Neural Beam Field (NBF): Hybrid RSRP Prediction

• Neural Beam Field (NBF) is a hybrid neural–physical framework that predicts spatial beam-level RSRP by learning a site-specific multipath representation from sparse measurements.
• It combines a Transformer-based neural network for MCPP estimation with analytical beamforming models to infer mean and variance of normalized RSRP efficiently.
• Empirical evaluations show that NBF with a Pretrain-and-Calibrate strategy outperforms MLP and CKM baselines, achieving lower MAE and superior storage efficiency.

Neural Beam Field (NBF) is a hybrid neural–physical framework for predicting spatial beam-level reference signal received power (RSRP) statistics in dense wireless networks. It is formulated for downlink beam management in a dense, multi-user 5G/6G macrocell with multi-panel massive MIMO at the base station (BS) and single-antenna user equipments (UEs), where exhaustive measurement of all beams at all positions is infeasible because of measurement overhead, high beam counts, and fast channel variation. NBF addresses this by learning a site-specific multipath representation from sparse measurements and UE positions, then combining that representation with a closed-form analytical beam model to infer the mean and variance of normalized beam-level RSRP. Its central intermediate representation is the Multi-path Conditional Power Profile (MCPP), and its training procedure combines ray-tracing priors with on-site calibration through a Pretrain-and-Calibrate (PaC) strategy (Guo et al., 9 Aug 2025).

1. Problem formulation and system objective

The target quantity in NBF is the beam-level RSRP, defined as normalized received power per transmit power for a given UE location and transmit beam. The service region is discretized into spatial grids, and at each location the network seeks beam-level RSRP for beam selection, beam sweeping reduction, handover and inter-cell beam and panel coordination, and user scheduling and link adaptation. In the setting considered, dense deployment, many beams, many users, and 3D positions make direct measurement across the full location–beam space impractical; in addition, channel coherence time may be shorter than the time required to scan all beams for all users (Guo et al., 9 Aug 2025).

Traditional channel knowledge maps (CKMs) store pre-computed RSRP or pathloss values on spatial grids and interpolate between them. The paper identifies two central limitations: storage grows rapidly when separate maps must be maintained for many beams, carriers, and configurations, and interpolation methods such as inverse distance weighting (IDW), Kriging, and splines struggle with complex non-smooth multipath and blockage patterns. Pure black-box deep neural networks, such as multilayer perceptrons (MLPs) that map location and beam index directly to RSRP, face a different limitation: they require dense training data in a high-dimensional input space, entangle environment, array, and beam codebook in a single opaque mapping, and are difficult to adapt to new antenna or beam configurations (Guo et al., 9 Aug 2025).

NBF is defined as a learned function that predicts the statistics of normalized beam-level RSRP for any UE location $\mathbf{x}$ and transmit beamforming vector $\mathbf{w}_{\text{tx}}$:

$$\mathrm{NBF}(\mathbf{x},\mathbf{w}_{\text{tx}}) \triangleq \left\{ \mu_{\mathrm{RSRP}}\bigl(f_\vartheta(\mathbf{x}),\mathbf{w}_{\text{tx}}\bigr), \, \sigma^2_{\mathrm{RSRP}}\bigl(f_\vartheta(\mathbf{x}),\mathbf{w}_{\text{tx}}\bigr) \right\}.$$

Here $f_\vartheta(\mathbf{x})$ is a Transformer-based neural network that predicts MCPP at location $\mathbf{x}$, while the analytical module maps the MCPP and the beamforming vector to mean and variance of beam-level RSRP. This blackbox–whitebox decoupling is intended to learn only the site-specific multipath structure that is difficult to model analytically, while retaining an explicit analytical treatment of beamforming and array physics (Guo et al., 9 Aug 2025).

2. Multi-path Conditional Power Profile

The physical channel is modeled as a sum of $L$ significant paths. For isotropic transmit and receive sides, path $l$ is associated with a transmit direction unit vector $\boldsymbol{u}_{\text{tx},l}$, a receive direction unit vector $\boldsymbol{u}_{\text{rx},l}$, delay $\tau_l$, and complex gain $\mathbf{w}_{\text{tx}}$0. The instantaneous phase $\mathbf{w}_{\text{tx}}$1 is random, whereas the deterministic environment-specific quantity of interest is the path power:

$\mathbf{w}_{\text{tx}}$2

This quantity is termed Single-path Conditional Power (SCP) and is a function over a 7-D condition space comprising the 3D transmit direction, 3D receive direction, and scalar delay (Guo et al., 9 Aug 2025).

