arXiv CSopen access

Multimodal Large Language Model Enabled Robust Beamforming for HAP Downlink Communications

1

Multimodal Large Language Model Enabled Robust Beamforming for HAP Downlink Communications

arXiv:2604.09017v1 [cs.NI] 10 Apr 2026

Xiaoyu Xing, Graduate Student Member, IEEE, Peng Yang, Member, IEEE, Guoquan Tao, Dingyi Lu, Zehui Xiong, Senior Member, IEEE, and Xianbin Cao, Senior Member, IEEE

Abstract—Small changes in high altitude platform (HAP) attitude can cause significant deviations in HAP downlink beam directions, thereby severely degrading HAP downlink communication performance. In this paper, we develop a multimodal large language model (LLM) enabled beamforming framework to achieve robust HAP downlink communications. Specifically, we design a vision-language LLM (VL-LLM) that learns from multivariate flight telemetry to forecast short-term HAP attitudes under platform shaking and support delay-aware proactive beam steering. We design an offline forecast-error calibration procedure to obtain upper bounds on forecast errors and improve the reliability of proactive analog beam steering. Based on the attitude forecasts, we proactively update the analog beamformer and propose a QoS-driven beamforming and admission method with a lightweight feasibility-enforcement step to satisfy instantaneous transmit-power and QoS requirements. Simulation results indicate that the designed VL-LLM can accurately capture changes in the HAP attitude and the proposed beamforming method achieves a 22.1% higher user service ratio and a 12.5% higher sum-rate than representative baselines. The measured mean and p99 total latencies are 36.24 ms and 40.13 ms, respectively, supporting practical delay-aware deployment. Index Terms—High altitude platform (HAP), platform shaking, multimodal large language model, robust beamforming

This causes noticeable main-lobe gain loss for uniform planar arrays (UPAs) [12], [13]. Therefore, the steering angles must be adjusted in both azimuth and elevation using instantaneous HAP attitude information. Otherwise, narrow UPA beams can incur substantial main-lobe gain loss. A practical solution is to forecast short-term HAP attitudes and proactively steer the beam before misalignment degrades link performance. However, in practical HAP downlinks, beam updates are not instantaneous. Attitude forecasting, communication optimization, and actuation together introduce non-negligible delay. As a result, a practical forecasting framework must support delay-aware proactive beam steering rather than idealized zero-latency compensation. Moreover, forecast errors translate into pointing uncertainty. Thus, the challenge extends beyond forecasting future attitudes. It also requires translating imperfect forecasts into robust hybrid beamforming decisions under quality-of-service (QoS) and power constraints. In this paper, we investigate delay-aware robust hybrid beamforming for HAP downlink communications by leveraging short-term HAP attitude forecasting.

A. State of the Art Overall, HAP communications face a fundamental challenge I. I NTRODUCTION in maintaining beam alignment across wide-area coverage, IGH-ALTITUDE platforms (HAPs) operate quasiparticularly at mmWave and higher frequencies where narrow stationary in the lower stratosphere and are a promising beams are essential [14], [15]. Even small platform attitude enabler for wide-area, on-demand coverage in beyond-5G/6G shaking can reduce array gain and degrade reliability and networks [1]–[6]. Compared with low-altitude unmanned aerial throughput, thereby motivating robust beam management. vehicles (UAVs), HAPs offer a much larger footprint and Recent studies can be broadly grouped into short-term beam multi-month endurance, while their 20-25 km altitude keeps tracking, large language model (LLM)-enabled communication propagation delay within a few milliseconds and avoids the intelligence, and learning-assisted beamforming. high deployment cost of satellites [7], [8]. To support high Short-term beam tracking and forecasting for air-based data rates over long-distance downlinks, mmWave links with platforms: A large body of work focuses on short-term beam antenna arrays are highly attractive for HAP communications, tracking. In [16], a multi-armed bandit beam-tracking scheme where hybrid analog/digital beamforming can be explored to for HAPs sequentially learns the best beam and achieves reduce radio frequency (RF) chain cost and power consumption outage probability close to an ideal direction-of-arrival tracker [9]–[11]. while keeping complexity linear in the codebook size. In A key challenge in HAP beamforming is maintaining reliable [17], a lightweight global positioning system (GPS)-aided beam alignment under continuous platform attitude shaking. deep learning model extracts motion-pattern features via a However, HAPs are continuously perturbed by wind and convolutional neural network (CNN) and uses an encoderturbulence. Even small attitude shaking can tilt the array plane decoder gated recurrent unit (GRU) to forecast future beam and misalign the beam with the line-of-sight (LoS) direction. indices over multiple slots. In [18], an attitude compensationbased beam-tracking (ACBT) method iteratively performs X. Xing, P. Yang, D. Lu, and X. Cao are with the School of Electronic Information Engineering, Beihang University, Beijing 100191, China. G. Tao attitude compensation and beam tracking to cope with windis with the School of Institute of Unmanned Systems, Beihang University, induced UAV attitude shaking. Beijing 100191, China. Z. Xiong is with the School of Electronics, Electrical However, these works [16]–[18] mainly focus on beam Engineering and Computer Science, Queen’s University Belfast, Belfast BT7 1NN, U.K. tracking or forecasting accuracy, leaving the translation of fore-

