QueryFieldNet: A Query-Conditioned Surrogate Model for Multi-Beam Reflectarray Principal-Cut Gain Prediction

Abstract

This study proposes QueryFieldNet for predicting two orthogonal principal-cut wide-angle gain curves of fixed-$\theta$-grid multi-beam reflectarrays. The balanced seven-point $\theta$-grid dataset $\{-45°,-30°,-15°,0°,15°,30°,45°\}$ contains 19 × 19 phase maps, 1- to 5-bit phase quantization, and one to four intended beams. QueryFieldNet combines periodic phase inputs, multi-scale 2D residual encoding, cross-attention decoding with 181 Fourier-$\theta$ queries, intended-beam metadata supervision, and beam-basis residual refinement. Three proposed metrics, Multi-Beam HPBW RMSE, Multi-Beam Main-Lobe RMSE, and Beam-Peak RMSE, quantify the local $-3$ dB, $-10$ dB, and peak-gain errors around each intended beam. Across three runs, the final QueryFieldNet obtains bit-wise macro-averaged RMSE $3.240\pm 0.143$, MAE $1.975\pm 0.134$, $R^2$ $0.99792\pm 0.00017$, Multi-Beam HPBW RMSE $1.872\pm 0.015$, Multi-Beam Main-Lobe RMSE $2.134\pm 0.047$, and Beam-Peak RMSE $1.730\pm 0.065$. The aggregate local-window values combine both principal cuts; for $\varphi =90°$, they represent target-centered local errors rather than physical intended-beam HPBW or peak measurements. These results concern an array-factor-level synthetic benchmark; full-wave and measurement-based validation remain future work.

1. Introduction

Reflectarray antennas combine spatial feeding and planar array structures to realize high-gain, low-profile, and relatively simple antenna systems [1,2,3]. However, the radiation pattern of a reflectarray is determined by the unit-cell phase distribution over the aperture, feed position, phase quantization, and beam-steering conditions. Reflectarray design therefore becomes a high-dimensional and nonlinear optimization problem [4,5]. In multi-beam conditions, multiple intended beams must be formed simultaneously from a single aperture, and the peak gain, main-lobe shape, beamwidth, and sidelobe structure of each beam change together.

Recently, machine-learning-based surrogate models have been increasingly used in antenna design to replace or assist expensive electromagnetic analysis [6,7,8]. Such models are useful for design exploration and optimization because they can rapidly approximate antenna responses without repeated full-wave simulation. Attention mechanisms were popularized by the transformer architecture [9], while antenna-response prediction and parametric modeling studies have adopted neural architectures such as CNN-LSTM models [10,11]; ensemble-learning approaches have also been explored for antenna surrogate modeling [12]. However, the fixed-$\theta$-grid multi-beam reflectarray principal-cut 1D gain-curve prediction problem considered in this study differs from general antenna surrogate settings in terms of input format, output format, and evaluation criteria. Because a single gain curve may contain multiple intended beams, a single-beam metric centered on the global maximum may not fully reflect the local prediction quality of each beam.

PhaseNet-style phase-map-based gain prediction baselines have shown the usefulness of combining 2D phase maps with angular embeddings [13]. This study extends that setting from single-beam-oriented vector regression to fixed-grid multi-beam prediction by using query-conditioned angular field decoding, intended-beam metadata-aware residual refinement, and local-window evaluation metrics. Appendix A 

Table A1. Task-level comparison between the adapted PhaseNet baseline and QueryFieldNet.

Aspect	Adapted PhaseNet	QueryFieldNet
Input representation	Two-dimensional phase map with condition features	sin/cos phase channels, spatial coordinates, and bit/cut conditions
Output definition	Direct regression of a 181-point curve	181 angle-conditioned query outputs
Angular decoding	Fixed output-vector decoder	Fourier-angle queries, cross-attention, and dilated 1D convolution
Intended-beam metadata	Not used	Auxiliary occupancy/count head and metadata-gated residual branch
Training objective	Full-curve MSE	Composite full-curve, local-region, shape, peak, and metadata losses
Evaluation	Global and local-window metrics under the common protocol	Same output and metric definitions

summarizes the task-level differences between the adapted PhaseNet baseline and QueryFieldNet. Nevertheless, prior studies that directly match the input, output, and evaluation setting of this study remain limited. Therefore, data generation, model architecture, and evaluation metrics need to be adapted for multi-beam reflectarray modeling.

To address these issues, this paper proposes QueryFieldNet. QueryFieldNet jointly encodes the spatial structure and phase periodicity of a 2D phase map and formulates gain prediction as angular field prediction using 181 angular query tokens rather than fixed-length vector regression. In addition, an intended-beam metadata head and a beam-basis residual branch are used to represent design-target occupancy, beam count, and local residual structure, with the aim of improving prediction accuracy around multi-beam peaks and main-lobe regions.

The novelty claimed here is not the use of attention or Fourier features in isolation. It is the task-specific coupling of three elements: output angles act as explicit queries that retrieve spatial phase-map features; intended-beam occupancy and count supervise a beam-aware representation without being supplied as inference inputs; and prediction quality is measured separately around every intended beam rather than only around the global maximum. Generic attention-based surrogates typically map a global latent representation to a fixed response vector, whereas QueryFieldNet preserves the correspondence between each angular coordinate, the aperture tokens it attends to, and the intended-beam-local evaluation regions.

The main contributions of this paper are as follows.

• A fixed-$\theta$-grid multi-beam reflectarray dataset is constructed for phase quantization levels from 1-bit to 5-bit. For each bit depth, 2000 phase maps and two orthogonal principal-cut wide-angle gain curves are generated.
• Multi-Beam HPBW RMSE, Multi-Beam Main-Lobe RMSE, and Beam-Peak RMSE are defined to address the limitation that conventional HPBW RMSE and Main-Lobe RMSE are primarily suited to single-beam evaluation.
• QueryFieldNet is proposed to predict wide-angle gain fields by combining 2D phase maps and angular queries. The proposed model incorporates local multi-beam structures through an intended-beam metadata head and a beam-basis residual branch.
• QueryFieldNet is compared with adapted baselines reconstructed from published machine-learning and antenna surrogate modeling approaches for the phase-map-based 181-point gain-curve prediction setting. Detailed bit-depth, beam-count, and $\theta$-position analyses are provided under fixed-$\theta$-grid multi-beam conditions.

The remainder of this paper is organized as follows. Section 2 describes the dataset generation process, proposed evaluation metrics, and QueryFieldNet architecture. Section 3 presents the experimental settings, quantitative performance comparison, and qualitative visualization results. Section 4 discusses the experimental results and the role of each model component. Section 5 concludes the paper.

2. Method

2.1. Problem Formulation

The objective of this study is to predict two orthogonal principal-cut wide-angle gain curves from 2D phase maps of reflectarray antennas. Let the input phase map be $P\in {\mathbb{R}}^{H\times W}$. In this study, the phase map size is $H=W=19$. The model receives the phase map, bit depth, cut-plane condition, and angular query as inputs and predicts the gain field as follows:

$$\widehat{G}=f(P,b,\varphi ,\theta ),$$

(1)

where b is the phase quantization bit depth, $\varphi$ is the principal-cut condition, and $\theta$ is the output angular coordinate. The final output is a 181-point gain curve from −90° to 90° sampled at 1° intervals. In this study, the gain curves are treated as two orthogonal principal-cut responses corresponding to the $\varphi =0°$ and $\varphi =90°$ cuts of the same generated phase map. All intended beams are steered in the $\varphi =0°$ azimuth plane, while both principal-cut planes are retained in the output dataset.

Although the entire gain curve can be directly regressed as a single vector, this study formulates gain prediction as an angular field prediction problem. Each $\theta$ location is represented as a query token that attends to phase-map features and predicts the gain at that angle. This formulation is related to coordinate-based implicit neural representations, such as NeRF-style coordinate networks and local implicit image functions [14,15], as well as query-based decoders such as DETR and Perceiver IO [16,17]. Unlike these general formulations, QueryFieldNet uses antenna-specific conditioning through Fourier-$\theta$ queries and bit-depth/cut-plane embeddings to predict fixed-grid principal-cut gain fields from phase maps.

2.2. Multi-Beam Reflectarray Dataset Generation

This study generates a balanced fixed-$\theta$-grid dataset for gain prediction of multi-beam reflectarray antennas. The reflectarray is represented by a square 19 × 19 phase-map grid with a 20.1 mm unit-cell pitch at 8 GHz, and the feed is fixed at (0, 0, 250) mm. The 19 cell centers span 361.8 mm, whereas including one full pitch gives a 381.9 mm grid extent. A continuous square aperture mask with side length 370 mm is used in the MRA forward model and in the efficiency calculation; thus, 370 mm denotes the mask dimension rather than the 19-pitch grid extent, and all 19 × 19 cell centers are retained in the aperture summation.

The dataset includes bit depths from 1-bit to 5-bit. For each bit depth, 2000 phase maps are generated, and each phase map has one to four intended beams. The beam-count distribution is balanced by construction, with 500 phase maps for each beam count at every bit depth. Beam $\theta$ values are selected from the seven-point steering grid $\{-45°,-30°,-15°,0°,15°,30°,45°\}$, and all intended beams are steered in the $\varphi =0°$ azimuth plane. Compared with the earlier five-point grid that included $\theta =\pm 90°$, the present dataset excludes grazing-direction intended beams and instead adds $\pm 15°$ and $\pm 30°$ to provide a denser steering range within $\pm 45°$. For each phase map, two principal-cut gain curves are generated for $\varphi =0°$ and $\varphi =90°$. Each output gain curve consists of 181 values sampled from $-90°$ to 90° at 1° intervals. Therefore, each bit-depth CSV file contains 4000 rows.

Data generation is performed by calling a multi-beam reflectarray analysis (MRA) calculation workflow through a Python (version 3.10) wrapper. Following standard reflectarray aperture-summation and directivity formulations [2,3,5], the quantized phase value of each unit cell is combined with the feed-to-cell propagation term and the observation-direction phase term to form an array-factor-level far-field response. The equations below express those standard relations in the notation and discretization of the present implementation rather than reproducing a formula verbatim from one source. To avoid confusion with the signed output cut angle $\theta \in [-90°,90°]$, $\vartheta$ denotes the polar observation angle used in the MRA normalization and radiation-power calculation:

$$F(\vartheta ,\varphi )={\displaystyle \sum _{u=1}^{19}}{\displaystyle \sum _{v=1}^{19}}{a}_{u,v}\left(\vartheta \right)\exp \left\{j\left(P_{u,v}^{\left(q\right)}+{\psi }_{u,v}^{\mathrm{feed}}+{\psi }_{u,v}^{\mathrm{obs}}(\vartheta ,\varphi )\right)\right\},$$

(2)

Here, the signed cut coordinate is connected to the polar-grid notation by the implementation convention $\theta \ge 0:(\vartheta ,\varphi )=\left(\right|\theta |,{\varphi }_{\mathrm{cut}})$ and $\theta <0:(\vartheta ,\varphi )=\left(\right|\theta |,{\varphi }_{\mathrm{cut}}+180°)$. Thus, the 181-point signed cut curve is obtained from opposite azimuthal directions of the nonnegative polar observation grid. Here, u and v index the aperture cells, $(x_u,y_v)$ are their planar coordinates, $P_{u,v}^{\left(q\right)}$ is the quantized phase assigned to the $(u,v)$ cell, $r_{u,v}$ is the feed-to-cell distance, and $k_0=2\pi /\lambda$ is the free-space wavenumber. The feed and observation phase terms are ${\psi }_{u,v}^{\mathrm{feed}}=-k_0{r}_{u,v}$ and ${\psi }_{u,v}^{\mathrm{obs}}=k_0[x_u\sin \vartheta \cos \varphi +y_v\sin \vartheta \sin \varphi ]$, respectively. For cells inside the 370 mm square aperture mask, the implemented amplitude factor is

$$a_{u,v}\left(\vartheta \right)={\displaystyle \frac{{\cos }^{Q_f}\left(\vartheta \right){\cos }^{2Q}\left(\vartheta \right)}{\max (r_{u,v},{10}^{-12})}},Q=3,Q_f=7.$$

(3)

Here, Q is the aperture element-pattern exponent controlling ${\cos }^{2Q}\left(\vartheta \right)$, whereas $Q_f$ is the feed-pattern exponent controlling ${\cos }^{Q_f}\left(\vartheta \right)$.

The feed-pattern term in the implemented amplitude factor is evaluated at the observation angle as an implementation-level simplification. A more detailed reflectarray model would instead evaluate feed taper using a cell-specific feed-to-cell angle ${\vartheta }_f(u,v)$. No separately measured element pattern or full-wave unit-cell response is applied. The magnitude is normalized over the sampled two-dimensional observation grid as

$$\tilde{F}(\vartheta ,\varphi )={\displaystyle \frac{\left|F\right(\vartheta ,\varphi \left)\right|}{\max \left({\max }_{\vartheta ,\varphi }\left|F(\vartheta ,\varphi )\right|,{10}^{-12}\right)}}.$$

(4)

The normalized response is converted to directivity and then to the stored dB response by

$$\begin{array}{cc}\hfill P_{\mathrm{rad}}& =\sum _{\vartheta ,\varphi }{\tilde{F}}^2(\vartheta ,\varphi )\sin \vartheta \Delta \vartheta \Delta \varphi ,\hfill \\ \hfill D(\vartheta ,\varphi )& ={\displaystyle \frac{4\pi {\tilde{F}}^2(\vartheta ,\varphi )}{P_{\mathrm{rad}}+{10}^{-20}}},\hfill \\ \hfill G(\vartheta ,\varphi )& =10{\log }_{10}\left({\eta }_t{\eta }_{s}D(\vartheta ,\varphi )+{10}^{-30}\right),\hfill \end{array}$$

(5)

where $\Delta \vartheta$ and $\Delta \varphi$ are the angular sampling intervals, $P_{\mathrm{rad}}$ is the sampled radiation integral, D is the directivity proxy, and ${\eta }_t$ and ${\eta }_s$ are the taper and spillover efficiency factors. With ${\theta }_e={\tan }^{-1}[(\sqrt{2}L/2)/z_f]$, $L=370$ mm, and $z_f=250$ mm, the implementation uses

$$\begin{array}{cc}\hfill {\eta }_t& ={\displaystyle \frac{{\left[\frac{1-{\cos }^{Q_f+1}{\theta }_e}{Q_f+1}+\frac{1-{\cos }^{Q_f}{\theta }_e}{Q_f}\right]}^2}{2{\tan }^2{\theta }_e\left(\frac{1-{\cos }^{2Q_f+1}{\theta }_e}{2Q_f+1}\right)}},\hfill \\ \hfill {\eta }_s& =1-{\cos }^{2Q_f+1}{\theta }_e.\hfill \end{array}$$

(6)

Thus, the stored curves are array-factor-level gain proxies in dB rather than independently validated absolute dBi gains. All learning and evaluation are performed on these stored responses.

For each bit depth, the 2000 phase maps are partitioned at the phase-map level into 1600 training, 200 validation, and 200 test samples, corresponding to an 80/10/10 ratio. Because each phase map provides paired $\varphi =0°$ and $\varphi =90°$ curves, the corresponding CSV row counts are 3200, 400, and 400. Both rows originating from the same phase map remain in one partition, and the generated files are organized by bit depth and cut-plane condition.

Representative phase-map examples and their corresponding 3D radiation patterns are shown in Figure 1 and Figure 2, respectively.

In Figure 1, panels (a)–(d) show how the quantized aperture phase pattern becomes progressively more superposed as the number of intended directions increases. In Figure 2, the corresponding panels show one to four design-target directions and the resulting directional response. Quantization, constructive/destructive interference, and overlap between neighboring lobes mean that the number of visually resolved maxima need not equal the intended-beam count. Together, the two figures illustrate why beam count cannot be inferred from a single global peak and why both spatial phase features and intended-beam-local evaluation are needed.

Table 1. Dataset configuration for the balanced theta7 experiments.