The Multi-path Conditional Power Profile is then defined as the set of SCPs for all significant paths at a given UE location:

$\mathbf{w}_{\text{tx}}$3

MCPP is therefore a per-location description of the powers of all significant paths, jointly conditioned on their directions and delays. The paper emphasizes that the geometry-dependent quantities $\mathbf{w}_{\text{tx}}$4, $\mathbf{w}_{\text{tx}}$5, and $\mathbf{w}_{\text{tx}}$6 can be obtained or approximated via ray tracing or site-survey CKMs and are relatively stable in a quasi-static environment, whereas the small-scale phases $\mathbf{w}_{\text{tx}}$7 are excluded from MCPP and treated probabilistically in the analytical module (Guo et al., 9 Aug 2025).

This representation is central because it separates environment-dependent multipath structure from antenna and beam configuration. MCPP is independent of a particular beam codebook or panel rotation, while beamforming-specific effects enter later through the analytical array and radiation model. A direct implication of this separation, stated in the paper, is that changes in beam codebooks, arrays, or carrier frequency can reuse the white-box formulas, with only modest adaptation needed in the MCPP learner (Guo et al., 9 Aug 2025).

3. Transformer-based black-box component

The neural component $\mathbf{w}_{\text{tx}}$8 takes UE location $\mathbf{w}_{\text{tx}}$9 or $$\mathrm{NBF}(\mathbf{x},\mathbf{w}_{\text{tx}}) \triangleq \left\{ \mu_{\mathrm{RSRP}}\bigl(f_\vartheta(\mathbf{x}),\mathbf{w}_{\text{tx}}\bigr), \, \sigma^2_{\mathrm{RSRP}}\bigl(f_\vartheta(\mathbf{x}),\mathbf{w}_{\text{tx}}\bigr) \right\}.$$0 as input and outputs a parametric representation of MCPP. For each of $$\mathrm{NBF}(\mathbf{x},\mathbf{w}_{\text{tx}}) \triangleq \left\{ \mu_{\mathrm{RSRP}}\bigl(f_\vartheta(\mathbf{x}),\mathbf{w}_{\text{tx}}\bigr), \, \sigma^2_{\mathrm{RSRP}}\bigl(f_\vartheta(\mathbf{x}),\mathbf{w}_{\text{tx}}\bigr) \right\}.$$1 candidate paths, the network predicts eight continuous features representing an SCP, typically including $$\mathrm{NBF}(\mathbf{x},\mathbf{w}_{\text{tx}}) \triangleq \left\{ \mu_{\mathrm{RSRP}}\bigl(f_\vartheta(\mathbf{x}),\mathbf{w}_{\text{tx}}\bigr), \, \sigma^2_{\mathrm{RSRP}}\bigl(f_\vartheta(\mathbf{x}),\mathbf{w}_{\text{tx}}\bigr) \right\}.$$2, $$\mathrm{NBF}(\mathbf{x},\mathbf{w}_{\text{tx}}) \triangleq \left\{ \mu_{\mathrm{RSRP}}\bigl(f_\vartheta(\mathbf{x}),\mathbf{w}_{\text{tx}}\bigr), \, \sigma^2_{\mathrm{RSRP}}\bigl(f_\vartheta(\mathbf{x}),\mathbf{w}_{\text{tx}}\bigr) \right\}.$$3, delay $$\mathrm{NBF}(\mathbf{x},\mathbf{w}_{\text{tx}}) \triangleq \left\{ \mu_{\mathrm{RSRP}}\bigl(f_\vartheta(\mathbf{x}),\mathbf{w}_{\text{tx}}\bigr), \, \sigma^2_{\mathrm{RSRP}}\bigl(f_\vartheta(\mathbf{x}),\mathbf{w}_{\text{tx}}\bigr) \right\}.$$4, and power $$\mathrm{NBF}(\mathbf{x},\mathbf{w}_{\text{tx}}) \triangleq \left\{ \mu_{\mathrm{RSRP}}\bigl(f_\vartheta(\mathbf{x}),\mathbf{w}_{\text{tx}}\bigr), \, \sigma^2_{\mathrm{RSRP}}\bigl(f_\vartheta(\mathbf{x}),\mathbf{w}_{\text{tx}}\bigr) \right\}.$$5 in normalized or encoded form, together with a binary indicator $$\mathrm{NBF}(\mathbf{x},\mathbf{w}_{\text{tx}}) \triangleq \left\{ \mu_{\mathrm{RSRP}}\bigl(f_\vartheta(\mathbf{x}),\mathbf{w}_{\text{tx}}\bigr), \, \sigma^2_{\mathrm{RSRP}}\bigl(f_\vartheta(\mathbf{x}),\mathbf{w}_{\text{tx}}\bigr) \right\}.$$6 that denotes whether the path exists at that location. Collecting all paths, the output is the predicted MCPP and existence mask (Guo et al., 9 Aug 2025).