H

2

cast errors into analog beamforming uncertainty and multiuser QoS-aware beamforming decisions largely unaddressed. This motivates short-term proactive beam steering with robustness considerations over the next few slots. Foundation models and LLMs for communications: Recent works have begun to explore LLMs for communicationrelated tasks. In [19], the authors discuss key challenges in enabling existing pre-trained LLMs to interpret and reason over communication system data. In [20], the authors construct a technical-document dataset to facilitate domain adaptation and knowledge injection for communications-oriented LLM applications. In [21], the authors show that relatively small language models, when augmented with retrieval-augmented generation (RAG), can already capture and utilize communication knowledge effectively. In [22], the authors further refine the RAG method for specialized technical corpora to improve retrieval reliability and downstream reasoning performance for telecommunications. In [23], the authors introduced a generative AI agent framework for satellite networks, in which LLMs and retrieval-augmented generation (RAG) were used for customized system modeling and a mixture-of-experts (MoE) method was employed for transmission optimization. In [24], the authors studied the use of large models for aerial edges through an integrated air-ground edge-cloud framework, with an emphasis on bandwidth-aware task allocation, data transmission, and model evolution for UAV-based multimodal inference. In [25], the authors combined channel extrapolation and generative AI for CSI generation in dynamic wireless environments, and showed that diffusion-based generative models improved robustness in time-varying communication tasks. However, these works [19]–[25] mainly target communication knowledge utilization, customized network modeling, edge-cloud large-model collaboration, or security-oriented CSI generation. They do not study multivariate HAP attitude forecasting for proactive beam steering or its coupling with robust hybrid beamforming. Learning-assisted beamforming: Beamforming problems are commonly addressed using two classes of algorithmic tools. The first class relies on model-based iterative solvers, such as weighted minimum mean-square error (WMMSE) and successive convex approximation (SCA). Recent WMMSEbased designs still involve multi-block alternating optimization with repeated variable updates [26]–[28], while SCA-based designs require iteratively forming and solving convex surrogate subproblems [29], [30]. As a result, runtime is often dominated by iteration counts and repeated numerical solves, which is challenging for online deployment, especially as the number of users and antennas grows. The second class uses learningbased forecasters for fast inference. However, without explicit feasibility guarantees, such forecasters may output infeasible admission decisions or beamformers, leading to QoS violations, which can compromise reliability in online deployment [31], [32]. To tackle the above issue, recent works have blended learning with optimization to accelerate resource allocation while preserving the feasibility of solutions. In [31], the authors introduced a knowledge distillation framework where an optimization routine guided a neural forecaster. In [32],

the authors demonstrated karush-kuhn-tucker (KKT) guided dual learning, where learning a small number of dual variables enabled structured beamformer reconstruction with lower computational complexity. In [33], differentiable projection modules were used to project network outputs onto constraint sets. To avoid infeasible solutions, [34] developed homeomorphic polar learning using an invertible mapping that strictly satisfied convex or non-convex constraints. However, most learning-assisted beamforming methods [31]– [34] assume that the analog beam setting is fixed or perfectly known. They do not explicitly model the beam pointing errors caused by imperfect forecasts. This is critical in forecast-aided hybrid beamforming for HAP downlinks. In summary, prior works typically address beam tracking, communication intelligence, dynamic CSI generation, or learning-based beamforming in isolation. The joint design of multivariate HAP attitude forecasting and robust hybrid beamforming remains largely unexplored. This includes the explicit modeling of forecast-error-induced beam pointing uncertainty and its impact on multiuser QoS guarantees. B. Motivations and Contributions Overall, most existing schemes remain reactive, and beam correction is activated only after misalignment occurs, introducing correction delay and transient array-gain loss. This motivates a forecast-then-beamform method. We forecast shortterm HAP attitudes and proactively perform beamforming, thereby reducing beam misalignment before it causes noticeable link-performance degradation. Accurate attitude forecasting from multivariate flight telemetry is nontrivial and is naturally a time-series forecasting problem. In the time-series domain, recent works [35]–[38] show that LLMs can be repurposed for forecasting when inputs are appropriately encoded. This design reuses the strong sequence modeling ability of LLMs. It also allows domain instructions or priors to be injected through prompts with limited extra parameters. Researchers [39], [40] have converted time series into image-like representations for vision-based modeling. This transformation exposes temporal structures as spatial patterns. It allows CNN/vision transformer (ViT) [41] to extract hierarchical features that capture latent temporal relations. However, LLM-based methods [36]–[38] face a modality gap between continuous time-series and discrete tokens, and they are not explicitly pre-trained for fine-grained temporal patterns, which can limit forecasting accuracy. Vision-based methods [39], [40] often provide limited semantic interpretability, which makes it difficult to incorporate domain knowledge or task-specific priors. This motivates a vision-language (VL) forecasting model that combines visual feature extraction with LLM sequence modeling to improve short-term attitude forecasting for proactive beam steering. Moreover, proactive attitude-aware beam steering must update the analog beam alignment for the next few time slots. Meanwhile, the downlink still requires online multiuser digital beamforming and user admission to satisfy transmit-power and QoS constraints. More broadly, recent robustness-oriented studies in Internet of Things