Item	Configuration
Aperture/frequency	Square reflectarray phase-map grid, 8 GHz
Forward-model aperture mask	Continuous square mask with side length $L=370$ mm; all 19 × 19 cell centers are retained in the aperture summation
Cell/feed	20.1 mm unit-cell pitch, feed = (0, 0, 250) mm
Phase map/bit depth	19 × 19 phase map, 1-bit to 5-bit
Beam condition	1 to 4 beams, $\theta \in \{-45°,-30°,-15°,0°,15°,30°,45°\}$, intended-beam azimuth $\varphi =0°$
Balanced sampling	500 phase maps per beam count for each bit depth; $\theta$ positions sampled as evenly as possible
Output condition	principal cuts $\varphi =0°$ and $\varphi =90°$, output $\theta =-90°$ to $90°$, 181 points
Dataset size	2000 phase maps per bit, 4000 CSV rows per bit
Split	80/10/10 at the phase-map level: 1600/200/200 phase maps (3200/400/400 cut rows)

summarizes the main dataset generation conditions.

2.3. Algorithmic Description of Dataset Generation

Instead of using a fixed phase-level set for each bit depth, this process generates a different combination of $2^b$ phase states for each sample. This increases phase quantization diversity within the same bit depth. The LUT generator samples distinct phase states and rejects duplicated sorted phase-state sets, including the 1-bit case; therefore, a sample is not assigned two identical phase states. No additional uniform-spacing constraint is imposed after this distinct-state check.

The beam steering angles are selected from a fixed seven-point grid to enable stable beam-position analysis while avoiding grazing-direction intended beams. All intended beams are steered in the $\varphi =0°$ azimuth plane, and output curves are generated for the $\varphi =0°$ and $\varphi =90°$ principal cuts.

Algorithms 1–3 form one sequential workflow. Algorithm 1 supplies a sample-specific phase-level LUT for the selected bit depth, Algorithm 2 supplies a beam-count-balanced set of intended directions, and Algorithm 3 combines both outputs to synthesize the quantized phase map and its paired $\varphi =0°$ and $\varphi =90°$ principal-cut curves. Each generated phase map is assigned to one train, validation, or test partition before the two cuts are expanded into separate rows. During model development, the phase map and cut condition form the prediction input, while the stored beam count and $\theta$ list provide auxiliary metadata targets and the masks required by the proposed multi-beam metrics.

These generated inputs and metadata are subsequently consumed by the QueryFieldNet architecture shown in Figure 3.

Algorithm 1 Sample-Specific b-bit Phase-Level LUT Generation

Require: bit depth b, target sample count N, LUT template Ensure: N sample-specific b-bit phase-level lookup tables   1: $m\leftarrow 2^b$   2: $U\leftarrow \varnothing$   3: while  $\left|U\right|<N$  do   4:       $z\leftarrow$ sample m distinct phase values from $[-180°,180°]$   5:       $z\leftarrow \mathrm{sort}\left(z\right)$   6:       if $z\in U$ then   7:            continue   8:       end if   9:       $U\leftarrow U\cup \left\{z\right\}$ 10:       generate LUT by pairing template dimensions with z 11:       save phase-level metadata 12: end while 13: return  U

Algorithm 2 Balanced Fixed-Grid Multi-Beam Target Sampling

Require: theta grid $\mathsf{\Theta }=\{-45°,-30°,-15°,0°,15°,30°,45°\}$, beam-count range $\{1,2,3,4\}$ Ensure: multi-beam target list B   1: Select $K\in \{1,2,3,4\}$ according to the balanced beam-count quota.   2: $B\leftarrow \varnothing$   3: while  $\left|B\right|<K$  do   4:       $\theta \leftarrow$ sample one angle from $\mathsf{\Theta }$ using the theta-balancing sampler.   5:       $\varphi \leftarrow 0°$   6:       if $\theta$ is not already selected then   7:             $B\leftarrow B\cup \left\{\right(\theta ,\varphi \left)\right\}$   8:       end if   9: end while 10: return  B

Algorithm 3 Multi-Beam Phase Map and Gain Curve Generation

Require: bit depth b, phase-level LUT, beam target list B, MRA array-factor-level reflectarray simulator Ensure: phase maps, principal-cut gain curves, and sample metadata       Fixed parameters:           frequency = 8 GHz; square reflectarray phase-map grid; cell pitch = 20.1 mm           aperture-mask side length $L=370$ mm; all $19\times 19$ cell centers retained           feed position $=(0,0,250)$ mm; phase-map size $=19\times 19$           output $\theta =-90°$ to $90$; output cut $\varphi \in \{0°,90°\}$ 1: for each sample do 2:       Generate continuous phase distribution for target beams. 3:       Quantize phase distribution using the sample-specific b-bit LUT. 4:       Save the $19\times 19$ phase map. 5:       Simulate the gain curve at $\varphi =0°$. 6:       Simulate the gain curve at $\varphi =90°$. 7:       Save ${\mathrm{gain}}_{-90},\dots ,{\mathrm{gain}}_{90}$. 8:       Save metadata: beam count, beam $\theta$ list, phase states, and peak gain. 9: end for

2.4. HPBW, Main-Lobe, and Proposed Multi-Beam Metrics

In antenna radiation patterns, a key region is generally the neighborhood of the main beam where the maximum gain occurs. The half-power beamwidth (HPBW) is the angular width between two points where the power drops to half of the peak value. In the dB scale, half power corresponds to $-10{\log }_{10}2\approx -3.01$ dB relative to the peak gain, which is commonly approximated as −3 dB. Therefore, HPBW is usually defined by the points at which the gain decreases by 3 dB from the peak. The main-lobe region is used to evaluate the principal radiation region more broadly than the HPBW; in this study, it is defined as the region within −10 dB of the local peak.

For single-beam evaluation, HPBW and main-lobe regions can be defined around the global maximum of the entire radiation pattern. However, a multi-beam reflectarray may contain multiple intended beams in a single gain curve. Using only the global maximum can therefore fail to reflect the prediction quality of the remaining beams. In this study, a local peak and its −3 dB and −10 dB regions are first computed for each intended beam. The per-beam regions are then merged into a sample-level union mask so that overlapping angular samples are counted only once.

Let the test sample index be $n=1,\dots ,N$, and let $K_n$ be the number of intended beams in sample n. The ground-truth and predicted gain curves are denoted by $G_n\left({\theta }_i\right)$ and ${\widehat{G}}_n\left({\theta }_i\right)$, respectively, where ${\theta }_i\in \{-90°,-89°,\dots ,90°\}$.

The standard global metrics are computed as follows.

$$\mathrm{RMSE}=\sqrt{{\displaystyle \frac{1}{NT}}{\displaystyle \sum _{n=1}^N}{\displaystyle \sum _{i=1}^T}{\left(G_n\left({\theta }_i\right)-{\widehat{G}}_n\left({\theta }_i\right)\right)}^2}.$$

(7)

$$\mathrm{MAE}={\displaystyle \frac{1}{NT}}{\displaystyle \sum _{n=1}^N}{\displaystyle \sum _{i=1}^T}\left|G_n\left({\theta }_i\right)-{\widehat{G}}_n\left({\theta }_i\right)\right|.$$

(8)

$$\begin{array}{cc}\hfill R^2& =1-{\displaystyle \frac{{\sum }_{n=1}^N{\sum }_{i=1}^T{\left(G_n\left({\theta }_i\right)-{\widehat{G}}_n\left({\theta }_i\right)\right)}^2}{{\sum }_{n=1}^N{\sum }_{i=1}^T{\left(G_n\left({\theta }_i\right)-\overline{G}\right)}^2}}.\hfill \end{array}$$

(9)

Here, $T=181$ and $\overline{G}$ is the mean gain value over the entire test set.

2.4.1. Multi-Beam HPBW RMSE

For the kth intended beam of sample n, let the target beam center be ${\theta }_{n,k}^{\left(0\right)}$. This study fixes the local search window as $W\left({\theta }_{n,k}^{\left(0\right)}\right)=\{\theta \mid |\theta -{\theta }_{n,k}^{\left(0\right)}|\le 12°\}$. The $\pm 12°$ window is chosen to cover the local peak and its neighboring main-lobe region for the theta7 steering range, including broadened lobes near $\pm 45°$, while reducing contamination from distant sidelobe regions. The same window is applied to all models and all bit depths. Because the intended beam centers are within $[-45°,45°]$, these windows do not exceed the $[-90°,90°]$ output range. The ground-truth local peak location and peak gain in this local window are defined as follows.

$${\theta }_{n,k}^{\left(p\right)}=\arg \underset{{\theta }_i\in W\left({\theta }_{n,k}^{\left(0\right)}\right)}{\max }{G}_n\left({\theta }_i\right).$$

(10)

$$p_{n,k}=G_n\left({\theta }_{n,k}^{\left(p\right)}\right).$$

(11)

The HPBW region of the kth beam is defined as the angular set within −3 dB of the local peak:

$$H_{n,k}=\{{\theta }_i\mid {\theta }_i\in W\left({\theta }_{n,k}^{\left(0\right)}\right),G_n\left({\theta }_i\right)\ge p_{n,k}-3\}.$$

(12)

For samples with multiple nearby intended beams, the evaluated high-gain mask is the union of the per-beam sets,

$$H_n^{\cup }={\displaystyle \bigcup _{k=1}^{K_n}}{H}_{n,k}.$$

(13)

The Multi-Beam HPBW RMSE over all samples is computed as follows.

$$\begin{array}{cc}\hfill \mathrm{MB}-\mathrm{HPBW}\mathrm{RMSE}& =\sqrt{{\displaystyle \frac{A_{\mathrm{HPBW}}}{B_{\mathrm{HPBW}}}}},\hfill \\ \hfill A_{\mathrm{HPBW}}& ={\displaystyle \sum _{n=1}^N}\sum _{{\theta }_i\in H_n^{\cup }}{\left(G_n\left({\theta }_i\right)-{\widehat{G}}_n\left({\theta }_i\right)\right)}^2,\hfill \\ \hfill B_{\mathrm{HPBW}}& ={\displaystyle \sum _{n=1}^N}\left|H_n^{\cup }\right|.\hfill \end{array}$$

(14)

This metric evaluates prediction error in the core high-gain region of each intended beam, namely the −3 dB HPBW region.

2.4.2. Multi-Beam Main-Lobe RMSE

The main-lobe region is defined as the region within −10 dB of the local peak:

$$M_{n,k}=\{{\theta }_i\mid {\theta }_i\in W\left({\theta }_{n,k}^{\left(0\right)}\right),G_n\left({\theta }_i\right)\ge p_{n,k}-10\}.$$

(15)

The union mask for sample n is

$$M_n^{\cup }={\displaystyle \bigcup _{k=1}^{K_n}}{M}_{n,k}.$$

(16)

The Multi-Beam Main-Lobe RMSE over all samples is computed as follows.

$$\begin{array}{cc}\hfill \mathrm{MB}-\mathrm{Main}-\mathrm{Lobe}\mathrm{RMSE}& =\sqrt{{\displaystyle \frac{A_{\mathrm{ML}}}{B_{\mathrm{ML}}}}},\hfill \\ \hfill A_{\mathrm{ML}}& ={\displaystyle \sum _{n=1}^N}\sum _{{\theta }_i\in M_n^{\cup }}{\left(G_n\left({\theta }_i\right)-{\widehat{G}}_n\left({\theta }_i\right)\right)}^2,\hfill \\ \hfill B_{\mathrm{ML}}& ={\displaystyle \sum _{n=1}^N}\left|M_n^{\cup }\right|.\hfill \end{array}$$

(17)

Because this metric evaluates a wider main-lobe shape than the HPBW region, it reflects prediction quality not only near the peak but also along the lobe skirt and neighboring radiation region.

2.4.3. Beam-Peak RMSE

Beam-Peak RMSE directly evaluates the difference between the ground-truth peak gain and predicted peak gain around each intended beam. The predicted peak gain is computed within the same local search window as the ground-truth beam:

$${\widehat{p}}_{n,k}=\underset{{\theta }_i\in W\left({\theta }_{n,k}^{\left(0\right)}\right)}{\max }{\widehat{G}}_n\left({\theta }_i\right).$$

(18)

$$\mathrm{Beam}-\mathrm{Peak}\mathrm{RMSE}=\sqrt{{\displaystyle \frac{{\sum }_{n=1}^N{\sum }_{k=1}^{K_n}{(p_{n,k}-{\widehat{p}}_{n,k})}^2}{{\sum }_{n=1}^N{K}_n}}}.$$

(19)

Thus, Multi-Beam HPBW RMSE evaluates gain error in the union of the local $-3$ dB regions around intended beam centers; Multi-Beam Main-Lobe RMSE evaluates gain error in the union of the local $-10$ dB regions; and Beam-Peak RMSE evaluates peak-gain accuracy for each intended-beam window. The reported implementation follows this union-mask definition, so overlapping angular samples are counted once in the region-based metrics. Beam-Peak RMSE is still computed independently for each intended beam, so closely spaced or merged peaks can be assigned to more than one intended-beam window; peak-position matching and unresolved-beam statistics are left for future work.

All reported global metrics are computed over both $\varphi =0°$ and $\varphi =90°$ cut rows. The physical intended-beam interpretation of MB-HPBW RMSE, MB-Main-Lobe RMSE, and Beam-Peak RMSE is tied to the $\varphi =0°$ cut, where the design targets are defined. In the current aggregate result files, the same local-window computation is also applied to the orthogonal $\varphi =90°$ cut rows; those values should therefore be interpreted only as target-centered local-window errors, not as physical HPBW or beam-peak measurements for that orthogonal cut.

2.5. QueryFieldNet Architecture

QueryFieldNet is a query-conditioned neural surrogate model that predicts wide-angle gain curves from 2D phase maps. The overall architecture consists of phase input encoding, a periodic 2D encoder, a query and conditioning module, a query field decoder, an intended-beam metadata head, and a beam-basis residual branch. Figure 3 shows the overall architecture and the flow from phase-map encoding to angular query decoding.

2.5.1. Periodic Phase Input Encoding

The input phase map is stored in degrees and first converted to radians, $P^{\left(\mathrm{rad}\right)}=\pi P^{\left(\mathrm{deg}\right)}/180$. It is then transformed into $\sin \left(P^{\left(\mathrm{rad}\right)}\right)$ and $\cos \left(P^{\left(\mathrm{rad}\right)}\right)$ to reflect phase periodicity. This alleviates the discontinuity between −180° and 180°. Normalized spatial coordinates x and y are linearly scaled to $[-1,1]$ and then added to construct a 4-channel input:

$$X=[\sin \left(P^{\left(\mathrm{rad}\right)}\right),\cos \left(P^{\left(\mathrm{rad}\right)}\right),x,y].$$

(20)

2.5.2. Periodic Multi-Scale 2D Encoder

The periodic 2D encoder is composed of a multi-scale residual CNN structure. The implementation uses a 4-to-64 stem, 64-channel residual blocks, stride-2 downsampling to 128 channels, and a second stride-2 downsampling stage to 256 channels. Mean-pooled features from the three scales are concatenated and projected to a 256-dimensional global context vector. The final spatial feature map is flattened into spatial tokens, which are used as keys and values in the cross-attention module of the query field decoder. The global context is used by the intended-beam metadata head and beam-basis residual branch.

2.5.3. Query and Conditioning Module

The query and conditioning module combines bit embedding, cut-$\varphi$ embedding, and Fourier-$\theta$ query to generate 181 angular query tokens. Let ${\theta }_{\mathrm{deg}}\in [-90,90]$ denote the query angle in degrees and let ${\theta }_{\mathrm{rad}}=\pi {\theta }_{\mathrm{deg}}/180$. The Fourier-angle query is

$$q_{\theta }=\left[{\theta }_{\mathrm{deg}}/90,\sin {\theta }_{\mathrm{rad}},\cos {\theta }_{\mathrm{rad}},\sin \left(2{\theta }_{\mathrm{rad}}\right),\cos \left(2{\theta }_{\mathrm{rad}}\right),\sin \left(4{\theta }_{\mathrm{rad}}\right),\cos \left(4{\theta }_{\mathrm{rad}}\right)\right].$$

(21)

The bit and cut-$\varphi$ embeddings distinguish phase-quantization and principal-cut conditions. Bit-aware FiLM modulates the global context as $c^{\prime }=c\odot (1+0.1\tanh {\gamma }_b)+0.1{\beta }_b$, where $({\gamma }_b,{\beta }_b)$ is generated from the bit embedding.