The location input is encoded using random Fourier features into a 256-dimensional vector,

$$\mathrm{NBF}(\mathbf{x},\mathbf{w}_{\text{tx}}) \triangleq \left\{ \mu_{\mathrm{RSRP}}\bigl(f_\vartheta(\mathbf{x}),\mathbf{w}_{\text{tx}}\bigr), \, \sigma^2_{\mathrm{RSRP}}\bigl(f_\vartheta(\mathbf{x}),\mathbf{w}_{\text{tx}}\bigr) \right\}.$$7

which is described as similar to NeRF and ViT-style sinusoidal or Fourier positional embeddings. In addition, the model introduces $$\mathrm{NBF}(\mathbf{x},\mathbf{w}_{\text{tx}}) \triangleq \left\{ \mu_{\mathrm{RSRP}}\bigl(f_\vartheta(\mathbf{x}),\mathbf{w}_{\text{tx}}\bigr), \, \sigma^2_{\mathrm{RSRP}}\bigl(f_\vartheta(\mathbf{x}),\mathbf{w}_{\text{tx}}\bigr) \right\}.$$8 learnable target tokens,

$$\mathrm{NBF}(\mathbf{x},\mathbf{w}_{\text{tx}}) \triangleq \left\{ \mu_{\mathrm{RSRP}}\bigl(f_\vartheta(\mathbf{x}),\mathbf{w}_{\text{tx}}\bigr), \, \sigma^2_{\mathrm{RSRP}}\bigl(f_\vartheta(\mathbf{x}),\mathbf{w}_{\text{tx}}\bigr) \right\}.$$9

which serve as location-independent slots for potential paths. These tokens are concatenated with the UE token and passed through $f_\vartheta(\mathbf{x})$0 layers of a Transformer encoder containing Multi-Head Self-Attention, feed-forward MLP blocks, LayerNorm, and residual connections. The attention mechanism allows path tokens to condition jointly on the UE location and on one another, thereby modeling global correlations among paths such as consistent angular clusters and shared delays (Guo et al., 9 Aug 2025).

At the output, each path token is processed by two heads: a two-layer regression head that produces eight real-valued outputs for SCP parameters, and a one-layer classification head that predicts the existence indicator using binary cross-entropy. In pretraining, the path set is permutation invariant, so one-to-one matching between predicted and true paths is performed with the Hungarian algorithm before applying regression loss. If the ground-truth MCPP contains fewer than $f_\vartheta(\mathbf{x})$1 paths, the existence head handles absent paths; if it contains more than $f_\vartheta(\mathbf{x})$2 paths, representative paths are obtained through clustering, for example weighted K-means. The overall pretraining loss is a weighted sum of MAE for matched SCP features and BCE for path existence:

$f_\vartheta(\mathbf{x})$3

In end-to-end calibration with only RSRP labels, the loss is applied directly to predicted mean RSRP values and backpropagated through the analytical block into $f_\vartheta(\mathbf{x})$4 (Guo et al., 9 Aug 2025).

The use of a Transformer is motivated in the source by two properties: it naturally handles variable set-like structures corresponding to multipath components, and self-attention captures non-local spatial dependencies in MCPP structure. This suggests that the architectural choice is aligned with the path-set character of the latent representation rather than with direct beam prediction (Guo et al., 9 Aug 2025).