3

(IoT) systems also reinforce the view that robustness should be treated as a primary design objective in dynamic networked systems [42]–[45]. These observations suggest that the core challenge is not forecasting alone, nor beamforming alone, but the conversion from imperfect attitude forecasts to robust and feasible hybrid beamforming actions. This motivates our forecast-thenbeamform framework. Specifically, a VL forecasting model generates short-term attitude forecasts from multivariate flight telemetry, offline calibration converts forecast errors into analog-pointing robustness margins, and a QoS-driven digital beamforming and admission stage ensures instantaneous online feasibility. The main contributions of this paper are summarized as follows:

TABLE I K EY S YMBOLS U SED IN THE P ROPOSED F RAMEWORK Symbol ∆T Tlat d Hpred Tt At+h|t Aτ ∆ω τ δω ρ, ρs αk rkmin QAR NRF

Meaning Slot duration End-to-end processing latency Decision delay Forecast horizon Target-slot set Forecast-based analog beamformer scheduled for slot t + h Analog beamformer actually applied in target-slot τ Pointing mismatch induced by forecasting error Offline-calibrated pointing-error bound Confidence parameters in the certificates Admission indicator of user k Minimum rate requirement of user k QoS admission ratio Number of RF chains

Frequently used paper-specific symbols are summarized in Delay-aware forecast-then-beamform framework for Table I. HAP downlinks. We propose a multimodal framework that couples short-term HAP attitude forecasting with II. S YSTEM M ODEL AND P ROBLEM F ORMULATION robust hybrid beamforming. The forecast attitudes provide timely input for proactive analog beam steering under A. System Model practical forecast-to-actuation delay. As shown in Fig. 1, we consider an HAP downlink • Vision-language forecasting model for short-term communication system where an HAP equipped with a attitude forecasting. We develop a VL-LLM forecasting Mx × My UPA (M = Mx My ) serves K single-antenna ground model that combines vision-based feature extraction and users indexed by K = {1, . . . , K}. The HAP is located at LLM-based sequence modeling, leveraging their comple- rHAP = (xHAP , yHAP , hHAP ) and the user k is located at mentary strengths to improve multivariate HAP attitude rk = (xk , yk , 0). However, stratospheric winds may perturb forecasting for proactive beam steering. the HAP attitude, leading to misalignment between the desired • Offline calibration and QoS-driven beamforming with beam pointing and the actual UPA beam. We use a slotted lightweight feasibility enforcement. We design an offline time model and assume that the HAP attitude remains constant calibration procedure that estimates a high-confidence within each slot. bound on the attitude forecasting error from an indepen1) Forecast horizon model: As shown in Fig. 2, we consider dent validation set. The calibrated bound is used to account a slotted timeline. The figure shows successive forecast horizons for pointing uncertainty in analog steering, while the generated at slot t and at the subsequent slot. At the beginning proposed QoS-driven digital beamforming and admission of slot t, the forecasting model outputs an Hpred -step attitude stage ensures instantaneous transmit-power and user-rate forecast. Due to the end-to-end processing time Tlat , the feasibility online. updated beamformer cannot take effect immediately. We define • Finally, we compare the proposed methods with represen- the resulting decision delay as   tative forecasting and beamforming baselines and conduct Tlat comprehensive ablation and robustness studies. Simulation , (1) d≜ ∆T results show that the proposed framework improves both attitude forecasting accuracy and communication service where ∆T is the slot duration. Therefore, only the forecasts for performance over representative baselines. horizons h ∈ {d+1, . . . , Hpred } are available for computing N otation: Bold uppercase and lowercase letters (for exam- the analog beamformer sequence. Accordingly, we define the target-slot set as ple, A and x) denote matrices and column vectors, respectively. √ Plain letters denote scalars. The imaginary unit is j = −1. Tt ≜ {t+d+1, . . . , t+Hpred }, |Tt | = Hpred − d. (2) Rp×q and Cp×q denote the sets of real and complex matrices. (·)⊤ and (·)H denote transpose and Hermitian transpose. | · | which includes the future slots within the forecast horizon that denotes the absolute value, ∥·∥2 the Euclidean norm, and ∥·∥F remain available after the decision delay. We then compute an analog beamformer sequence the Frobenius norm. E{·} denotes expectation and ≜ denotes Hpred definition. ⊗ denotes the Kronecker product and diag(·) forms {At+h|t }h=d+1 for τ = t + h ∈ Tt . If a slot τ belongs to a diagonal matrix. For vectors, min(·, ·) and max(·, ·) operate multiple target-slot sets Tt′ , we use the beamformer computed elementwise, and [·]+ denotes elementwise projection onto R+ . at the most recent forecast time, i.e., CN (µ, σ 2 ) denotes a circularly symmetric complex Gaussian Aτ = Aτ |t⋆ (τ ) , t⋆ (τ ) ≜ max{t′ : τ ∈ Tt′ }. (3) distribution with mean µ and variance σ 2 . atan2(·, ·) denotes the four-quadrant arctangent. For a rotation matrix R ∈ SO(3), 2) Hybrid beamforming signal model: Hybrid beamforming log(R) ∈ so(3) denotes the matrix logarithm mapping to the is widely adopted to reduce the number of RF chains and Lie algebra of SO(3), and vee(·) : so(3) → R3 maps a skew- the associated cost and power consumption [10], [11]. We symmetric matrix to its corresponding vector representation. adopt an analog beamformer A ∈ CM ×NRF and a digital •

