into sensing channel priors using CADM. The resulting likelihood model supports maximum-likelihood target localization in NLoS environments, and the paper’s CRLB analysis gives $3\times 3$6, reflecting the information increase from multipath combinations (Wu et al., 4 Jul 2025). "Channel Knowledge Map-Enabled NLoS ISAC Localization" (Hong et al., 8 Apr 2026) instead learns AoA-ToA path signatures offline, maps each path to a candidate scatterer, matches online observations to CKM entries, and uses weighted nonlinear least squares for joint UE and scatterer estimation; the reported error CDF reaches nearly $3\times 3$7 reliability at $3\times 3$8 m, and the RMSE is about $3\times 3$9 m when $4\times$0 (Hong et al., 8 Apr 2026).

Beam management is another major application. "Leveraging Channel Knowledge Map for Multi-User Hierarchical Beam Training Under Position Uncertainty" (Shi et al., 28 Nov 2025) defines BeamCKM as a beamforming-codebook-oriented map of equivalent gain over a spatial grid, models position uncertainty explicitly, and formulates hierarchical beam search as a pruned binary tree guided by CKM-derived beam potential. The paper reports roughly $4\times$1 overhead reduction in single-user trials, about $4\times$2 spectral-efficiency improvement over baselines at fixed overhead, and nearly $4\times$3 of multi-user trials with total overhead around $4\times$4 for the proposed MU method (Shi et al., 28 Nov 2025). In high-mobility networks, CKM also couples coordinate-domain EKF tracking with beam-domain Markov tracking, so that location estimates produce AoA priors and predictive beamforming minimizes AoA-estimation CRB pathwise (Du et al., 28 Jun 2025). This suggests that neural CKM has evolved from a passive map-construction problem into an active component of online control, scheduling, and tracking loops.

6. Limitations, maintenance, and emerging research directions

A recurrent limitation is environment specificity. Cross-AP inference assumes that APs are deployed in the same physical region and that their CKMs are correlated because propagation is shaped by the same obstacles and geometry; the same paper notes that the reported experiments are on simulated RadioMapSeer CGMs and do not address real-world measurement noise or hardware variability (Dai et al., 2024). Similar constraints appear in beam-aware, SCM, and generative work, where the training and evaluation pipelines are predominantly simulation-based and the papers explicitly or implicitly leave domain adaptation, calibration, and real-scene generalization open (Wang et al., 22 May 2025, Huang et al., 6 Jan 2026, Zhu et al., 2 Dec 2025). A related misconception addressed by the dataset and beam-training papers is that CKM can simply replace CSI; the more precise statement is that CKM provides probabilistic prior information that must often be fused with real-time measurements and updated when the environment changes (Wu et al., 14 Apr 2025, Shi et al., 28 Nov 2025).

Freshness and sampling efficiency have consequently become central themes. "Update Strategy for Channel Knowledge Map in Complex Environments" (Wang et al., 17 Dec 2025) introduces the Map Efficacy Function

$4\times$5

models both gradual aging and abrupt environmental transitions, and formulates update scheduling as fractional programming. The paper derives Delta-P, which guarantees global optimality, Delta-L, which achieves near-optimal performance with near-linear complexity, and a threshold rule stating that immediate updates are optimal when

$4\times$6

In parallel, "Active Learning for Channel Knowledge Map Construction via Bayesian Inference Diffusion Models" (Zhu et al., 29 Jun 2026) turns sparse CGM construction into an active learning problem by estimating epistemic uncertainty from a latent diffusion model through a last-layer Laplace approximation and then selecting new measurement locations using both uncertainty and spatial-uniformity criteria. The paper reports better reconstruction performance than baseline methods on both static and dynamic CGM datasets and shows that uncertainty concentrates near building boundaries and LOS/NLOS transitions (Zhu et al., 29 Jun 2026).

Taken together, these developments indicate a shift in neural CKM research from one-shot map reconstruction toward four coupled problems: representation design, uncertainty-aware sensing, continual updating, and downstream integration. A plausible implication is that future neural CKM systems will be judged less by isolated image-reconstruction metrics than by how reliably they support communication, sensing, scheduling, and control under spatial uncertainty, environmental drift, and multimodal channel structure.