4. Physics-inspired white-box beam inference

Given MCPP and a beamforming vector $f_\vartheta(\mathbf{x})$5, the instantaneous normalized received signal is modeled as a sum over the set of significant paths. The normalized beam RSRP per unit transmit power is

$f_\vartheta(\mathbf{x})$6

with

$f_\vartheta(\mathbf{x})$7

where $f_\vartheta(\mathbf{x})$8 is the transmit antenna element radiation pattern and $f_\vartheta(\mathbf{x})$9 is the transmit array response (Guo et al., 9 Aug 2025).

The analytical module assumes a narrowband channel, far-field propagation for each path, and i.i.d. random phases $\mathbf{x}$0 uniformly distributed in $\mathbf{x}$1. Under these assumptions, $\mathbf{x}$2 is random because of small-scale fading, and Proposition 1 in the paper gives the mean and variance:

$\mathbf{x}$3

$\mathbf{x}$4

where

$\mathbf{x}$5

is the average power contribution of path $\mathbf{x}$6, and

$\mathbf{x}$7

is the array factor for that path (Guo et al., 9 Aug 2025).

For DFT beams on a uniform planar array (UPA), the beamforming vector is separable across the two panel dimensions, and the transmit array response is similarly separable. The squared magnitude of the array factor decomposes as

$\mathbf{x}$8

where the array factor sum is

$\mathbf{x}$9

The spatial frequencies $L$0 and $L$1 are determined by the path directions, carrier frequency, panel axes, and element spacings. The consequence is that, once path powers, directions, and delays are estimated through MCPP, beam-dependent RSRP statistics follow analytically without retraining a separate neural model for each beam configuration (Guo et al., 9 Aug 2025).

This analytical structure is also the basis of the framework’s interpretability. The individual terms $L$2 identify pathwise contributions to mean beam gain, and changes in beam, array size, panel orientation, or carrier frequency enter explicitly through $L$3, $L$4, and $L$5 rather than through an opaque latent mapping (Guo et al., 9 Aug 2025).

5. Pretrain-and-Calibrate strategy and training pipeline

The paper introduces a two-stage Pretrain-and-Calibrate (PaC) strategy to improve convergence and adaptation. In the pretraining stage, ray-tracing tools such as SionnaRT and standardized models such as QuaDRiGa are used to generate synthetic, site-specific MCPP labels on a spatial grid. SionnaRT provides deterministic multipath from city geometry, while QuaDRiGa adds random perturbations to approximate unknown environmental factors. The MCPP learner is then pretrained using the permutation-invariant loss described above so that predicted MCPP matches the synthetic labels. This stage is intended to place the model near a realistic MCPP manifold and mitigate poor local minima during later end-to-end training (Guo et al., 9 Aug 2025).

In the calibration stage, real RSRP measurements paired with UE positions and beams are collected from SSB or CSI-RS in the live network. The model computes MCPP from location, passes it through the white-box formulas to obtain mean beam RSRP, and optimizes an end-to-end loss on the measured RSRP, using smooth-$L$6 or MAE between predicted and measured mean RSRP in dB. According to the paper, this stage adapts the pretrained model to actual site conditions, including structures not included in the ray-tracing environment and other unmodeled factors, while using relatively sparse on-site measurements (Guo et al., 9 Aug 2025).

The overall training pipeline comprises four steps. First, synthetic ray-tracing outputs provide MCPP labels on a UE-position grid, and calibration data provide measured RSRP aggregated into stable statistics using a grid-based weighted scheme combining IDW and temporal decay. Second, UE coordinates are mapped to 256-dimensional random Fourier positional embeddings, while beam configuration is injected only through the white-box module. Third, labels differ by stage: full MCPP with per-path powers, directions, delays, and existence for pretraining, and scalar mean RSRP, with optionally variance, for calibration. Fourth, optimization uses Adam with OneCycleLR, with MCPP MAE plus BCE during pretraining and smooth-$L$7 loss on mean RSRP in dB during calibration (Guo et al., 9 Aug 2025).

The model’s data-efficiency claim follows from this decomposition. The neural network learns only multipath structure over space, whereas the mapping from multipath to beam RSRP is handled analytically. The paper reports that this yields faster convergence and lower validation errors than an MLP baseline on identical sparse datasets and preserves more detail in spatial RSRP maps. In the evaluated configuration, the NBF model size is 8.21 MB, the MLP baseline is 10.57 MB, and a CKM table baseline using IDW with MCPP prior requires 27.86 MB of storage. Inference consists of one forward pass through a mid-size Transformer followed by a white-box per-beam cost of $L$8, with the latter described as trivial relative to the Transformer pass (Guo et al., 9 Aug 2025).