4

Fig. 1. Downlink beam misalignment for a HAP under windinduced attitude shaking. The HAP carries an Mx ×My UPA. Dashed lobes denote desired beams, and solid lobes denote actual beams under attitude errors.

Fig. 2. Forecast horizon with decision delay d and target-slot set Tt = {t+d+1, . . . , t+Hpred }. The figure shows the case Hpred = 2d.

beamformer D ∈ CNRF ×K . At the beginning of slot t, we Accordingly, we model the channel of user k in slot t as r r compute an analog beamformer sequence for Tt based on the κk 1 LoS HAP attitude forecasts, and then optimize D online in each hk (t) = hk (t) + hNLoS (t), (9) k κ + 1 κ k k +1 target-slot using the instantaneous effective channel Heff (τ ) after applying the analog beamformer. For clarity, we set where κk is the Rician factor, hLoS (t) is determined by the k NRF = K,and the case NRF < K can be handled by user attitude-dependent steering geometry, and hNLoS (t) captures k scheduling/grouping and serving fewer data streams. The trans- the diffuse component. H mitted signal is x(t) = ADs(t) with E{s(t)s (t)} = IK . Let We next specify how the attitude forecast is translated into D = [d1 , . . . , dK ], where dk is the digital beamformer for user phase-only analog beamforming in each target-slot. Consider k. The instantaneous transmit power satisfies ∥AD∥2F ≤ Pmax , a target-slot τ = t + h with h ∈ {d+1, . . . , Hpred }. Given and a constant circuit power Pc is included for energy efficiency the forecast attitude (ψb b b t+h|t , θt+h|t , ϕt+h|t ), we map the LoS (EE) evaluation. direction of user k into the corresponding body frame as [10], Let hk (t) ∈ CM denote the downlink channel from [11] the HAP array to user k in slot t, and define H(t) ≜ (B) uk,t+h|t = RW→B (ψbt+h|t , θbt+h|t , ϕbt+h|t ) ek , (10) [h1 (t), . . . , hK (t)] ∈ CM ×K . The received signal at user k is yk (t) = hH k (t)ADs(t) + nk (t),

(4)

where nk (t) ∼ CN (0, σ 2 ) denotes noise. The instantaneous SINR of user k is SINRk (t) = P

j̸=k

hH k (t)Adk

2

hH k (t)Adj

2

+ σ2

,

(5)

and the achievable rate is
Rk (t) = B log2 1 + SINRk (t) ,

(6)

where B is the system bandwidth. 3) Modeling of impact of HAP shaking: Let W and B denote the world frame and body frame, respectively. The ZYX attitude at time t is (ψt , θt , ϕt ) and the corresponding rotation matrices are [46] RB→W (t) = Rz (ψt ) Ry (θt ) Rx (ϕt ), RW→B (t) = R⊤ B→W (t).