2.5.4. Query Field Decoder

The query field decoder uses cross-attention between angular query tokens and phase-map spatial tokens. The angular token sequence provides the query matrix Q, and the encoder spatial tokens provide the key and value matrices K and V. The decoder uses multi-head attention with eight heads, followed by a feed-forward block. A dilated 1D convolution stack with dilation factors 1, 2, 4, and 8 is then applied to refine adjacent gain responses along the angular direction. This branch predicts the base component of the final gain curve:

$${\widehat{G}}_{\mathrm{base}}\left(\theta \right)=D_{\mathrm{query}}(Q_{\theta },T_{\mathrm{spatial}}).$$

(22)

2.5.5. Intended-Beam Metadata Head and Beam-Basis Residual

The intended-beam metadata head predicts intended beam occupancy and intended beam count, not the number of resolved local maxima in the realized gain curve. Beam occupancy indicates whether a design target exists at each position in the fixed $\theta$ grid $\{-45°,-30°,-15°,0°,15°,30°,15°\}$, and beam count indicates the number of intended beams from one to four. During training, this branch uses design metadata as auxiliary supervision through binary occupancy loss and count classification loss to encourage beam-aware representations.

The beam-basis residual branch uses the predicted occupancy probabilities and encoder features to generate residual gain around fixed intended-beam centers. For each grid center, the branch predicts an amplitude, width, and skew parameter for a metadata-gated Gaussian-like basis function; the residual is the sum of these basis functions weighted by the predicted occupancy probabilities. During inference, the residual branch uses only predicted occupancy probabilities. During training, metadata teacher forcing can temporarily replace predicted occupancy with ground-truth metadata for a subset of batches to stabilize early training. After the query field decoder predicts the overall base gain curve, the beam-basis residual branch refines local peak and main-lobe shapes around fixed $\theta$ centers. The final output is computed as the sum of the base gain and residual gain:

$$\widehat{G}\left(\theta \right)={\widehat{G}}_{\mathrm{base}}\left(\theta \right)+{\widehat{G}}_{\mathrm{residual}}\left(\theta \right).$$

(23)

2.6. QueryFieldNet Training Objective

QueryFieldNet is trained using both the full gain-curve prediction loss and beam-aware auxiliary losses. The gain curve is z-score normalized using the angle-wise mean and standard deviation computed from the train split, and all reported metrics are computed in the dB domain after inverse transformation.

The base gain prediction loss is Smooth L1 loss on the normalized gain curve. This loss is combined with a dB-domain global loss, high-gain HPBW-region loss, high-gain main-lobe-region loss, intended-beam-region loss, peak loss, derivative shape loss, and intended-beam metadata auxiliary loss. The total loss is defined as follows:

$$\begin{array}{cc}\hfill L=& L_{\mathrm{gain}}+{\lambda }_{\mathrm{global}}{L}_{\mathrm{global}}+{\lambda }_{\mathrm{hpbw}}{L}_{\mathrm{hpbw}}+{\lambda }_{\mathrm{lobe}}{L}_{\mathrm{lobe}}\hfill \\ \hfill & +{\lambda }_{\mathrm{beam}}{L}_{\mathrm{beam}-\mathrm{region}}+{\lambda }_{\mathrm{peak}}{L}_{\mathrm{peak}}+{\lambda }_{\mathrm{shape}}{L}_{\mathrm{shape}}+{\lambda }_{\mathrm{metadata}}{L}_{\mathrm{metadata}}.\hfill \end{array}$$

(24)

Let $Z_n\left(\theta \right)$ and ${\widehat{Z}}_n\left(\theta \right)$ denote the normalized target and prediction, and let $G_n\left(\theta \right)$ and ${\widehat{G}}_n\left(\theta \right)$ denote the corresponding dB-domain curves after inverse normalization. The individual loss terms used in implementation are

$$\begin{array}{cc}\hfill L_{\mathrm{gain}}& =\mathrm{SmoothL}1(\widehat{Z},Z),\hfill \\ \hfill L_{\mathrm{global}}& ={\displaystyle \frac{1}{100NT}}\sum _{n,i}{\left({\widehat{G}}_n\left({\theta }_i\right)-G_n\left({\theta }_i\right)\right)}^2,\hfill \\ \hfill L_{\mathrm{hpbw}}& ={\displaystyle \frac{1}{|H^g|}}\sum _{(n,i)\in H^g}{\left({\widehat{Z}}_n\left({\theta }_i\right)-Z_n\left({\theta }_i\right)\right)}^2,\hfill \\ \hfill L_{\mathrm{lobe}}& ={\displaystyle \frac{1}{|M^g|}}\sum _{(n,i)\in M^g}{\left({\widehat{Z}}_n\left({\theta }_i\right)-Z_n\left({\theta }_i\right)\right)}^2,\hfill \\ \hfill L_{\mathrm{beam}-\mathrm{region}}& ={\displaystyle \frac{1}{\left|B\right|}}\sum _{(n,i)\in B}{\left({\widehat{Z}}_n\left({\theta }_i\right)-Z_n\left({\theta }_i\right)\right)}^2,\hfill \\ \hfill L_{\mathrm{peak}}& ={\displaystyle \frac{1}{100N}}\sum _n{\left(\underset{i}{\max }{\widehat{G}}_n\left({\theta }_i\right)-\underset{i}{\max }{G}_n\left({\theta }_i\right)\right)}^2,\hfill \\ \hfill L_{\mathrm{shape}}& =\mathrm{SmoothL}1({\Delta }_{\theta }\widehat{Z},{\Delta }_{\theta }Z),\hfill \\ \hfill L_{\mathrm{metadata}}& =\mathrm{BCE}(\widehat{o},o)+\mathrm{CE}(\widehat{c},c).\hfill \end{array}$$

(25)

Here, $H^g$ and $M^g$ are global high-gain masks defined by the $-3$ dB and $-10$ dB regions around the maximum of each training curve, B is the union of training-only intended-beam regions computed from design beam-center metadata with a 14° radius, ${\Delta }_{\theta }$ is the first-order angular finite difference, $\widehat{o}$ and o are predicted and target intended-beam occupancy vectors, and $\widehat{c}$ and c are predicted and target beam-count classes. The evaluation metrics in Section 2.4 use intended-beam local-window masks, while the training objective combines global high-gain emphasis and intended-beam-region emphasis. The 14° training radius is slightly wider than the $\pm 12°$ evaluation window to provide a soft optimization margin around intended-beam neighborhoods.

3. Experimental Results

3.1. Experimental Settings

The final QueryFieldNet is evaluated with five baseline models configured for the present 181-point prediction task: TabNet [18], SKi-Le-BNN [19], PhaseNet [13], 1D GP-HetCNN [20], and BOBE [12]. All methods use the balanced theta7 source dataset, the same 181-point output definition, and the same evaluation code. Every experiment is repeated with seeds 42, 43, and 44, and the reported results are mean ± standard deviation.

QueryFieldNet and QueryFieldNet-Lite are each trained as one joint model on the pooled 1-bit to 5-bit training split, whereas each baseline is trained independently for each bit depth. For the joint models, angle-wise z-score statistics are computed from the pooled training split, and per-bit results are obtained from the corresponding test subsets. Across the five bit depths, the pooled QueryFieldNet data contain 8000/1000/1000 phase maps, or 16,000/2000/2000 cut rows, for training/validation/test. The balanced sampler accounts for bit depth, beam count, maximum steering magnitude, and cut-$\varphi$. The maximum training budget is 15,000 epochs, but validation early stopping terminates the three final runs after 312–589 epochs; their selected checkpoints occur at epochs 62–339. Checkpoints minimize

$$\begin{array}{cc}\hfill S_{\mathrm{val}}=& \mathrm{RMSE}+\mathrm{MB}-\mathrm{HPBW}\mathrm{RMSE}\hfill \\ \hfill & +\mathrm{MB}-\mathrm{Main}-\mathrm{Lobe}\mathrm{RMSE}+0.85{\mathrm{HPBW}}_{\mathrm{worst}}.\hfill \end{array}$$

(26)

Here, ${\mathrm{HPBW}}_{\mathrm{worst}}$ is the worst validation HPBW RMSE among the bit, beam-count, steering-magnitude, and combined subgroups. Additional verified QueryFieldNet settings are listed in Appendix A 

Table A2. Verified training settings for the final QueryFieldNet experiments.

Setting	Value
Training scope	One joint model trained on the pooled 1-bit to 5-bit training split
Optimizer/batch size	AdamW/256
Initial learning rate/weight decay	$8\times {10}^{-4}$/$1\times {10}^{-4}$
Maximum budget/early stopping	15,000 epochs/patience 250; final runs stopped after 312–589 epochs
Learning-rate scheduler	ReduceLROnPlateau with patience 60
Loss weights	${\lambda }_{\mathrm{global}}=1.4$, ${\lambda }_{\mathrm{hpbw}}=3.5$, ${\lambda }_{\mathrm{lobe}}=3.5$, ${\lambda }_{\mathrm{beam}}=2.8$, ${\lambda }_{\mathrm{peak}}=0.5$, ${\lambda }_{\mathrm{shape}}=0.12$, and ${\lambda }_{\mathrm{metadata}}=0.25$
Curriculum/teacher forcing	180-epoch loss warm-up; occupancy teacher forcing decays from 0.2 to 0 over 500 epochs
Training/evaluation radius	$14°$ intended-beam training region/$\pm 12°$ evaluation window
Checkpoint criterion	Minimum composite validation score in Equation (26)

.

TabNet uses its regression objective with AdamW, learning rate $1\times {10}^{-3}$, weight decay $1\times {10}^{-4}$, batch size 1024, virtual batch size 128, and a 15,000-epoch maximum budget. SKi-Le-BNN uses MSE plus a KL term with ${\beta }_{\mathrm{KL}}=1\times {10}^{-5}$, AdamW, learning rate $1\times {10}^{-3}$, weight decay $1\times {10}^{-4}$, and full-batch training. PhaseNet uses full-curve MSE with the same optimizer, learning rate, weight decay, and full-batch mode. SKi-Le-BNN and PhaseNet use ReduceLROnPlateau with factor 0.5 and patience 1000 and select checkpoints by validation global RMSE.

The 1D GP-HetCNN uses Smooth L1 loss, Adam, learning rate $1\times {10}^{-3}$, zero weight decay, batch size 256, a 15,000-epoch maximum budget, early-stopping patience 250, and ReduceLROnPlateau with factor 0.1 and patience 50; its checkpoint is selected by validation HPBW RMSE. BOBE combines feature-engineered XGBoost, Random Forest, Extra Trees, and CatBoost regressors [21,22,23,24]; 200 Optuna trials minimize validation RMSE over normalized blending weights [25]. Neural optimizer, learning rate, batch size, and epoch count do not apply to this tree ensemble.

A common upper budget of 15,000 epochs is applied to the neural models. The initial learning rate is $1\times {10}^{-3}$ for the neural baselines and $8\times {10}^{-4}$ for QueryFieldNet. Model-specific losses, optimizers, and batch configurations are retained to preserve the intended learning behavior of each architecture, while training termination and best-checkpoint selection use validation performance only. QueryFieldNet uses AdamW, weight decay $1\times {10}^{-4}$, batch size 256, early-stopping patience 250, and ReduceLROnPlateau with factor 0.5 and patience 60. BOBE is optimized with 200 Optuna trials instead of an epoch-based budget, and its ensemble is selected by validation RMSE. Accordingly, the benchmark is presented as a common-source, validation-controlled, task-adapted comparison using an equal output dimensionality, identical evaluation code, and the same three reporting seeds.

3.2. Performance Comparison by Bit Depth

Table 2. Bit-depth performance on the balanced theta7 dataset. Entries are mean ± standard deviation over three runs.