6. Empirical evaluation and comparative performance

The reported evaluation uses a 3.5 GHz outdoor scenario with an $L$9 UPA, $l$0 paths, and a $l$1 area. Performance is measured primarily by MAE in mean RSRP, expressed in dB, together with model or table storage. The comparison includes end-to-end NBF without prior, an MLP black-box baseline, an IDW CKM baseline with MCPP prior, and NBF with PaC (Guo et al., 9 Aug 2025).

Method	MAE in mean RSRP	Storage
End-to-end NBF (no prior)	3.547 dB	8.21 MB
MLP black-box baseline	4.105 dB	10.57 MB
IDW CKM (with MCPP prior, table-based)	3.328 dB	27.86 MB
NBF with PaC (MCPP prior + calibration)	2.416 dB	8.21 MB

These results support several specific comparisons. Even without prior information, NBF outperforms the MLP baseline while using less storage. With PaC, NBF reduces MAE from 3.547 dB to 2.416 dB and surpasses both the MLP baseline and the table-based CKM baseline in accuracy, while remaining much more storage-efficient than CKM (Guo et al., 9 Aug 2025).

The paper further reports that training and validation loss curves show faster convergence and lower final loss for NBF than for the MLP baseline under the same data regime, and that PaC further improves convergence speed and validation performance. Qualitatively, spatial RSRP heatmaps for a representative beam show NBF tracking ground-truth RSRP with fine spatial details, while the MLP output is over-smoothed and misses local variations such as blockage edges, and IDW CKM remains limited by grid resolution and interpolation assumptions (Guo et al., 9 Aug 2025).

A common misunderstanding would be to interpret these outputs as instantaneous channel realizations. The framework instead predicts beam-RSRP statistics, specifically mean and variance, and the source explicitly states that it focuses on large-scale beam gains shaped by multipath geometry rather than on instantaneous small-scale fading samples (Guo et al., 9 Aug 2025).

7. Interpretability, assumptions, and possible extensions

NBF’s interpretability arises from both the latent representation and the inference mechanism. MCPP explicitly encodes path powers, directions, and delays, and the paper notes that the neural output can be visualized as angular power spectra versus location or as delay–angle clusters. The white-box equations then show exactly how these pathwise quantities map into beam RSRP through antenna patterns and array factors. This enables diagnosis of whether an error originates in the neural estimate of MCPP or in the physical beam and antenna parameterization. The paper also reports agreement between analytical mean and variance and Monte Carlo simulation under QuaDRiGa, supporting the claim that the white-box statistics capture small-scale fading behavior correctly under the stated assumptions (Guo et al., 9 Aug 2025).

The framework is also defined by several explicit assumptions and limits. It uses a narrowband channel model and considers a single carrier frequency rather than wideband frequency-selective behavior. It adopts a far-field geometric multipath model and does not explicitly model near-field or extreme large-array effects. It uses a fixed number of significant paths $l$2, with an existence head and clustering to mitigate variable path counts. PaC depends on a reasonably accurate virtual environment model for pretraining; the paper notes that very inaccurate priors could mislead pretraining, although calibration helps. The reported evaluations are confined to one representative scenario: 3.5 GHz, an $l$3 UPA, and a single site, so scaling to very large arrays, ultra-dense multi-cell deployments, or strong mobility and time variation remains to be studied systematically (Guo et al., 9 Aug 2025).

Several extensions are identified directly in the source. A wideband extension would model per-subcarrier or per-delay-tap MCPP, leading to frequency-dependent beam RSRP or full channel state information. The white-box module could be generalized to arbitrary beamforming matrices, including hybrid analog/digital beamforming or multi-layer precoding. MCPP together with analytical mapping could be used to produce full channel covariance or impulse responses rather than only RSRP. By learning temporal dynamics of MCPP, the framework could be extended to beam tracking. The MCPP concept could also be generalized to multi-BS paths for multi-cell coordination and interference-aware beam fields (Guo et al., 9 Aug 2025).