(B)

where uk,t+h|t = [ux , uy , uz ]⊤ . The corresponding steering angles are
(ϑek,t+h|t , φ ek,t+h|t ) = arccos(uz ), atan2(uy , ux ) . (11) For the UPA, we use inter-element spacings (dx , dy ) and index elements by (m, n) with m = 0, . . . , Mx − 1 and n = 0, . . . , My − 1. The phase-only analog beam for user k is implemented by per-element phase shifters following standard hybrid beamforming and array-response constructions [10], [11] as   1 2π  wk,t+h|t m,n = √ exp j mdx sin ϑek,t+h|t cos φ ek,t+h|t λc M !  e + ndy sin ϑk,t+h|t sin φ ek,t+h|t . (12)

(7)

We represent the attitude by a rotation matrix R(t) ∈ SO(3), where λc is the carrier wavelength. With NRF RF chains, the which avoids the discontinuities and coupling issues of Euler- analog beamformer used in target-slot t + h is formed by stacking user beams [10], [11] as angle parameterizations [47].   Over the short forecast horizon, we assume a quasi-static At+h|t = w1,t+h|t , . . . , wK,t+h|t ∈ CM ×NRF , geometry. Ground users are stationary. Within each slot, the 1 LoS direction and distance are treated as constant and are given |[At+h|t ]p,q | = √ . (13) M by ek ≜

rk − rHAP , ∥rk − rHAP ∥

H

dk ≜ ∥rk − rHAP ∥.

(8)

pred During online operation, {At+h|t }h=d+1 is applied across the target-slots in Tt according to (3), while the digital

5

beamformer is updated per target-slot using the instantaneous channel estimate. For a target-slot τ , the analog beamformer is fixed as Aτ |t⋆ (τ ) . At the beginning of slot τ , the HAP estimates the instantaneous channel and forms Heff (τ ) = HH (τ )Aτ |t⋆ (τ ) .

(14)

Based on Heff (τ ), the digital beamformer is computed online under the instantaneous power and rate constraints. b τ |t⋆ (τ ) and In target-slot τ , denote the forecast attitude by R the realized attitude by Rτ . We capture the overall pointing error relevant to analog beamforming by the relative rotation 
 b ⊤ ⋆ Rτ ∈ R3 , ∆ω τ ≜ vee log R (15) τ |t (τ ) which captures the pointing mismatch caused by attitude forecast errors. In online digital beamforming, ∆ω τ is treated as an uncertainty. In offline calibration, it is computed from ground-truth attitudes. We focus on the pointing mismatch caused by attitude forecast errors. For user k, the normalized detuning vector ξ k = [ξx,k , ξy,k ]⊤ admits a first-order model   ξx,k ≈ Jξ,k ∆ω τ , (16) ξy,k where ξ k is a 2D detuning in the direction-cosine domain and Jξ,k ∈ R2×3 is the corresponding Jacobian evaluated at the operating point associated with Aτ |t⋆ (τ ) . Under small detuning, the normalized array factor yields a quadratic main-lobe gain loss 2 2 ∆GUPA,k ≈ cx ξx,k + cy ξy,k , 2 cx = π3 (Mx2 − 1),

2 cy = π3 (My2 − 1).

If ∆ω τ has mean µω and covariance Σω , then   ξ E x,k = Jξ,k µω , Σξ,k = Jξ,k Σω J⊤ ξ,k . ξy,k We further define a scalar uncertainty proxy
2 σ bξ,k ≜ tr Σξ,k ,

(17)

(18)

(19)

B. Problem Formulation We focus on a snapshot at the target-slot τ . When serving all users is infeasible under the instantaneous power budget Pmax , we introduce binary admission indicators αk ∈ {0, 1}, where αk = 1 indicates that user k is admitted and must satisfy its QoS target. The QoS admission ratio (QAR) is defined as K

1 X αk . K k=1

h∈{d+1,...,Hpred }

where ∆ω t+h|t is the pointing mismatch induced by using the h-step-ahead forecast generated at time t, 
 b ⊤ Rt+h ∈ R3 , ∆ω t+h|t ≜ vee log R t+h|t h ∈ {d+1, . . . , Hpred }.