Bit Depth	Model (Year)	RMSE	MAE	${\mathit{R}}^2$	MB-HPBW RMSE	MB-Main-Lobe RMSE	Beam-Peak RMSE
1	TabNet (2021)	3.313 ± 0.018	2.169 ± 0.009	0.99791 ± 0.00002	2.501 ± 0.103	2.584 ± 0.091	2.250 ± 0.048
	SKi-Le-BNN (2025)	5.060 ± 0.063	3.560 ± 0.084	0.99513 ± 0.00012	6.097 ± 0.097	5.459 ± 0.062	5.093 ± 0.141
	PhaseNet (2025)	3.342 ± 0.054	2.125 ± 0.037	0.99788 ± 0.00007	2.666 ± 0.067	2.661 ± 0.032	2.446 ± 0.203
	1D GP-HetCNN (2026)	15.296 ± 1.799	10.934 ± 0.471	0.95511 ± 0.01069	6.922 ± 0.832	6.919 ± 0.820	7.250 ± 0.952
	BOBE (2026)	3.172 ± 0.036	1.996 ± 0.035	0.99809 ± 0.00004	2.190 ± 0.140	2.231 ± 0.136	2.066 ± 0.181
	QueryFieldNet-Lite	2.987 ± 0.036	1.884 ± 0.033	0.99830 ± 0.00004	1.481 ± 0.085	1.749 ± 0.062	1.468 ± 0.115
	QueryFieldNet	2.456 ± 0.200	1.486 ± 0.173	0.99885 ± 0.00018	1.211 ± 0.061	1.422 ± 0.085	1.041 ± 0.076
2	TabNet (2021)	4.311 ± 0.032	3.007 ± 0.025	0.99640 ± 0.00005	3.504 ± 0.077	3.633 ± 0.092	3.108 ± 0.098
	SKi-Le-BNN (2025)	5.836 ± 0.018	4.448 ± 0.022	0.99339 ± 0.00004	7.798 ± 0.055	6.868 ± 0.050	6.226 ± 0.042
	PhaseNet (2025)	4.289 ± 0.045	2.999 ± 0.023	0.99643 ± 0.00008	3.490 ± 0.046	3.561 ± 0.051	2.981 ± 0.018
	1D GP-HetCNN (2026)	16.620 ± 2.590	12.958 ± 1.839	0.94552 ± 0.01607	10.615 ± 2.660	10.364 ± 2.417	10.964 ± 2.611
	BOBE (2026)	3.728 ± 0.147	2.536 ± 0.117	0.99729 ± 0.00023	2.770 ± 0.285	2.815 ± 0.227	2.528 ± 0.244
	QueryFieldNet-Lite	3.702 ± 0.054	2.432 ± 0.047	0.99734 ± 0.00008	2.028 ± 0.069	2.346 ± 0.053	2.038 ± 0.055
	QueryFieldNet	3.277 ± 0.170	2.074 ± 0.151	0.99791 ± 0.00021	1.709 ± 0.129	1.992 ± 0.128	1.721 ± 0.150
3	TabNet (2021)	4.390 ± 0.049	2.995 ± 0.023	0.99623 ± 0.00008	3.715 ± 0.224	3.769 ± 0.145	3.211 ± 0.104
	SKi-Le-BNN (2025)	5.623 ± 0.382	4.309 ± 0.315	0.99380 ± 0.00085	6.457 ± 1.283	5.976 ± 1.022	5.090 ± 0.812
	PhaseNet (2025)	4.326 ± 0.081	2.963 ± 0.031	0.99634 ± 0.00014	3.609 ± 0.157	3.677 ± 0.125	2.938 ± 0.048
	1D GP-HetCNN (2026)	18.780 ± 3.291	14.488 ± 1.719	0.92956 ± 0.02383	11.184 ± 2.068	10.784 ± 2.156	11.376 ± 2.292
	BOBE (2026)	3.730 ± 0.041	2.436 ± 0.059	0.99727 ± 0.00006	2.677 ± 0.171	2.792 ± 0.162	2.431 ± 0.102
	QueryFieldNet-Lite	3.937 ± 0.078	2.570 ± 0.066	0.99697 ± 0.00012	2.440 ± 0.090	2.716 ± 0.054	2.223 ± 0.041
	QueryFieldNet	3.618 ± 0.159	2.264 ± 0.130	0.99744 ± 0.00022	2.194 ± 0.012	2.480 ± 0.068	2.030 ± 0.049
4	TabNet (2021)	3.975 ± 0.084	2.586 ± 0.046	0.99690 ± 0.00013	3.218 ± 0.099	3.313 ± 0.091	2.748 ± 0.070
	SKi-Le-BNN (2025)	4.405 ± 0.217	3.176 ± 0.209	0.99619 ± 0.00038	3.945 ± 0.373	4.094 ± 0.289	3.111 ± 0.239
	PhaseNet (2025)	4.026 ± 0.031	2.633 ± 0.037	0.99682 ± 0.00005	3.341 ± 0.055	3.466 ± 0.086	2.832 ± 0.084
	1D GP-HetCNN (2026)	19.279 ± 3.919	14.266 ± 2.533	0.92509 ± 0.03092	9.758 ± 2.632	9.402 ± 2.472	9.885 ± 2.693
	BOBE (2026)	3.637 ± 0.028	2.232 ± 0.007	0.99740 ± 0.00004	2.634 ± 0.152	2.738 ± 0.161	2.400 ± 0.060
	QueryFieldNet-Lite	3.710 ± 0.042	2.362 ± 0.058	0.99730 ± 0.00006	2.420 ± 0.088	2.694 ± 0.077	2.261 ± 0.059
	QueryFieldNet	3.475 ± 0.078	2.098 ± 0.097	0.99763 ± 0.00011	2.229 ± 0.097	2.468 ± 0.041	1.982 ± 0.054
5	TabNet (2021)	3.676 ± 0.139	2.240 ± 0.046	0.99734 ± 0.00020	2.584 ± 0.101	2.804 ± 0.097	2.314 ± 0.111
	SKi-Le-BNN (2025)	3.683 ± 0.122	2.399 ± 0.125	0.99733 ± 0.00018	2.749 ± 0.172	2.998 ± 0.177	2.248 ± 0.126
	PhaseNet (2025)	3.705 ± 0.022	2.303 ± 0.024	0.99730 ± 0.00003	2.518 ± 0.048	2.820 ± 0.059	2.235 ± 0.068
	1D GP-HetCNN (2026)	24.217 ± 3.765	17.433 ± 2.546	0.88302 ± 0.03477	9.329 ± 0.322	9.052 ± 0.302	9.393 ± 0.346
	BOBE (2026)	3.195 ± 0.111	1.824 ± 0.061	0.99799 ± 0.00014	2.247 ± 0.248	2.353 ± 0.117	1.839 ± 0.107
	QueryFieldNet-Lite	3.594 ± 0.085	2.204 ± 0.047	0.99746 ± 0.00012	2.042 ± 0.138	2.375 ± 0.115	2.041 ± 0.026
	QueryFieldNet	3.375 ± 0.130	1.954 ± 0.127	0.99776 ± 0.00017	2.018 ± 0.076	2.307 ± 0.105	1.876 ± 0.046
Mean	TabNet (2021)	3.933 ± 0.034	2.599 ± 0.008	0.99696 ± 0.00005	3.104 ± 0.009	3.221 ± 0.019	2.726 ± 0.015
	SKi-Le-BNN (2025)	4.922 ± 0.091	3.579 ± 0.085	0.99517 ± 0.00020	5.409 ± 0.294	5.079 ± 0.231	4.354 ± 0.136
	PhaseNet (2025)	3.938 ± 0.033	2.605 ± 0.018	0.99695 ± 0.00005	3.125 ± 0.029	3.237 ± 0.016	2.687 ± 0.042
	1D GP-HetCNN (2026)	18.838 ± 1.950	14.016 ± 1.043	0.92766 ± 0.01517	9.562 ± 0.934	9.304 ± 0.837	9.774 ± 0.980
	BOBE (2026)	3.492 ± 0.003	2.205 ± 0.015	0.99761 ± 0.00001	2.504 ± 0.081	2.586 ± 0.038	2.253 ± 0.050
	QueryFieldNet-Lite	3.586 ± 0.050	2.290 ± 0.049	0.99747 ± 0.00007	2.082 ± 0.078	2.376 ± 0.065	2.006 ± 0.038
	QueryFieldNet	3.240 ± 0.143	1.975 ± 0.134	0.99792 ± 0.00017	1.872 ± 0.015	2.134 ± 0.047	1.730 ± 0.065

Entries are mean ± standard deviation over three runs. All models use the same 181-point output and metric definitions. In the mean block, each run is first averaged over the relevant subgroups, and then the displayed mean and standard deviation are computed over the three runs.

reports bit-depth-wise performance on the balanced theta7 dataset. For each bit depth, values are mean ± standard deviation over three runs on 181-point principal-cut gain curves. RMSE, MAE, MB-HPBW RMSE, MB-Main-Lobe RMSE, and Beam-Peak RMSE measure curve-level and local-window errors, while $R^2$ measures curve-level goodness of fit. The aggregate local-window values include both principal cuts; for the orthogonal $\varphi =90°$ cut, they denote target-centered local-window errors rather than physical intended-beam HPBW or peak measurements. Bold and underline indicate the best and second best values within each group. Error metrics are lower-is-better, whereas $R^2$ is higher-is-better. QueryFieldNet-Lite is included with the five baselines and full QueryFieldNet in all bit-depth, beam-count, and steering-position performance tables so that the capacity-control result can be inspected at both aggregate and subgroup levels.

In the bit-wise macro average in Table 2, the final QueryFieldNet variant reports RMSE 3.240, MAE 1.975, $R^2$ 0.99792, MB-HPBW RMSE 1.872, MB-Main-Lobe RMSE 2.134, and Beam-Peak RMSE 1.730. Relative to BOBE, these values correspond to reductions of 7.2% in RMSE, 10.4% in MAE, 25.2% in MB-HPBW RMSE, 17.5% in MB-Main-Lobe RMSE, and 23.2% in Beam-Peak RMSE. The advantage is not uniform: on 5-bit data, BOBE has 5.6% lower global RMSE and 2.0% lower Beam-Peak RMSE, whereas QueryFieldNet retains 10.2% lower MB-HPBW RMSE and 2.0% lower MB-Main-Lobe RMSE.

3.3. Beam Count-Based Performance Comparison

Table 3. (a) Beam-count subgroup performance (1-bit data). Entries are mean ± standard deviation over three runs. (b) Beam-count subgroup performance (2-bit data). Entries are mean ± standard deviation over three runs. (c) Beam-count subgroup performance (3-bit data). Entries are mean ± standard deviation over three runs. (d) Beam-count subgroup performance (4-bit data). Entries are mean ± standard deviation over three runs. (e) Beam-count subgroup performance (5-bit data). Entries are mean ± standard deviation over three runs.