(22)

The radius δω is obtained offline on an independent validation/calibration split by taking an empirical (1 − ρ)-quantile of the target-window maxima, following standard forecasting practice and recent conformal methods for multi-step timeseries forecasting [48]. Assumption A2 (Effective angular region). Let (ϑk , φk ) denote the forecast body-frame steering angles used to form the analog beam in the target-slot, i.e., (ϑk , φk ) ≜ (ϑek,τ |t⋆ (τ ) , φ ek,τ |t⋆ (τ ) ). For each user k, there exists an angular region Sk such that
P (ϑk , φk ) ∈ Sk ≥ 1 − ρs , (23) where ρs ∈ (0, 1) is a small confidence parameter. The region Sk is specified offline from the service geometry and validation data, consistent with sector-based descriptions used in aerial beamforming [49]. For small detuning, the main-lobe gain loss can be approximated by the quadratic model in (17). Combining (16) and (17), we can write ∆GUPA,k ≈ ∆ω ⊤ τ Qk (ϑk , φk ) ∆ω τ ,

which summarizes the total detuning variance along the two cosine-domain. These error measures will be used to impose robustness constraints in the problem formulation below.

QAR ≜

1) Robustness modeling and offline-calibrated certificates: Such pointing errors can reduce the main-lobe gain of narrow UPA beams. We describe ∆ω τ using an offline-calibrated error model and use it to derive tractable robustness constraints. We consider a bounded-error model and a moment-based model. We introduce a design tolerance ϵ > 0 that specifies the maximum tolerable normalized main-lobe gain loss due to pointing errors. Assumption A1 (Offline-calibrated target-window pointing bound). There exist a confidence level 1 − ρ with ρ ∈ (0, 1) and a radius δω > 0 such that, for each forecast time t,   P max ∆ω t+h|t 2 ≤ δω ≥ 1 − ρ, (21)

(20)

Qk (ϑ, φ) ≜ Jξ,k (ϑ, φ)⊤ diag(cx , cy )Jξ,k (ϑ, φ).

(24)

To obtain a conservative constant that is valid over the angular region Sk , we take a worst-case spectral bound
L2k ≜ max λmax Qk (ϑ, φ) . (25) (ϑ,φ)∈Sk

Lemma 1. Offline-calibrated pointing-feasibility certificate. Under Assumptions A1-A2, define L2k by (25). On the event  maxh∈{d+1,...,Hpred } ∥∆ω t+h|t ∥2 ≤ δω ∩ {(ϑk , φk ) ∈ Sk }, if L2k δω2 ≤ ϵ, then the quadratic approximation of the mainlobe gain loss satisfies ∆GUPA,k ≤ ϵ for user k over all target-slot horizons h ∈ {d+1, . . . , Hpred } programmed at time t. Consequently, the certificate holds with probability at least 1 − (ρ + ρs ). Proof. Please refer to Appendix 1 in the supplementary material.

6

Lemma 1 provides an offline-calibrated pointing-feasibility certificate. In particular, enforcing L2k δω2 ≤ ϵ guarantees ∆GUPA,k ≤ ϵ on the calibrated target-window event, with probability at least 1 − (ρ + ρs ). This motivates the sufficient robustness constraint in (27d), which introduces no additional online decision variables once Aτ is fixed. In addition to the bounded-error certificate above, we also consider a moment-based condition. If ∆ω τ has mean µω and covariance Σω , then E[∆GUPA,k ] = µ⊤ ω Qk (ϑk , φk )µω + tr(Qk (ϑk , φk )Σω ) ≤ ϵ, (26) which follows from standard second-moment analysis [50]. 2) QoS-driven two-stage optimization: We adopt a QoSdriven policy. We first maximize QAR and then refine the EE on the admitted set. Combining QoS, power, the analog constantmodulus structure, and the above certificate, the snapshot constraints are written as Rk (Aτ , D) ≥ αk rkmin , ∀k,

(27a)

∥Aτ D∥2F

≤ Pmax , (27b) n Aτ ∈ Apred = Aτ ∈ CM ×NRF : |[Aτ ]p,q | = √1M , o ∀p = 1, . . . , M, q = 1, . . . , NRF , (27c) αk L2k δω2 ≤ αk ϵ, ∀k.

(27d)

Here, α = [α1 , . . . , αK ]⊤ collects the binary admission indicators. The analog beamformer Aτ is generated from the HAP attitude via (10)-(13) and therefore satisfies the constantmodulus constraint (27c) by construction. The certificate parameter δω is calibrated offline from forecast errors over the target-slot horizons h ∈ {d+1, . . . , Hpred }. Given the analog beamformer Aτ fixed for the target-slot τ , we adopt a QoS-driven two-stage formulation. Stage I maximizes QAR and is formulated as

Fig. 3. Overall VL-LLM method for forecast-aided analog beamforming in HAP downlinks.

hardness, the analog beamformer must be set in advance based on the forecast attitudes over the horizon, whereas the digital beamformer must be computed online using instantaneous CSI. Meanwhile, attitude forecast errors introduce pointing uncertainty, and robustness-aware design is needed without incurring high online complexity. These gaps motivate the learning-assisted reconstruction and strict feasibility repair developed in Sec. IV, while Sec. III specifies how VL-LLM Hpred produces {At+h|t }h=d+1 and how forecast errors over the target-slot window calibrate the certificate parameter used above.