(a)
Beam Count	Model (Year)	RMSE	MAE	${\mathit{R}}^{\mathbf{2}}$	MB-HPBW RMSE	MB-Main-Lobe RMSE	Beam-Peak RMSE
1	TabNet (2021)	3.008 ± 0.075	1.975 ± 0.039	0.99827 ± 0.00009	2.295 ± 0.135	2.390 ± 0.061	1.920 ± 0.023
	SKi-Le-BNN (2025)	5.194 ± 0.067	3.642 ± 0.081	0.99484 ± 0.00013	9.422 ± 0.113	8.597 ± 0.108	6.848 ± 0.142
	PhaseNet (2025)	2.875 ± 0.018	1.826 ± 0.040	0.99842 ± 0.00002	2.132 ± 0.148	2.189 ± 0.136	1.811 ± 0.097
	1D GP-HetCNN (2026)	15.176 ± 1.871	10.866 ± 0.519	0.95599 ± 0.01071	7.428 ± 0.298	7.339 ± 0.321	7.505 ± 0.428
	BOBE (2026)	2.510 ± 0.274	1.529 ± 0.236	0.99878 ± 0.00027	1.949 ± 0.204	1.917 ± 0.103	1.663 ± 0.127
	QueryFieldNet-Lite	2.693 ± 0.070	1.671 ± 0.057	0.99861 ± 0.00007	1.127 ± 0.121	1.295 ± 0.118	1.149 ± 0.162
	QueryFieldNet	2.350 ± 0.083	1.431 ± 0.063	0.99894 ± 0.00007	1.041 ± 0.082	1.232 ± 0.095	1.043 ± 0.051
2	TabNet (2021)	3.435 ± 0.079	2.264 ± 0.040	0.99776 ± 0.00010	2.428 ± 0.162	2.521 ± 0.187	2.267 ± 0.163
	SKi-Le-BNN (2025)	5.321 ± 0.061	3.768 ± 0.080	0.99462 ± 0.00012	6.658 ± 0.087	5.901 ± 0.051	5.593 ± 0.145
	PhaseNet (2025)	3.298 ± 0.038	2.171 ± 0.007	0.99793 ± 0.00005	2.636 ± 0.119	2.551 ± 0.084	2.271 ± 0.092
	1D GP-HetCNN (2026)	15.782 ± 1.833	11.230 ± 0.452	0.95246 ± 0.01114	7.248 ± 0.644	7.175 ± 0.612	7.515 ± 0.733
	BOBE (2026)	3.467 ± 0.148	2.149 ± 0.183	0.99772 ± 0.00020	2.341 ± 0.295	2.319 ± 0.129	1.988 ± 0.263
	QueryFieldNet-Lite	3.111 ± 0.031	1.996 ± 0.014	0.99816 ± 0.00004	1.536 ± 0.019	1.800 ± 0.044	1.530 ± 0.078
	QueryFieldNet	2.781 ± 0.045	1.757 ± 0.020	0.99853 ± 0.00005	1.385 ± 0.111	1.668 ± 0.125	1.342 ± 0.094
3	TabNet (2021)	3.604 ± 0.029	2.353 ± 0.030	0.99754 ± 0.00004	2.873 ± 0.064	2.848 ± 0.030	2.408 ± 0.048
	SKi-Le-BNN (2025)	5.011 ± 0.068	3.553 ± 0.087	0.99524 ± 0.00013	5.761 ± 0.095	5.016 ± 0.064	4.868 ± 0.150
	PhaseNet (2025)	3.780 ± 0.278	2.422 ± 0.125	0.99728 ± 0.00041	3.098 ± 0.120	2.978 ± 0.069	2.785 ± 0.158
	1D GP-HetCNN (2026)	15.295 ± 1.762	10.960 ± 0.513	0.95504 ± 0.01076	6.841 ± 0.768	6.819 ± 0.784	7.225 ± 0.885
	BOBE (2026)	3.343 ± 0.250	2.193 ± 0.199	0.99787 ± 0.00032	2.137 ± 0.340	2.252 ± 0.274	2.089 ± 0.369
	QueryFieldNet-Lite	3.190 ± 0.080	2.037 ± 0.050	0.99807 ± 0.00010	1.641 ± 0.162	1.935 ± 0.112	1.566 ± 0.163
	QueryFieldNet	2.949 ± 0.137	1.854 ± 0.085	0.99835 ± 0.00015	1.427 ± 0.107	1.783 ± 0.106	1.422 ± 0.168
4	TabNet (2021)	3.270 ± 0.056	2.146 ± 0.045	0.99797 ± 0.00007	2.342 ± 0.093	2.480 ± 0.102	2.225 ± 0.059
	SKi-Le-BNN (2025)	4.549 ± 0.060	3.185 ± 0.095	0.99608 ± 0.00010	4.317 ± 0.123	3.928 ± 0.082	4.070 ± 0.140
	PhaseNet (2025)	3.544 ± 0.205	2.204 ± 0.030	0.99761 ± 0.00028	2.529 ± 0.035	2.624 ± 0.093	2.450 ± 0.543
	1D GP-HetCNN (2026)	14.919 ± 1.742	10.681 ± 0.407	0.95693 ± 0.01024	6.676 ± 1.142	6.750 ± 1.093	7.056 ± 1.274
	BOBE (2026)	3.263 ± 0.106	2.095 ± 0.135	0.99798 ± 0.00014	2.178 ± 0.183	2.219 ± 0.222	2.128 ± 0.229
	QueryFieldNet-Lite	3.021 ± 0.073	1.898 ± 0.040	0.99827 ± 0.00008	1.431 ± 0.062	1.713 ± 0.051	1.451 ± 0.124
	QueryFieldNet	2.767 ± 0.117	1.723 ± 0.099	0.99855 ± 0.00012	1.194 ± 0.021	1.519 ± 0.016	1.265 ± 0.056
(b)
Beam Count	Model (Year)	RMSE	MAE	${\mathit{R}}^{\mathbf{2}}$	MB-HPBW RMSE	MB-Main-Lobe RMSE	Beam-Peak RMSE
1	TabNet (2021)	3.869 ± 0.014	2.720 ± 0.020	0.99708 ± 0.00002	3.295 ± 0.173	3.256 ± 0.100	3.161 ± 0.222
	SKi-Le-BNN (2025)	5.810 ± 0.022	4.292 ± 0.031	0.99341 ± 0.00005	12.192 ± 0.196	10.941 ± 0.190	8.002 ± 0.130
	PhaseNet (2025)	3.767 ± 0.083	2.609 ± 0.058	0.99723 ± 0.00012	2.606 ± 0.126	2.827 ± 0.069	2.489 ± 0.165
	1D GP-HetCNN (2026)	16.533 ± 2.543	13.010 ± 1.912	0.94550 ± 0.01560	11.297 ± 2.662	10.832 ± 2.420	11.346 ± 2.817
	BOBE (2026)	3.534 ± 0.047	2.373 ± 0.081	0.99754 ± 0.00007	2.565 ± 0.920	2.684 ± 0.650	2.521 ± 0.812
	QueryFieldNet-Lite	3.465 ± 0.107	2.279 ± 0.086	0.99765 ± 0.00015	1.758 ± 0.188	1.999 ± 0.143	2.016 ± 0.273
	QueryFieldNet	3.229 ± 0.039	2.081 ± 0.009	0.99796 ± 0.00005	1.655 ± 0.190	1.947 ± 0.140	1.766 ± 0.335
2	TabNet (2021)	4.206 ± 0.105	2.943 ± 0.071	0.99658 ± 0.00017	3.045 ± 0.107	3.291 ± 0.164	2.797 ± 0.091
	SKi-Le-BNN (2025)	6.011 ± 0.011	4.526 ± 0.008	0.99301 ± 0.00003	8.279 ± 0.072	7.283 ± 0.071	6.657 ± 0.055
	PhaseNet (2025)	4.273 ± 0.047	2.979 ± 0.026	0.99647 ± 0.00008	3.201 ± 0.152	3.389 ± 0.187	2.870 ± 0.181
	1D GP-HetCNN (2026)	16.691 ± 2.832	12.983 ± 1.979	0.94519 ± 0.01789	11.089 ± 2.826	10.747 ± 2.553	11.477 ± 2.803
	BOBE (2026)	3.858 ± 0.307	2.562 ± 0.197	0.99710 ± 0.00048	2.613 ± 0.337	2.696 ± 0.330	2.378 ± 0.265
	QueryFieldNet-Lite	3.751 ± 0.021	2.465 ± 0.029	0.99728 ± 0.00003	1.932 ± 0.094	2.373 ± 0.101	2.019 ± 0.074
	QueryFieldNet	3.475 ± 0.077	2.267 ± 0.054	0.99766 ± 0.00010	1.898 ± 0.120	2.245 ± 0.071	2.020 ± 0.102
3	TabNet (2021)	4.395 ± 0.165	3.051 ± 0.104	0.99625 ± 0.00028	3.318 ± 0.114	3.443 ± 0.204	2.932 ± 0.074
	SKi-Le-BNN (2025)	5.823 ± 0.024	4.470 ± 0.042	0.99343 ± 0.00006	7.189 ± 0.057	6.377 ± 0.054	5.969 ± 0.027
	PhaseNet (2025)	4.406 ± 0.024	3.096 ± 0.031	0.99624 ± 0.00004	3.554 ± 0.125	3.618 ± 0.035	2.976 ± 0.101
	1D GP-HetCNN (2026)	16.567 ± 2.554	12.879 ± 1.810	0.94569 ± 0.01609	10.421 ± 2.594	10.242 ± 2.359	10.792 ± 2.490
	BOBE (2026)	3.834 ± 0.218	2.669 ± 0.117	0.99713 ± 0.00033	2.958 ± 0.216	2.980 ± 0.107	2.664 ± 0.131
	QueryFieldNet-Lite	3.764 ± 0.082	2.472 ± 0.040	0.99725 ± 0.00012	1.977 ± 0.081	2.271 ± 0.013	2.125 ± 0.131
	QueryFieldNet	3.589 ± 0.127	2.333 ± 0.092	0.99750 ± 0.00018	1.927 ± 0.029	2.232 ± 0.077	2.035 ± 0.079
4	TabNet (2021)	4.681 ± 0.041	3.281 ± 0.046	0.99576 ± 0.00007	3.912 ± 0.095	4.010 ± 0.037	3.356 ± 0.167
	SKi-Le-BNN (2025)	5.713 ± 0.020	4.495 ± 0.012	0.99369 ± 0.00004	6.441 ± 0.080	5.716 ± 0.052	5.744 ± 0.115
	PhaseNet (2025)	4.611 ± 0.109	3.269 ± 0.060	0.99589 ± 0.00019	3.769 ± 0.053	3.743 ± 0.072	3.131 ± 0.111
	1D GP-HetCNN (2026)	16.680 ± 2.472	12.961 ± 1.717	0.94567 ± 0.01510	10.343 ± 2.621	10.146 ± 2.391	10.731 ± 2.548
	BOBE (2026)	3.686 ± 0.099	2.527 ± 0.129	0.99738 ± 0.00015	2.709 ± 0.391	2.736 ± 0.341	2.429 ± 0.508
	QueryFieldNet-Lite	3.797 ± 0.038	2.500 ± 0.045	0.99721 ± 0.00006	2.183 ± 0.101	2.479 ± 0.079	1.972 ± 0.072
	QueryFieldNet	3.717 ± 0.071	2.403 ± 0.047	0.99733 ± 0.00010	2.001 ± 0.064	2.293 ± 0.027	1.962 ± 0.159
(c)
Beam Count	Model (Year)	RMSE	MAE	${\mathit{R}}^{\mathbf{2}}$	MB-HPBW RMSE	MB-Main-Lobe RMSE	Beam-Peak RMSE
1	TabNet (2021)	4.205 ± 0.128	2.949 ± 0.068	0.99651 ± 0.00021	3.404 ± 0.266	3.708 ± 0.287	3.117 ± 0.300
	SKi-Le-BNN (2025)	5.110 ± 0.534	3.861 ± 0.378	0.99481 ± 0.00110	8.539 ± 3.428	8.036 ± 2.815	6.600 ± 1.814
	PhaseNet (2025)	3.927 ± 0.080	2.707 ± 0.043	0.99696 ± 0.00012	2.728 ± 0.063	3.084 ± 0.041	2.590 ± 0.037
	1D GP-HetCNN (2026)	17.997 ± 3.748	13.826 ± 1.864	0.93483 ± 0.02622	12.619 ± 1.342	11.860 ± 1.287	12.532 ± 1.669
	BOBE (2026)	3.627 ± 0.170	2.440 ± 0.182	0.99739 ± 0.00024	2.330 ± 0.360	2.502 ± 0.242	2.331 ± 0.574
	QueryFieldNet-Lite	3.550 ± 0.133	2.396 ± 0.087	0.99751 ± 0.00019	1.532 ± 0.102	2.030 ± 0.037	1.613 ± 0.112
	QueryFieldNet	3.313 ± 0.041	2.204 ± 0.037	0.99783 ± 0.00005	1.543 ± 0.124	2.027 ± 0.170	1.730 ± 0.098
2	TabNet (2021)	4.422 ± 0.172	2.991 ± 0.092	0.99617 ± 0.00029	3.941 ± 0.373	4.037 ± 0.211	3.330 ± 0.211
	SKi-Le-BNN (2025)	5.785 ± 0.361	4.493 ± 0.308	0.99343 ± 0.00083	7.024 ± 1.309	6.620 ± 1.065	5.576 ± 0.922
	PhaseNet (2025)	4.334 ± 0.131	2.946 ± 0.067	0.99632 ± 0.00022	3.508 ± 0.203	3.747 ± 0.244	3.027 ± 0.130
	1D GP-HetCNN (2026)	18.890 ± 3.167	14.545 ± 1.650	0.92844 ± 0.02386	11.523 ± 2.214	11.136 ± 2.286	11.630 ± 2.391
	BOBE (2026)	3.803 ± 0.078	2.433 ± 0.056	0.99716 ± 0.00011	2.740 ± 0.199	2.806 ± 0.259	2.563 ± 0.287
	QueryFieldNet-Lite	4.028 ± 0.079	2.609 ± 0.075	0.99682 ± 0.00013	2.552 ± 0.122	2.844 ± 0.100	2.483 ± 0.141
	QueryFieldNet	4.012 ± 0.093	2.543 ± 0.082	0.99685 ± 0.00015	2.452 ± 0.160	2.768 ± 0.152	2.381 ± 0.227
3	TabNet (2021)	4.424 ± 0.052	3.017 ± 0.038	0.99618 ± 0.00009	3.615 ± 0.195	3.721 ± 0.157	3.237 ± 0.060
	SKi-Le-BNN (2025)	5.609 ± 0.381	4.322 ± 0.313	0.99384 ± 0.00084	6.043 ± 1.166	5.620 ± 0.937	4.758 ± 0.716
	PhaseNet (2025)	4.466 ± 0.102	3.074 ± 0.060	0.99610 ± 0.00018	3.626 ± 0.115	3.680 ± 0.116	2.955 ± 0.169
	1D GP-HetCNN (2026)	18.945 ± 3.403	14.667 ± 1.872	0.92823 ± 0.02446	10.825 ± 2.178	10.474 ± 2.225	11.076 ± 2.354
	BOBE (2026)	3.752 ± 0.132	2.406 ± 0.085	0.99724 ± 0.00020	2.576 ± 0.479	2.672 ± 0.268	2.345 ± 0.193
	QueryFieldNet-Lite	3.926 ± 0.110	2.558 ± 0.079	0.99699 ± 0.00017	2.349 ± 0.095	2.645 ± 0.101	2.228 ± 0.042
	QueryFieldNet	3.786 ± 0.117	2.452 ± 0.089	0.99720 ± 0.00017	2.179 ± 0.074	2.633 ± 0.121	2.195 ± 0.117
4	TabNet (2021)	4.453 ± 0.064	3.012 ± 0.033	0.99614 ± 0.00011	3.728 ± 0.278	3.672 ± 0.226	3.142 ± 0.148
	SKi-Le-BNN (2025)	5.835 ± 0.305	4.448 ± 0.277	0.99336 ± 0.00070	5.886 ± 0.721	5.393 ± 0.561	4.752 ± 0.591
	PhaseNet (2025)	4.446 ± 0.069	3.051 ± 0.024	0.99615 ± 0.00012	3.809 ± 0.237	3.741 ± 0.108	2.937 ± 0.065
	1D GP-HetCNN (2026)	19.241 ± 2.939	14.908 ± 1.634	0.92678 ± 0.02139	10.876 ± 2.195	10.533 ± 2.344	11.150 ± 2.432
	BOBE (2026)	3.762 ± 0.055	2.493 ± 0.042	0.99723 ± 0.00007	2.815 ± 0.167	2.945 ± 0.159	2.444 ± 0.257
	QueryFieldNet-Lite	4.126 ± 0.035	2.674 ± 0.048	0.99669 ± 0.00006	2.608 ± 0.153	2.827 ± 0.084	2.177 ± 0.115
	QueryFieldNet	3.963 ± 0.092	2.495 ± 0.048	0.99694 ± 0.00014	2.452 ± 0.167	2.645 ± 0.098	2.231 ± 0.098
(d)
Beam Count	Model (Year)	RMSE	MAE	${\mathit{R}}^{\mathbf{2}}$	MB-HPBW RMSE	MB-Main-Lobe RMSE	Beam-Peak RMSE
1	TabNet (2021)	3.621 ± 0.064	2.361 ± 0.045	0.99740 ± 0.00009	2.194 ± 0.303	2.318 ± 0.198	2.019 ± 0.326
	SKi-Le-BNN (2025)	3.740 ± 0.087	2.621 ± 0.089	0.99723 ± 0.00013	3.117 ± 0.326	3.252 ± 0.292	2.272 ± 0.205
	PhaseNet (2025)	3.701 ± 0.153	2.362 ± 0.087	0.99728 ± 0.00022	3.130 ± 0.190	3.051 ± 0.191	2.692 ± 0.237
	1D GP-HetCNN (2026)	18.737 ± 4.048	13.691 ± 2.425	0.92835 ± 0.03091	12.007 ± 3.468	11.417 ± 3.326	11.751 ± 3.639
	BOBE (2026)	3.628 ± 0.197	2.286 ± 0.181	0.99737 ± 0.00030	2.139 ± 0.184	2.584 ± 0.037	2.312 ± 0.274
	QueryFieldNet-Lite	3.458 ± 0.031	2.184 ± 0.055	0.99763 ± 0.00004	1.799 ± 0.155	2.049 ± 0.186	1.998 ± 0.290
	QueryFieldNet	3.321 ± 0.081	2.036 ± 0.026	0.99782 ± 0.00011	1.856 ± 0.143	2.137 ± 0.197	1.883 ± 0.178
2	TabNet (2021)	4.192 ± 0.109	2.751 ± 0.047	0.99654 ± 0.00018	3.652 ± 0.321	3.581 ± 0.325	3.109 ± 0.075
	SKi-Le-BNN (2025)	4.504 ± 0.261	3.307 ± 0.272	0.99600 ± 0.00047	4.063 ± 0.375	4.058 ± 0.258	3.276 ± 0.092
	PhaseNet (2025)	4.176 ± 0.072	2.743 ± 0.052	0.99657 ± 0.00012	3.863 ± 0.113	3.715 ± 0.173	3.188 ± 0.139
	1D GP-HetCNN (2026)	19.018 ± 3.936	14.093 ± 2.510	0.92666 ± 0.03130	9.990 ± 2.893	9.637 ± 2.740	10.030 ± 3.044
	BOBE (2026)	4.008 ± 0.066	2.451 ± 0.083	0.99685 ± 0.00012	3.448 ± 0.429	3.425 ± 0.280	2.984 ± 0.445
	QueryFieldNet-Lite	3.966 ± 0.083	2.574 ± 0.058	0.99691 ± 0.00013	2.916 ± 0.062	2.898 ± 0.053	2.711 ± 0.114
	QueryFieldNet	3.847 ± 0.090	2.398 ± 0.049	0.99709 ± 0.00014	2.581 ± 0.250	2.654 ± 0.165	2.438 ± 0.211
3	TabNet (2021)	3.984 ± 0.043	2.628 ± 0.023	0.99689 ± 0.00007	3.396 ± 0.040	3.577 ± 0.032	3.038 ± 0.206
	SKi-Le-BNN (2025)	4.573 ± 0.287	3.327 ± 0.262	0.99589 ± 0.00052	3.956 ± 0.401	4.170 ± 0.307	3.095 ± 0.238
	PhaseNet (2025)	4.022 ± 0.097	2.673 ± 0.087	0.99683 ± 0.00015	3.342 ± 0.089	3.680 ± 0.052	3.029 ± 0.076
	1D GP-HetCNN (2026)	19.620 ± 3.903	14.590 ± 2.569	0.92301 ± 0.03084	9.488 ± 2.539	9.124 ± 2.410	9.662 ± 2.627
	BOBE (2026)	3.532 ± 0.256	2.152 ± 0.134	0.99754 ± 0.00035	2.545 ± 0.318	2.735 ± 0.221	2.474 ± 0.255
	QueryFieldNet-Lite	3.677 ± 0.055	2.335 ± 0.081	0.99735 ± 0.00008	2.391 ± 0.246	2.873 ± 0.154	2.222 ± 0.129
	QueryFieldNet	3.570 ± 0.110	2.218 ± 0.078	0.99750 ± 0.00015	2.023 ± 0.137	2.601 ± 0.186	1.963 ± 0.082
4	TabNet (2021)	4.065 ± 0.212	2.608 ± 0.111	0.99679 ± 0.00033	3.081 ± 0.034	3.174 ± 0.046	2.521 ± 0.142
	SKi-Le-BNN (2025)	4.684 ± 0.217	3.408 ± 0.216	0.99574 ± 0.00040	4.062 ± 0.369	4.206 ± 0.293	3.201 ± 0.322
	PhaseNet (2025)	4.163 ± 0.085	2.737 ± 0.077	0.99664 ± 0.00014	3.115 ± 0.057	3.284 ± 0.111	2.572 ± 0.049
	1D GP-HetCNN (2026)	19.710 ± 3.867	14.686 ± 2.677	0.92242 ± 0.03099	9.208 ± 2.328	8.929 ± 2.142	9.446 ± 2.293
	BOBE (2026)	3.409 ± 0.158	2.086 ± 0.117	0.99774 ± 0.00021	2.312 ± 0.315	2.368 ± 0.245	2.021 ± 0.211
	QueryFieldNet-Lite	3.728 ± 0.068	2.362 ± 0.059	0.99730 ± 0.00010	2.304 ± 0.098	2.591 ± 0.120	2.132 ± 0.056
	QueryFieldNet	3.518 ± 0.123	2.149 ± 0.067	0.99760 ± 0.00017	2.084 ± 0.019	2.395 ± 0.038	2.109 ± 0.061
(e)
Beam Count	Model (Year)	RMSE	MAE	${\mathit{R}}^{\mathbf{2}}$	MB-HPBW RMSE	MB-Main-Lobe RMSE	Beam-Peak RMSE
1	TabNet (2021)	3.710 ± 0.132	2.200 ± 0.020	0.99726 ± 0.00020	2.091 ± 0.102	2.528 ± 0.144	2.386 ± 0.090
	SKi-Le-BNN (2025)	3.586 ± 0.076	2.209 ± 0.065	0.99744 ± 0.00011	2.476 ± 0.288	2.739 ± 0.175	2.529 ± 0.156
	PhaseNet (2025)	3.698 ± 0.082	2.204 ± 0.041	0.99728 ± 0.00012	2.127 ± 0.106	2.694 ± 0.078	2.148 ± 0.062
	1D GP-HetCNN (2026)	23.585 ± 3.879	16.843 ± 2.638	0.88743 ± 0.03510	11.579 ± 0.522	11.054 ± 0.466	11.323 ± 0.452
	BOBE (2026)	3.568 ± 0.277	2.113 ± 0.207	0.99746 ± 0.00039	1.850 ± 0.154	2.288 ± 0.394	1.661 ± 0.307
	QueryFieldNet-Lite	3.733 ± 0.104	2.255 ± 0.062	0.99723 ± 0.00015	1.951 ± 0.099	2.414 ± 0.066	2.294 ± 0.108
	QueryFieldNet	3.592 ± 0.035	2.098 ± 0.039	0.99743 ± 0.00005	1.911 ± 0.053	2.289 ± 0.014	2.303 ± 0.156
2	TabNet (2021)	3.587 ± 0.172	2.156 ± 0.065	0.99747 ± 0.00024	2.712 ± 0.123	3.054 ± 0.083	2.506 ± 0.132
	SKi-Le-BNN (2025)	3.633 ± 0.086	2.372 ± 0.121	0.99741 ± 0.00012	2.767 ± 0.174	3.179 ± 0.154	2.420 ± 0.120
	PhaseNet (2025)	3.586 ± 0.071	2.224 ± 0.041	0.99748 ± 0.00010	2.564 ± 0.112	3.026 ± 0.116	2.594 ± 0.189
	1D GP-HetCNN (2026)	24.162 ± 3.812	17.515 ± 2.604	0.88409 ± 0.03574	9.776 ± 0.425	9.519 ± 0.373	9.754 ± 0.437
	BOBE (2026)	3.174 ± 0.147	1.772 ± 0.116	0.99801 ± 0.00020	2.203 ± 0.424	2.530 ± 0.311	2.054 ± 0.409
	QueryFieldNet-Lite	3.581 ± 0.093	2.179 ± 0.026	0.99748 ± 0.00013	2.236 ± 0.202	2.688 ± 0.139	2.293 ± 0.014
	QueryFieldNet	3.450 ± 0.061	2.021 ± 0.031	0.99766 ± 0.00008	2.188 ± 0.019	2.738 ± 0.077	2.271 ± 0.054
3	TabNet (2021)	3.932 ± 0.166	2.407 ± 0.055	0.99697 ± 0.00025	2.761 ± 0.144	2.854 ± 0.158	2.036 ± 0.203
	SKi-Le-BNN (2025)	3.870 ± 0.192	2.550 ± 0.179	0.99706 ± 0.00030	2.848 ± 0.182	2.968 ± 0.205	2.021 ± 0.147
	PhaseNet (2025)	3.918 ± 0.148	2.458 ± 0.069	0.99699 ± 0.00023	2.684 ± 0.112	2.776 ± 0.112	1.900 ± 0.051
	1D GP-HetCNN (2026)	24.375 ± 3.713	17.553 ± 2.503	0.88164 ± 0.03409	9.019 ± 0.366	8.744 ± 0.367	9.090 ± 0.453
	BOBE (2026)	3.099 ± 0.107	1.740 ± 0.080	0.99812 ± 0.00013	2.345 ± 0.229	2.441 ± 0.275	1.770 ± 0.510
	QueryFieldNet-Lite	3.625 ± 0.056	2.260 ± 0.037	0.99742 ± 0.00008	1.929 ± 0.185	2.095 ± 0.137	1.566 ± 0.084
	QueryFieldNet	3.601 ± 0.071	2.155 ± 0.025	0.99746 ± 0.00010	1.798 ± 0.071	2.017 ± 0.066	1.475 ± 0.068
4	TabNet (2021)	3.530 ± 0.119	2.239 ± 0.044	0.99756 ± 0.00016	2.517 ± 0.138	2.688 ± 0.097	2.337 ± 0.154
	SKi-Le-BNN (2025)	3.684 ± 0.167	2.501 ± 0.149	0.99734 ± 0.00024	2.747 ± 0.201	2.967 ± 0.199	2.202 ± 0.126
	PhaseNet (2025)	3.666 ± 0.057	2.368 ± 0.042	0.99737 ± 0.00008	2.487 ± 0.093	2.753 ± 0.049	2.229 ± 0.040
	1D GP-HetCNN (2026)	24.734 ± 3.682	17.822 ± 2.469	0.87886 ± 0.03436	8.667 ± 0.364	8.476 ± 0.333	8.883 ± 0.398
	BOBE (2026)	2.887 ± 0.073	1.654 ± 0.068	0.99837 ± 0.00007	2.214 ± 0.680	2.127 ± 0.461	1.776 ± 0.393
	QueryFieldNet-Lite	3.442 ± 0.099	2.138 ± 0.066	0.99768 ± 0.00013	2.006 ± 0.098	2.330 ± 0.126	2.071 ± 0.051
	QueryFieldNet	3.363 ± 0.036	1.992 ± 0.034	0.99779 ± 0.00005	1.897 ± 0.105	2.232 ± 0.079	1.989 ± 0.158

Entries are mean ± standard deviation over three runs. All models use the same 181-point output and metric definitions.

a–e compare performance across one- to four-beam cases at each bit depth. These tables use the same three-run mean ± standard deviation format as Table 2.

The beam-count subgroup results show that QueryFieldNet is strongest in local-window metrics across beam-count groups. Across the 20 bit-depth–beam-count combinations, QueryFieldNet improves on BOBE in 19 combinations for MB-HPBW RMSE and in 16 combinations for both MB-Main-Lobe RMSE and Beam-Peak RMSE, but in only 10 combinations for global RMSE. The global metrics do not follow a monotonic trend with beam count, indicating that prediction difficulty depends not only on the number of intended beams but also on steering-angle combinations, peak overlap, and bit-dependent phase distributions.

3.4. Model Complexity

Table 4. Parameter count and estimated FLOPs for one 181-point gain-curve prediction.

Model	Parameters (M)	FLOPs (M)
TabNet (2021)	0.66	1.93
SKi-Le-BNN (2025)	0.29	0.52
PhaseNet (2025)	0.18	1.50
1D GP-HetCNN (2026)	0.33	71.66
BOBE (2026)	338.09	4.99
QueryFieldNet-Lite	0.62	56.01
QueryFieldNet	6.06	542.59

reports parameter counts and forward-computation estimates for the five baselines and two QueryFieldNet variants. Neural and tree-ensemble FLOPs use model-specific primitive operations and are therefore not directly equivalent.

QueryFieldNet-Lite is obtained from full QueryFieldNet by reducing the three encoder-stage widths from 64/128/256 to 20/40/80 channels, the global/query hidden dimension from 256 to 80, and the cross-attention heads from eight to four, while retaining the periodic encoding, query decoder, intended-beam metadata head, beam-basis residual branch, and training protocol. It uses 0.617M parameters, close to TabNet’s 0.663M, and outperforms similarly sized neural baselines on every macro metric. It uses 99.82% fewer stored parameters than BOBE but has a higher neural FLOP estimate; full QueryFieldNet remains the accuracy-oriented model.

3.5. Beam-Center |θ|-Based Performance Comparison

Table 5. (a) Beam-center $\left|\theta \right|$ subgroup performance (1-bit data). Entries are mean ± standard deviation over three runs. (b) Beam-center $\left|\theta \right|$ subgroup performance (2-bit data). Entries are mean ± standard deviation over three runs. (c) Beam-center $\left|\theta \right|$ subgroup performance (3-bit data). Entries are mean ± standard deviation over three runs. (d) Beam-center $\left|\theta \right|$ subgroup performance (4-bit data). Entries are mean ± standard deviation over three runs. (e) Beam-center $\left|\theta \right|$ subgroup performance (5-bit data). Entries are mean ± standard deviation over three runs.