III. VL-LLM FOR F ORECAST-A IDED A NALOG B EAMFORMING Recent time-series forecasting methods for multivariate HAP flight telemetry can be broadly categorized into visionbased methods [39], [40] and LLM-based methods [36]– [38]. However, LLM-based methods face a modality gap between continuous-valued time series and discrete tokens, and they are not explicitly pre-trained for fine-grained temporal patterns, which can limit forecasting accuracy. Vision-based K 1 X maximize αk (28a) methods often provide limited semantic interpretability, which D, α K makes it difficult to incorporate domain knowledge or taskk=1 subject to (27a), (27b), (27d), (28b) specific priors. These limitations motivate a vision-language forecasting model that combines visual feature extraction αk ∈ {0, 1}, ∀k. (28c) with LLM sequence modeling to improve short-term attitude Let α⋆ be an optimal admission decision of (28). Stage II forecasting for proactive analog beamforming under decision delay. To address these limitations, we combine vision-based maximizes EE over the admitted set and is formulated as feature extraction with prompt-based LLM conditioning. The PK ⋆ k=1 αk Rk (Aτ , D) vision module renders multivariate flight telemetry as visual maximize (29a) D ∥Aτ D∥2F + Pc representations so that a vision backbone can capture local subject to Rk (Aτ , D) ≥ αk⋆ rkmin , ∀k, (29b) dynamics, cross-variable coupling, and periodic structures. (27b). (29c) The language module acts as a prompt-conditioned sequence forecaster that injects domain-aware task priors into multi-step The resulting problems are non-convex and include binary attitude forecasting. admission variables. Even with fixed Aτ , the SINR/rate conAs shown in Fig. 3, the input to VL-LLM is a length-Lwin straints are non-convex due to multiuser interference coupling multivariate telemetry window Xt ∈ RNvar ×Lwin . The model in (5). Moreover, Stage I introduces binary admissions, leading contains three complementary modules. First, a telemetry-toto a mixed-integer non-convex program (MINCP), while Stage vision module converts Xt into vision tokens for capturing II features a fractional EE objective under instantaneous QoS local temporal dynamics, periodic structures, and frequencyand power constraints. Such coupled formulations are widely sensitive patterns. Second, a small set of learned cross-variable recognized as challenging to solve at scale under stringent real- adapter (CVA) tokens summarizes inter-channel coupling and time processing constraints [51], [52]. Beyond computational provides a compact token-level representation of cross-variable

7

dependencies. Third, a text module injects task instructions and lightweight window statistics into the frozen LLM. The resulting multimodal token sequence is then fed into a frozen LLM with lightweight adaptation. This frozen LLM acts as a prompt-conditioned forecaster and outputs an Hpred -step forecast of yaw, pitch, and roll. Due to the decision delay d, only the target-slot horizons h ∈ {d+1, . . . , Hpred } are used to form the analog beamformer sequence for the target-slot set Tt . In each target-slot τ ∈ Tt , the digital beamformer is still optimized online from the instantaneous channel estimate.

A. Telemetry-to-Vision Module The vision module renders multivariate flight telemetry into a pseudo-image so that local transitions, cross-channel structures, and frequency-sensitive patterns can be encoded by a vision backbone. Given Xt ∈ RNvar ×Lwin , where Nvar is the number of variables and Lwin is the look-back window length, we apply channel-wise reversible instance normalization (RevIN) [53] as e t = RevIN(Xt ). X (30) We form first-order differences along the time axis. We set e t [:, ℓ] − X e t [:, ℓ − 1] for ∆Xt [:, 1] = 0 and define ∆Xt [:, ℓ] = X ℓ = 2, . . . , Lwin . We compute the real fast Fourier transform (FFT) magnitude of each variable along the time axis and interpolate it to length Lwin .
e t ) = Interp→L |rFFT(X e t )| . F(X win

(31)