(a)
Beam Center $\mathbf{\left|}\mathit{\theta }\mathbf{\right|}$	Model (Year)	RMSE	MAE	${\mathit{R}}^{\mathbf{2}}$	MB-HPBW RMSE	MB-Main-Lobe RMSE	Beam-Peak RMSE
$0°$	TabNet (2021)	3.355 ± 0.050	2.208 ± 0.033	0.99785 ± 0.00006	2.329 ± 0.207	2.491 ± 0.122	2.197 ± 0.165
	SKi-Le-BNN (2025)	4.812 ± 0.061	3.462 ± 0.090	0.99558 ± 0.00011	5.204 ± 0.166	4.648 ± 0.135	7.006 ± 0.232
	PhaseNet (2025)	3.497 ± 0.184	2.214 ± 0.088	0.99766 ± 0.00024	2.734 ± 0.172	2.748 ± 0.078	2.727 ± 0.259
	1D GP-HetCNN (2026)	15.164 ± 1.738	10.876 ± 0.499	0.95566 ± 0.01050	7.129 ± 0.514	6.839 ± 0.511	7.791 ± 0.531
	BOBE (2026)	3.186 ± 0.081	2.050 ± 0.058	0.99806 ± 0.00010	1.988 ± 0.242	2.013 ± 0.204	2.045 ± 0.167
	QueryFieldNet-Lite	1.860 ± 0.103	1.216 ± 0.070	0.99933 ± 0.00008	0.456 ± 0.018	0.528 ± 0.040	0.480 ± 0.067
	QueryFieldNet	1.559 ± 0.212	0.953 ± 0.164	0.99952 ± 0.00013	0.271 ± 0.021	0.351 ± 0.039	0.257 ± 0.019
$15°$	TabNet (2021)	3.318 ± 0.001	2.143 ± 0.011	0.99790 ± 0.00000	2.305 ± 0.130	2.267 ± 0.079	2.030 ± 0.134
	SKi-Le-BNN (2025)	4.924 ± 0.059	3.439 ± 0.089	0.99539 ± 0.00011	6.032 ± 0.120	4.736 ± 0.070	5.612 ± 0.255
	PhaseNet (2025)	3.417 ± 0.073	2.166 ± 0.043	0.99778 ± 0.00009	2.517 ± 0.105	2.301 ± 0.035	2.150 ± 0.062
	1D GP-HetCNN (2026)	15.441 ± 1.673	11.078 ± 0.444	0.95415 ± 0.00999	5.665 ± 0.446	5.731 ± 0.401	5.905 ± 0.453
	BOBE (2026)	3.283 ± 0.057	2.082 ± 0.042	0.99794 ± 0.00008	2.090 ± 0.187	1.970 ± 0.098	1.991 ± 0.135
	QueryFieldNet-Lite	2.750 ± 0.061	1.763 ± 0.057	0.99855 ± 0.00007	0.697 ± 0.050	0.981 ± 0.082	0.657 ± 0.069
	QueryFieldNet	2.179 ± 0.240	1.336 ± 0.199	0.99908 ± 0.00020	0.580 ± 0.006	0.766 ± 0.064	0.556 ± 0.023
$30°$	TabNet (2021)	3.514 ± 0.044	2.341 ± 0.034	0.99766 ± 0.00006	2.780 ± 0.232	2.940 ± 0.219	2.208 ± 0.121
	SKi-Le-BNN (2025)	5.023 ± 0.063	3.547 ± 0.085	0.99522 ± 0.00012	6.721 ± 0.076	5.963 ± 0.061	3.998 ± 0.049
	PhaseNet (2025)	3.647 ± 0.144	2.355 ± 0.029	0.99747 ± 0.00020	2.669 ± 0.098	2.778 ± 0.127	2.258 ± 0.176
	1D GP-HetCNN (2026)	15.226 ± 1.809	10.875 ± 0.495	0.95552 ± 0.01072	6.769 ± 0.837	7.043 ± 0.936	6.870 ± 1.038
	BOBE (2026)	3.384 ± 0.055	2.176 ± 0.028	0.99782 ± 0.00006	2.335 ± 0.148	2.361 ± 0.077	1.950 ± 0.181
	QueryFieldNet-Lite	3.103 ± 0.013	1.981 ± 0.010	0.99817 ± 0.00001	1.140 ± 0.117	1.333 ± 0.076	1.010 ± 0.091
	QueryFieldNet	2.609 ± 0.155	1.584 ± 0.145	0.99870 ± 0.00015	1.018 ± 0.061	1.249 ± 0.015	0.879 ± 0.072
$45°$	TabNet (2021)	3.276 ± 0.053	2.139 ± 0.029	0.99796 ± 0.00007	2.720 ± 0.245	2.814 ± 0.117	2.514 ± 0.107
	SKi-Le-BNN (2025)	4.976 ± 0.069	3.467 ± 0.086	0.99531 ± 0.00013	6.325 ± 0.141	7.039 ± 0.185	4.053 ± 0.035
	PhaseNet (2025)	3.339 ± 0.091	2.092 ± 0.051	0.99788 ± 0.00012	2.896 ± 0.275	3.101 ± 0.207	2.660 ± 0.293
	1D GP-HetCNN (2026)	15.065 ± 1.871	10.734 ± 0.429	0.95628 ± 0.01117	8.192 ± 1.871	8.260 ± 1.673	8.354 ± 2.056
	BOBE (2026)	3.160 ± 0.160	2.018 ± 0.110	0.99810 ± 0.00020	2.443 ± 0.123	2.771 ± 0.236	2.207 ± 0.389
	QueryFieldNet-Lite	3.044 ± 0.047	1.908 ± 0.040	0.99825 ± 0.00005	1.095 ± 0.044	1.411 ± 0.056	1.058 ± 0.119
	QueryFieldNet	2.487 ± 0.222	1.509 ± 0.184	0.99882 ± 0.00021	0.918 ± 0.054	1.176 ± 0.059	0.883 ± 0.095
(b)
Beam Center $\mathbf{\left|}\mathit{\theta }\mathbf{\right|}$	Model (Year)	RMSE	MAE	${\mathit{R}}^{\mathbf{2}}$	MB-HPBW RMSE	MB-Main-Lobe RMSE	Beam-Peak RMSE
$0°$	TabNet (2021)	4.378 ± 0.046	3.070 ± 0.005	0.99629 ± 0.00008	2.811 ± 0.051	2.950 ± 0.082	2.613 ± 0.196
	SKi-Le-BNN (2025)	5.803 ± 0.006	4.520 ± 0.008	0.99348 ± 0.00001	7.968 ± 0.149	6.885 ± 0.105	8.561 ± 0.159
	PhaseNet (2025)	4.319 ± 0.064	3.050 ± 0.041	0.99639 ± 0.00011	2.769 ± 0.256	3.046 ± 0.128	2.503 ± 0.324
	1D GP-HetCNN (2026)	16.733 ± 2.555	13.131 ± 1.850	0.94507 ± 0.01574	12.402 ± 4.319	11.402 ± 4.049	12.906 ± 4.323
	BOBE (2026)	3.614 ± 0.175	2.449 ± 0.126	0.99747 ± 0.00025	2.340 ± 0.527	2.394 ± 0.373	2.311 ± 0.538
	QueryFieldNet-Lite	3.427 ± 0.234	2.305 ± 0.171	0.99767 ± 0.00032	0.293 ± 0.143	0.333 ± 0.106	0.280 ± 0.138
	QueryFieldNet	3.200 ± 0.235	2.068 ± 0.205	0.99797 ± 0.00030	0.127 ± 0.030	0.136 ± 0.020	0.157 ± 0.089
$15°$	TabNet (2021)	4.434 ± 0.010	3.073 ± 0.025	0.99618 ± 0.00002	3.262 ± 0.146	3.526 ± 0.065	2.605 ± 0.120
	SKi-Le-BNN (2025)	5.778 ± 0.028	4.398 ± 0.016	0.99351 ± 0.00006	8.072 ± 0.117	6.494 ± 0.089	7.095 ± 0.225
	PhaseNet (2025)	4.442 ± 0.042	3.105 ± 0.027	0.99616 ± 0.00007	3.384 ± 0.070	3.414 ± 0.081	2.568 ± 0.092
	1D GP-HetCNN (2026)	16.866 ± 2.475	13.183 ± 1.791	0.94376 ± 0.01570	9.481 ± 2.974	9.044 ± 2.432	9.662 ± 2.959
	BOBE (2026)	3.719 ± 0.193	2.527 ± 0.150	0.99730 ± 0.00029	2.653 ± 0.367	2.619 ± 0.340	2.054 ± 0.358
	QueryFieldNet-Lite	3.660 ± 0.105	2.450 ± 0.117	0.99737 ± 0.00015	0.918 ± 0.068	1.383 ± 0.083	0.865 ± 0.027
	QueryFieldNet	3.128 ± 0.163	2.003 ± 0.158	0.99807 ± 0.00020	0.840 ± 0.082	1.247 ± 0.141	0.768 ± 0.104
$30°$	TabNet (2021)	4.454 ± 0.046	3.110 ± 0.025	0.99616 ± 0.00008	3.790 ± 0.135	3.880 ± 0.202	3.159 ± 0.060
	SKi-Le-BNN (2025)	5.798 ± 0.014	4.452 ± 0.025	0.99349 ± 0.00003	8.015 ± 0.084	7.115 ± 0.057	4.826 ± 0.019
	PhaseNet (2025)	4.398 ± 0.081	3.094 ± 0.043	0.99626 ± 0.00014	3.709 ± 0.163	3.703 ± 0.086	3.004 ± 0.188
	1D GP-HetCNN (2026)	16.597 ± 2.596	12.904 ± 1.758	0.94597 ± 0.01604	8.657 ± 1.338	8.688 ± 1.173	8.696 ± 1.226
	BOBE (2026)	3.847 ± 0.099	2.652 ± 0.081	0.99713 ± 0.00016	2.890 ± 0.215	2.876 ± 0.172	2.605 ± 0.152
	QueryFieldNet-Lite	3.472 ± 0.054	2.243 ± 0.019	0.99766 ± 0.00007	1.147 ± 0.028	1.596 ± 0.060	0.994 ± 0.024
	QueryFieldNet	3.098 ± 0.151	1.930 ± 0.119	0.99813 ± 0.00018	0.941 ± 0.046	1.378 ± 0.050	0.800 ± 0.037
$45°$	TabNet (2021)	4.454 ± 0.066	3.113 ± 0.058	0.99617 ± 0.00011	3.999 ± 0.173	3.974 ± 0.059	3.689 ± 0.256
	SKi-Le-BNN (2025)	5.859 ± 0.026	4.533 ± 0.036	0.99337 ± 0.00006	6.265 ± 0.168	6.939 ± 0.181	4.456 ± 0.047
	PhaseNet (2025)	4.458 ± 0.062	3.134 ± 0.024	0.99616 ± 0.00011	3.961 ± 0.095	3.991 ± 0.065	3.544 ± 0.181
	1D GP-HetCNN (2026)	16.386 ± 2.634	12.655 ± 1.822	0.94719 ± 0.01609	12.399 ± 2.666	12.614 ± 2.723	12.876 ± 2.682
	BOBE (2026)	3.739 ± 0.203	2.576 ± 0.154	0.99729 ± 0.00031	3.113 ± 0.286	3.354 ± 0.252	3.002 ± 0.286
	QueryFieldNet-Lite	3.828 ± 0.062	2.530 ± 0.049	0.99717 ± 0.00009	1.420 ± 0.034	1.830 ± 0.023	0.990 ± 0.118
	QueryFieldNet	3.393 ± 0.182	2.162 ± 0.167	0.99777 ± 0.00024	1.381 ± 0.093	1.604 ± 0.077	1.017 ± 0.058
(c)
Beam Center $\mathbf{\left|}\mathit{\theta }\mathbf{\right|}$	Model (Year)	RMSE	MAE	${\mathit{R}}^{\mathbf{2}}$	MB-HPBW RMSE	MB-Main-Lobe RMSE	Beam-Peak RMSE
$0°$	TabNet (2021)	3.967 ± 0.057	2.670 ± 0.021	0.99697 ± 0.00009	2.217 ± 0.379	2.158 ± 0.237	2.092 ± 0.295
	SKi-Le-BNN (2025)	5.482 ± 0.385	4.126 ± 0.324	0.99419 ± 0.00082	4.931 ± 1.504	4.913 ± 1.120	4.603 ± 1.599
	PhaseNet (2025)	3.922 ± 0.007	2.649 ± 0.028	0.99703 ± 0.00001	2.071 ± 0.192	2.291 ± 0.189	1.954 ± 0.154
	1D GP-HetCNN (2026)	19.018 ± 3.468	14.777 ± 1.853	0.92850 ± 0.02496	14.505 ± 2.471	13.213 ± 2.384	14.748 ± 2.512
	BOBE (2026)	3.465 ± 0.079	2.242 ± 0.086	0.99766 ± 0.00011	1.008 ± 0.260	1.291 ± 0.341	1.034 ± 0.308
	QueryFieldNet-Lite	2.358 ± 0.055	1.534 ± 0.031	0.99891 ± 0.00005	0.199 ± 0.105	0.223 ± 0.086	0.216 ± 0.133
	QueryFieldNet	2.005 ± 0.286	1.222 ± 0.209	0.99920 ± 0.00023	0.108 ± 0.006	0.116 ± 0.015	0.122 ± 0.022
$15°$	TabNet (2021)	4.463 ± 0.094	3.046 ± 0.042	0.99608 ± 0.00017	3.677 ± 0.209	3.741 ± 0.117	2.561 ± 0.024
	SKi-Le-BNN (2025)	5.751 ± 0.337	4.401 ± 0.272	0.99347 ± 0.00077	7.980 ± 1.567	6.905 ± 1.074	6.527 ± 1.344
	PhaseNet (2025)	4.443 ± 0.078	3.070 ± 0.039	0.99611 ± 0.00013	3.565 ± 0.247	3.672 ± 0.127	2.470 ± 0.200
	1D GP-HetCNN (2026)	18.951 ± 3.061	14.743 ± 1.675	0.92804 ± 0.02231	10.277 ± 1.175	9.561 ± 1.119	10.400 ± 1.240
	BOBE (2026)	3.818 ± 0.082	2.501 ± 0.027	0.99713 ± 0.00012	2.397 ± 0.461	2.546 ± 0.382	1.617 ± 0.315
	QueryFieldNet-Lite	3.690 ± 0.237	2.498 ± 0.161	0.99726 ± 0.00036	0.499 ± 0.077	1.382 ± 0.014	0.518 ± 0.050
	QueryFieldNet	3.311 ± 0.226	2.173 ± 0.155	0.99779 ± 0.00030	0.455 ± 0.089	1.511 ± 0.056	0.405 ± 0.067
$30°$	TabNet (2021)	4.541 ± 0.079	3.080 ± 0.053	0.99598 ± 0.00014	4.022 ± 0.316	4.007 ± 0.215	3.513 ± 0.186
	SKi-Le-BNN (2025)	5.719 ± 0.362	4.400 ± 0.314	0.99361 ± 0.00082	5.994 ± 1.346	5.524 ± 1.109	4.325 ± 0.540
	PhaseNet (2025)	4.526 ± 0.114	3.089 ± 0.046	0.99600 ± 0.00020	4.011 ± 0.193	3.959 ± 0.244	3.140 ± 0.117
	1D GP-HetCNN (2026)	18.971 ± 3.074	14.643 ± 1.558	0.92856 ± 0.02221	8.771 ± 1.463	8.728 ± 1.730	8.611 ± 1.646
	BOBE (2026)	3.748 ± 0.054	2.436 ± 0.054	0.99726 ± 0.00007	3.076 ± 0.165	3.067 ± 0.079	2.790 ± 0.202
	QueryFieldNet-Lite	3.911 ± 0.067	2.527 ± 0.055	0.99703 ± 0.00010	1.424 ± 0.029	1.916 ± 0.150	1.209 ± 0.037
	QueryFieldNet	3.672 ± 0.214	2.268 ± 0.146	0.99737 ± 0.00031	1.174 ± 0.098	1.677 ± 0.099	1.045 ± 0.109
$45°$	TabNet (2021)	4.446 ± 0.052	3.042 ± 0.021	0.99614 ± 0.00009	4.070 ± 0.189	4.190 ± 0.136	3.827 ± 0.074
	SKi-Le-BNN (2025)	5.730 ± 0.356	4.432 ± 0.310	0.99356 ± 0.00081	5.771 ± 0.898	5.818 ± 0.956	4.295 ± 0.208
	PhaseNet (2025)	4.416 ± 0.088	3.049 ± 0.031	0.99618 ± 0.00015	3.807 ± 0.316	3.897 ± 0.258	3.470 ± 0.243
	1D GP-HetCNN (2026)	18.934 ± 3.297	14.516 ± 1.821	0.92831 ± 0.02421	12.146 ± 4.076	12.357 ± 4.152	12.481 ± 4.410
	BOBE (2026)	3.806 ± 0.094	2.501 ± 0.066	0.99716 ± 0.00014	3.196 ± 0.222	3.323 ± 0.113	3.115 ± 0.290
	QueryFieldNet-Lite	4.049 ± 0.064	2.653 ± 0.059	0.99679 ± 0.00010	1.858 ± 0.129	2.269 ± 0.045	1.258 ± 0.042
	QueryFieldNet	3.709 ± 0.134	2.333 ± 0.121	0.99731 ± 0.00019	2.009 ± 0.077	2.237 ± 0.065	1.279 ± 0.048
(d)
Beam Center $\mathbf{\left|}\mathit{\theta }\mathbf{\right|}$	Model (Year)	RMSE	MAE	${\mathit{R}}^{\mathbf{2}}$	MB-HPBW RMSE	MB-Main-Lobe RMSE	Beam-Peak RMSE
$0°$	TabNet (2021)	3.475 ± 0.084	2.222 ± 0.044	0.99766 ± 0.00011	1.324 ± 0.044	1.471 ± 0.074	1.313 ± 0.027
	SKi-Le-BNN (2025)	4.089 ± 0.257	2.905 ± 0.235	0.99675 ± 0.00041	2.655 ± 0.497	3.405 ± 0.380	2.271 ± 0.409
	PhaseNet (2025)	3.495 ± 0.045	2.273 ± 0.028	0.99763 ± 0.00006	1.431 ± 0.050	1.660 ± 0.116	1.265 ± 0.142
	1D GP-HetCNN (2026)	19.419 ± 4.077	14.413 ± 2.597	0.92511 ± 0.03185	13.202 ± 5.372	12.082 ± 5.092	13.335 ± 5.427
	BOBE (2026)	2.821 ± 0.182	1.712 ± 0.099	0.99845 ± 0.00020	0.549 ± 0.063	0.623 ± 0.081	0.473 ± 0.074
	QueryFieldNet-Lite	2.004 ± 0.091	1.173 ± 0.070	0.99921 ± 0.00007	0.191 ± 0.105	0.221 ± 0.086	0.193 ± 0.133
	QueryFieldNet	1.737 ± 0.089	0.980 ± 0.100	0.99940 ± 0.00006	0.108 ± 0.008	0.112 ± 0.014	0.115 ± 0.026
$15°$	TabNet (2021)	4.037 ± 0.088	2.618 ± 0.047	0.99679 ± 0.00014	2.670 ± 0.153	3.035 ± 0.107	1.799 ± 0.079
	SKi-Le-BNN (2025)	4.589 ± 0.231	3.311 ± 0.224	0.99585 ± 0.00042	3.914 ± 0.423	4.088 ± 0.304	2.556 ± 0.281
	PhaseNet (2025)	4.108 ± 0.093	2.688 ± 0.073	0.99668 ± 0.00015	2.654 ± 0.093	3.083 ± 0.006	1.936 ± 0.136
	1D GP-HetCNN (2026)	19.329 ± 3.736	14.451 ± 2.514	0.92447 ± 0.02971	9.516 ± 3.097	8.917 ± 2.790	9.646 ± 3.149
	BOBE (2026)	3.561 ± 0.111	2.180 ± 0.071	0.99749 ± 0.00017	2.135 ± 0.199	2.451 ± 0.377	1.240 ± 0.223
	QueryFieldNet-Lite	3.465 ± 0.078	2.165 ± 0.067	0.99759 ± 0.00011	0.449 ± 0.037	0.594 ± 0.041	0.443 ± 0.034
	QueryFieldNet	3.263 ± 0.076	1.946 ± 0.101	0.99786 ± 0.00010	0.320 ± 0.042	0.418 ± 0.035	0.298 ± 0.029
$30°$	TabNet (2021)	4.137 ± 0.171	2.749 ± 0.093	0.99664 ± 0.00028	3.981 ± 0.189	3.932 ± 0.159	3.405 ± 0.143
	SKi-Le-BNN (2025)	4.620 ± 0.222	3.407 ± 0.214	0.99582 ± 0.00040	4.463 ± 0.328	4.416 ± 0.259	3.619 ± 0.196
	PhaseNet (2025)	4.194 ± 0.079	2.807 ± 0.070	0.99655 ± 0.00013	4.036 ± 0.104	4.004 ± 0.073	3.343 ± 0.015
	1D GP-HetCNN (2026)	19.589 ± 3.745	14.545 ± 2.479	0.92345 ± 0.02976	7.937 ± 1.488	7.809 ± 1.396	7.729 ± 1.542
	BOBE (2026)	3.734 ± 0.126	2.299 ± 0.064	0.99728 ± 0.00019	3.290 ± 0.086	3.248 ± 0.129	3.268 ± 0.151
	QueryFieldNet-Lite	3.500 ± 0.008	2.278 ± 0.030	0.99760 ± 0.00001	1.506 ± 0.199	1.925 ± 0.069	1.019 ± 0.084
	QueryFieldNet	3.314 ± 0.083	2.061 ± 0.072	0.99784 ± 0.00011	1.572 ± 0.322	2.171 ± 0.438	1.089 ± 0.053
$45°$	TabNet (2021)	4.222 ± 0.116	2.730 ± 0.058	0.99653 ± 0.00019	3.681 ± 0.018	3.630 ± 0.049	3.368 ± 0.109
	SKi-Le-BNN (2025)	4.679 ± 0.239	3.406 ± 0.238	0.99573 ± 0.00044	4.106 ± 0.370	4.101 ± 0.289	3.481 ± 0.231
	PhaseNet (2025)	4.290 ± 0.056	2.819 ± 0.069	0.99641 ± 0.00009	4.025 ± 0.351	3.996 ± 0.402	3.625 ± 0.247
	1D GP-HetCNN (2026)	19.450 ± 4.137	14.336 ± 2.753	0.92353 ± 0.03303	9.457 ± 1.493	9.636 ± 1.528	9.833 ± 1.613
	BOBE (2026)	3.748 ± 0.079	2.308 ± 0.032	0.99725 ± 0.00012	2.826 ± 0.198	2.979 ± 0.021	2.739 ± 0.108
	QueryFieldNet-Lite	3.955 ± 0.067	2.551 ± 0.071	0.99695 ± 0.00010	1.406 ± 0.032	2.103 ± 0.042	1.226 ± 0.080
	QueryFieldNet	3.696 ± 0.090	2.254 ± 0.114	0.99733 ± 0.00013	1.510 ± 0.066	2.078 ± 0.160	1.121 ± 0.129
(e)
Beam Center $\mathbf{\left|}\mathit{\theta }\mathbf{\right|}$	Model (Year)	RMSE	MAE	${\mathit{R}}^{\mathbf{2}}$	MB-HPBW RMSE	MB-Main-Lobe RMSE	Beam-Peak RMSE
$0°$	TabNet (2021)	2.561 ± 0.127	1.529 ± 0.033	0.99873 ± 0.00012	1.049 ± 0.102	1.324 ± 0.134	1.077 ± 0.221
	SKi-Le-BNN (2025)	2.781 ± 0.176	1.771 ± 0.137	0.99850 ± 0.00019	1.616 ± 0.261	2.099 ± 0.350	1.224 ± 0.196
	PhaseNet (2025)	2.638 ± 0.018	1.633 ± 0.022	0.99866 ± 0.00002	1.070 ± 0.090	1.495 ± 0.125	0.967 ± 0.072
	1D GP-HetCNN (2026)	24.116 ± 3.829	17.313 ± 2.608	0.88628 ± 0.03373	10.917 ± 0.817	10.019 ± 0.748	11.014 ± 0.796
	BOBE (2026)	2.198 ± 0.080	1.217 ± 0.070	0.99907 ± 0.00007	0.317 ± 0.077	0.400 ± 0.102	0.254 ± 0.074
	QueryFieldNet-Lite	1.617 ± 0.092	0.860 ± 0.054	0.99949 ± 0.00006	0.192 ± 0.079	0.230 ± 0.066	0.188 ± 0.115
	QueryFieldNet	1.384 ± 0.097	0.730 ± 0.033	0.99962 ± 0.00005	0.116 ± 0.007	0.118 ± 0.016	0.120 ± 0.025
$15°$	TabNet (2021)	3.831 ± 0.119	2.417 ± 0.063	0.99708 ± 0.00018	2.378 ± 0.105	2.732 ± 0.057	1.742 ± 0.194
	SKi-Le-BNN (2025)	3.876 ± 0.131	2.606 ± 0.136	0.99701 ± 0.00020	2.695 ± 0.129	3.012 ± 0.166	1.736 ± 0.168
	PhaseNet (2025)	3.919 ± 0.063	2.504 ± 0.049	0.99695 ± 0.00010	2.474 ± 0.207	2.833 ± 0.152	1.999 ± 0.197
	1D GP-HetCNN (2026)	23.866 ± 3.542	17.328 ± 2.425	0.88544 ± 0.03292	9.460 ± 0.588	8.969 ± 0.424	9.503 ± 0.555
	BOBE (2026)	3.092 ± 0.094	1.766 ± 0.068	0.99811 ± 0.00011	1.659 ± 0.151	1.832 ± 0.117	0.839 ± 0.160
	QueryFieldNet-Lite	3.395 ± 0.119	2.105 ± 0.069	0.99767 ± 0.00016	0.477 ± 0.049	0.533 ± 0.055	0.476 ± 0.067
	QueryFieldNet	3.185 ± 0.101	1.912 ± 0.096	0.99795 ± 0.00013	0.281 ± 0.051	0.332 ± 0.058	0.245 ± 0.056
$30°$	TabNet (2021)	3.785 ± 0.142	2.352 ± 0.039	0.99719 ± 0.00021	2.899 ± 0.166	3.038 ± 0.097	2.584 ± 0.169
	SKi-Le-BNN (2025)	3.828 ± 0.163	2.547 ± 0.160	0.99713 ± 0.00025	2.922 ± 0.161	3.124 ± 0.146	2.512 ± 0.153
	PhaseNet (2025)	3.821 ± 0.041	2.420 ± 0.034	0.99714 ± 0.00006	2.701 ± 0.059	2.984 ± 0.103	2.345 ± 0.050
	1D GP-HetCNN (2026)	24.681 ± 3.878	17.786 ± 2.607	0.87921 ± 0.03626	7.547 ± 0.418	7.491 ± 0.331	7.291 ± 0.373
	BOBE (2026)	3.273 ± 0.119	1.887 ± 0.060	0.99790 ± 0.00016	2.540 ± 0.136	2.650 ± 0.158	2.246 ± 0.203
	QueryFieldNet-Lite	3.726 ± 0.189	2.253 ± 0.069	0.99728 ± 0.00028	1.024 ± 0.068	1.507 ± 0.095	1.087 ± 0.079
	QueryFieldNet	3.526 ± 0.207	2.013 ± 0.166	0.99756 ± 0.00028	0.943 ± 0.027	1.451 ± 0.047	1.083 ± 0.038
$45°$	TabNet (2021)	3.836 ± 0.167	2.388 ± 0.061	0.99709 ± 0.00025	3.131 ± 0.278	3.194 ± 0.177	2.924 ± 0.091
	SKi-Le-BNN (2025)	3.831 ± 0.117	2.567 ± 0.125	0.99711 ± 0.00018	3.272 ± 0.225	3.270 ± 0.191	2.769 ± 0.085
	PhaseNet (2025)	3.876 ± 0.026	2.475 ± 0.034	0.99704 ± 0.00004	3.068 ± 0.163	3.203 ± 0.081	2.767 ± 0.138
	1D GP-HetCNN (2026)	24.781 ± 3.730	17.768 ± 2.485	0.87774 ± 0.03516	9.873 ± 0.952	10.029 ± 0.963	10.137 ± 1.131
	BOBE (2026)	3.232 ± 0.104	1.854 ± 0.069	0.99795 ± 0.00014	2.932 ± 0.860	2.931 ± 0.423	2.433 ± 0.180
	QueryFieldNet-Lite	3.721 ± 0.067	2.346 ± 0.079	0.99728 ± 0.00010	1.253 ± 0.107	1.609 ± 0.012	1.217 ± 0.132
	QueryFieldNet	3.486 ± 0.114	2.064 ± 0.125	0.99761 ± 0.00016	1.281 ± 0.088	1.607 ± 0.085	1.110 ± 0.022

Entries are mean ± standard deviation over three runs. All models use the same 181-point output and metric definitions.

a–e compare performance across absolute beam-center steering positions. The subgroup value $\left|\theta \right|=45°$ combines samples containing either $-45°$ or $45°$ intended beams, and similarly for $15°$ and $30°$.

The steering-position analysis shows lower errors near broadside and generally larger errors toward $\left|\theta \right|=45°$. Across the 20 bit-depth–absolute-beam-center combinations reported in Table 5a–e, QueryFieldNet improves on BOBE in all 20 combinations for MB-HPBW RMSE, MB-Main-Lobe RMSE, and Beam-Peak RMSE, and in 17 combinations for global RMSE. The three global-RMSE exceptions occur at 5-bit with $\left|\theta \right|=15°$, $30°$, and $45°$. Thus, the local intended-beam advantage remains consistent across the reported absolute steering groups, while BOBE remains competitive for full-curve fitting at higher phase resolution.

3.6. Ablation Study

A component-level ablation was repeated with seeds 42, 43, and 44 on the balanced theta7 dataset. All variants use the same training setting, and exactly one component is added at each stage: A0 is the base 2D CNN, A1 adds periodic phase encoding, A2 adds angular query cross-attention, A3 adds the intended-beam metadata head, and A4 adds the beam-basis residual branch and is the final QueryFieldNet used in the main benchmark.

Table 6. Three-run component-level ablation study of QueryFieldNet on the balanced theta7 dataset. Entries are bit-wise macro means ± standard deviations.

Variant	Configuration	RMSE	MAE	${\mathit{R}}^2$	MB-HPBW RMSE	MB-Main-Lobe RMSE	Beam-Peak RMSE
A0	Base 2D CNN	3.5166 ± 0.0619	2.2375 ± 0.0611	0.997567 ± 0.000082	2.0580 ± 0.0675	2.3583 ± 0.0604	1.9514 ± 0.0418
A1	A0 + periodic phase encoding	3.5467 ± 0.1081	2.2292 ± 0.1007	0.997524 ± 0.000142	2.0620 ± 0.0730	2.3506 ± 0.0466	1.9561 ± 0.0469
A2	A1 + query cross-attention	3.2412 ± 0.1238	1.9592 ± 0.1052	0.997919 ± 0.000154	1.9581 ± 0.0289	2.2183 ± 0.0745	1.7915 ± 0.0472
A3	A2 + intended-beam metadata head	3.3835 ± 0.2616	2.1043 ± 0.2424	0.997733 ± 0.000331	1.9578 ± 0.1111	2.2586 ± 0.1422	1.9005 ± 0.1705
A4	Final QueryFieldNet: A3 + beam-basis residual	3.2400 ± 0.1433	1.9754 ± 0.1336	0.997918 ± 0.000174	1.8721 ± 0.0154	2.1339 ± 0.0471	1.7299 ± 0.0654

Bold indicates the best value and underline indicates the second-best value. Error metrics are lower-is-better, whereas $R^2$ is higher-is-better.

reports the resulting bit-wise macro averages. As summarized in Table 6, periodic encoding alone is non-monotonic, whereas query cross-attention gives the largest isolated improvement. The metadata head alone does not improve A2, but A4 reduces its three local errors by 4.4%, 3.8%, and 3.4%. This supports the joint metadata-conditioned residual path rather than an isolated metadata-head gain.

To verify that this gain is specific to multi-beam conditions rather than an overall performance shift,

Table 7. Beam-count-specific local metrics for the principal ablation variants. Entries are mean ± standard deviation over three runs.

Beam Count	Variant	Configuration	MB-HPBW RMSE	MB-Main-Lobe RMSE	Beam-Peak RMSE
1	A2	Query cross-attention	0.970 ± 0.032	1.188 ± 0.036	0.858 ± 0.036
1	A3	+ Metadata head	1.143 ± 0.112	1.338 ± 0.141	1.062 ± 0.167
1	A4	+ Beam-basis residual	1.015 ± 0.028	1.215 ± 0.038	0.907 ± 0.031
2	A2	Query cross-attention	1.423 ± 0.051	1.703 ± 0.089	1.261 ± 0.083
2	A3	+ Metadata head	1.508 ± 0.135	1.823 ± 0.170	1.391 ± 0.148
2	A4	+ Beam-basis residual	1.379 ± 0.007	1.664 ± 0.042	1.220 ± 0.039
3	A2	Query cross-attention	1.435 ± 0.045	1.753 ± 0.092	1.224 ± 0.064
3	A3	+ Metadata head	1.541 ± 0.161	1.885 ± 0.228	1.367 ± 0.169
3	A4	+ Beam-basis residual	1.395 ± 0.043	1.707 ± 0.083	1.219 ± 0.050
4	A2	Query cross-attention	1.463 ± 0.061	1.782 ± 0.088	1.263 ± 0.060
4	A3	+ Metadata head	1.551 ± 0.168	1.894 ± 0.214	1.417 ± 0.212
4	A4	+ Beam-basis residual	1.433 ± 0.101	1.745 ± 0.113	1.260 ± 0.130

decomposes the three intended-beam-local metrics of variants A2–A4 by intended-beam count. As shown in Table 7, A4 is slightly worse than A2 for the one-beam group, but it improves both lobe-region metrics for every two- to four-beam group, supporting a specifically multi-beam benefit.

3.7. Intended-Beam Metadata Head Diagnostics

The auxiliary head is evaluated on the common test split using three checkpoints, and the resulting diagnostics are reported in

Table 8. Intended-beam metadata-head performance. Entries are mean ± standard deviation over three runs.

Subset	Occupancy Exact Match	Occupancy Micro F1	Beam-Count Accuracy	Beam-Count MAE	Joint Topology Accuracy
All bits	0.981 ± 0.002	0.9938 ± 0.0009	0.977 ± 0.007	0.028 ± 0.007	0.971 ± 0.007
1-bit	0.920 ± 0.005	0.9718 ± 0.0028	0.912 ± 0.028	0.113 ± 0.026	0.887 ± 0.021
2-bit	0.985 ± 0.005	0.9951 ± 0.0020	0.975 ± 0.017	0.027 ± 0.020	0.968 ± 0.020
3–5-bit	1.000 ± 0.000	1.0000 ± 0.0000	1.000 ± 0.000	0.000 ± 0.000	1.000 ± 0.000

. Exact match requires all seven occupancy labels to be correct; joint accuracy additionally requires the beam-count class to be correct. As Table 8 indicates, the head learns the auxiliary task reliably; most errors occur at 1-bit, where coarse phase quantization reduces separation between beam combinations.

3.8. Joint Auxiliary-Loss Scale Sensitivity

All auxiliary loss weights were jointly scaled to 0.5, 1.0, and 1.5 times their reference values over three seeds, with all other settings fixed, and the corresponding test-set results are summarized in

Table 9. Three-run joint auxiliary-loss scale sensitivity of QueryFieldNet on the test set. Values are mean ± standard deviation.

Scale	RMSE	MAE	${\mathit{R}}^2$	MB-HPBW RMSE	MB-Main-Lobe RMSE	Beam-Peak RMSE
0.5×	3.251 ± 0.132	1.968 ± 0.116	0.99790 ± 0.00017	1.873 ± 0.045	2.165 ± 0.066	1.798 ± 0.083
1.0×	3.240 ± 0.143	1.975 ± 0.134	0.99792 ± 0.00017	1.872 ± 0.015	2.134 ± 0.047	1.730 ± 0.065
1.5×	3.248 ± 0.116	1.980 ± 0.120	0.99791 ± 0.00014	1.928 ± 0.050	2.203 ± 0.072	1.795 ± 0.074

. This experiment evaluates sensitivity to the overall auxiliary-loss scale and does not isolate the effect of each individual $\lambda$ value.

The 1.0× setting was fixed before test evaluation. The test-set sensitivity results in Table 9 are therefore reported only as a post-hoc robustness analysis: 1.0× gives the best mean result in five of six metrics, while 0.5× is lower only in MAE. No configuration was selected using the test results, and the small differences are interpreted as local robustness rather than global optimality.

The 1.0× setting was fixed before test evaluation. The test-set sensitivity results are reported only as a post-hoc robustness analysis: 1.0× gives the best mean result in five of six metrics, while 0.5× is lower only in MAE. No configuration was selected using the test results, and the small differences are interpreted as local robustness rather than global optimality.

3.9. Qualitative Model Comparison

Figure 4 shows a favorable 4-beam example in which QueryFieldNet follows the intended-beam peak and valley structure closely. It is qualitative; the preceding three-run tables provide the quantitative evidence.

Figure 5 provides a complementary case where BOBE has lower curve-level error, showing that QueryFieldNet’s advantage is aggregate intended-beam fidelity rather than superiority on every curve.

To move beyond a single selected failure curve,

Table 10. Subgroup-level QueryFieldNet wins over BOBE. Each entry is the number of groups with a better QueryFieldNet mean divided by the total number of compared groups.

Grouping	RMSE	MB-HPBW RMSE	MB-Main-Lobe RMSE	Beam-Peak RMSE
Bit depth	3/5	5/5	5/5	4/5
Bit depth × beam count	10/20	19/20	16/20	16/20
Bit depth × beam-center $\left|\theta \right|$	17/20	20/20	20/20	20/20

counts the subgroups in which the three-run mean of QueryFieldNet is better than that of BOBE. This is a descriptive analysis of the already reported subgroup results, not an additional training experiment.

The failure pattern is metric-dependent. Global-RMSE losses become frequent from 3-bit onward and occur for every beam count at 5-bit. In contrast, the only MB-HPBW loss among the 20 bit-depth–beam-count groups is the 5-bit one-beam group. MB-Main-Lobe and Beam-Peak losses are concentrated in the 4-bit four-beam group and the 5-bit one-, two-, and four-beam groups. In the complementary bit-depth–$\left|\theta \right|$ analysis, QueryFieldNet wins all 20 groups for each local metric, whereas the three global-RMSE losses occur at the nonzero 5-bit steering groups. Thus, the model’s aggregate advantage is robust for intended-beam neighborhoods but weaker for full-curve fitting at higher phase resolution, where BOBE can better fit sidelobes and low-gain regions not emphasized by the local objectives.

4. Discussion

Relative to BOBE, QueryFieldNet reduces MB-HPBW, MB-Main-Lobe, and Beam-Peak RMSE by 25.2%, 17.5%, and 23.2%, and wins 19 of 20 bit-depth–beam-count groups in MB-HPBW RMSE; BOBE remains competitive for high-bit full-curve error. Three-run ablation identifies query cross-attention as the strongest isolated component and shows that the complete metadata-conditioned residual improves two- to four-beam local prediction. QueryFieldNet-Lite supports the capacity-controlled comparison, and the post-hoc auxiliary-loss scale analysis shows local robustness around the pre-specified 1.0× weighting.

Although the query decoder is angle-conditioned, the current metadata head and beam-basis residual branch are defined on the fixed seven-point steering grid. Therefore, the present results should not be interpreted as validation of arbitrary unseen steering angles. The evidence also concerns the $19\times 19$ aperture, feed configuration, and array-factor-level simulation used here; other geometries require adaptation or retraining. Because full-wave coupling, material loss, fabrication tolerance, and measurement noise are not represented, a domain gap and corresponding accuracy degradation may occur in direct hardware application. Qualitatively, the intended-beam-local advantage is expected to persist under moderate domain shift because the query decoder and residual branch do not depend on the origin of the training curves. In contrast, absolute accuracy will degrade with unmodeled coupling, tolerance, and noise effects, and quantifying this degradation is part of the planned full-wave and measurement-based follow-up.

The sample-specific phase-level LUT is used as a synthetic diversity mechanism and does not represent one fabricated reflectarray with a fixed calibrated phase-state set. Deployment to a specific platform therefore requires evaluation or adaptation using its hardware-specific phase LUT.

5. Conclusions

This paper proposed QueryFieldNet for principal-cut wide-angle gain-curve prediction of fixed-$\theta$-grid multi-beam reflectarray antennas and defined three intended-beam-aware metrics. On the balanced theta7 benchmark, the three-run per-bit macro averages are RMSE $3.240\pm 0.143$, MAE $1.975\pm 0.134$, $R^2$ $0.99792\pm 0.00017$, MB-HPBW RMSE $1.872\pm 0.015$, MB-Main-Lobe RMSE $2.134\pm 0.047$, and Beam-Peak RMSE $1.730\pm 0.065$. The aggregate local metrics include both cuts; their $\varphi =90°$ component is a target-centered local-window error rather than a physical intended-beam HPBW or peak measurement. Relative to BOBE, the proposed model reduces the three local errors by 25.2%, 17.5%, and 23.2%, and it wins 19 of 20 bit-depth–beam-count groups in MB-HPBW RMSE. The 0.617M-parameter QueryFieldNet-Lite outperforms similarly sized neural baselines and, relative to BOBE, uses 99.82% fewer stored parameters while reducing all three beam-local errors, although its global RMSE and operation estimate are higher. Three-run ablation identifies angular query cross-attention as the largest contributor and shows that the metadata-conditioned residual is most useful for two- to four-beam local regions. The global advantage remains more limited. Broader steering grids and array configurations remain future work, together with full-wave or measured validation.