B. Cross-Variable Adapter Module While the telemetry-to-vision module is effective at extracting rendered local patterns, it does not explicitly provide a compact token-level summary of cross-variable coupling. To supplement this information, we introduce a lightweight CVA that aggregates inter-channel dependencies into a small set of learnable adapter tokens before the frozen LLM. This module utilizes a small set of learnable query tokens and multi-head cross-attention over per-variable features to output a fixed number of adapter tokens. Given Xt ∈ RNvar ×Lwin , we take the most recent Ltail steps (Ltail ≤ Lwin ) and form per-variable tail e tail [i, :] ∈ RLtail for i = 1, . . . , Nvar . Each variable patches X t patch is projected to a latent width dcva as e tail [i, :]+bvar ∈ Rdcva , vi = Wvar X t

i = 1, . . . , Nvar . (33)

where Wvar ∈ Rdcva ×Ltail and bvar ∈ Rdcva are shared learnable projection parameters. We stack Vvar = [v1 ; . . . ; vNvar ] and use Ncv learnable queries to attend over variables via multi-head cross-attention as Hcva = MHA(Q, Vvar , Vvar ) ∈ RNcv ×dcva .

(34)

where Q ∈ RNcv ×dcva denotes learnable query tokens, Vvar is used as both keys and values. By employing a fixed number of learnable query tokens Ncv , the adapter output length is independent of Nvar , which effectively bounds the additional computation introduced by the cross-variable module. A lightweight residual block refines the adapter tokens as
U = LN(Q + Hcva ), U ← LN U + FFN(U) , (35)

where LN(·) denotes layer normalization applied along the where rFFT(·) denotes the one-sided real FFT along the time feature dimension, and FFN(·) denotes a position-wise feedaxis, and Interp→Lwin (·) denotes interpolation that resamples forward network that preserves the token length and feature the one-sided magnitude spectrum to length Lwin for consistent dimension. We finally project them to the LLM width as stacking. c = UWout ∈ RNcv ×dLLM . M (36) We form a multivariate tensor by concatenating raw values, first-order differences, FFT magnitudes, and sinusoidal where Wout ∈ Rdcva ×dLLM is a learnable linear projection codes along the channel axis. This process is gated by that maps the CVA latent width dcva to the LLM embedding sraw , s∆ , sfft , sper ∈ {0, 1} and is expressed as width dLLM . Consequently, CVA produces Ncv tokens that are  appended to the LLM input sequence. Acting as a compact e t , s∆ ∆Xt , Xtvis =concatc sraw X bridge between raw telemetry and the frozen backbone, these  e t ), sper S e t , sper C e t ∈ RCin ×Nvar ×Lwin , (32) tokens encode inter-variable interactions. sfft F(X where concatc (·) denotes concatenation along the channel C. Text Module e t and C et Alongside the telemetry-to-vision and CVA modules, we use dimension and Cin = sraw + s∆ + sfft + 2sper , S represent adaptive per-variable sinusoidal codes. Raw values a text module to inject task priors and lightweight windowpreserve short-term motion states, while first-order differ- level summaries into the frozen LLM. This module provides ences emphasize local transitions. FFT magnitudes summarize structured task conditioning for short-term HAP attitude dominant frequency content, and sinusoidal codes provide forecasting rather than free-form language generation. Table II presents a representative prompt structure used in lightweight periodicity cues. These components are stacked to form a compact multi-channel visual tensor that integrates the text module. The conditioning comprises high-level task temporal, spectral, and periodic information. We convert Xtvis priors, structured output constraints for numerical forecasting, to a pseudo-RGB image using a lightweight convolutional stem and local context descriptors derived from the input window. which maps Cin channels to three channels for compatibility We summarize trend statistics using a per-channel least-squares et [i, ℓ] denote the RevINwith image backbones. We then resize the image to (H0 , W0 ) slope over the look-back window. Let X of channel i at index ℓP∈ {1, . . . , Lwin }, and apply the standard preprocessing required by the selected normalized value P Lwin Lwin e 1 1 vision backbone. and let ℓ̄ = Lwin ℓ=1 ℓ and X̄i = Lwin ℓ=1 Xt [i, ℓ]. To

8

The last-layer hidden states of the frozen LLM are given by  (L)  (L) (L) HLLM = HLLM,txt ; HLLM,cont ∈ R(Ttxt +Ncont )×dLLM , Component Content (42) (L) Role Expert HAP attitude forecaster where HLLM,txt corresponds to the retained text tokens Input Last {seq len} time steps (L) Task Forecast next {pred len} steps for proactive beam steering with length Ttxt after truncation, and HLLM,cont collects the Output format {enc in} real values per step, channel-ordered remaining content tokens, including the Ncv CVA tokens and Guidance Temporal consistency, trends, periodic patterns the Nvis vision tokens. A lightweight temporal mixer Π(·) Context Window statistics: slope and dominant period per channel models the content-token sequence into a compact context vector g for direct multi-step forecasting, formulated as instantiate the prompt in a lightweight and reproducible manner,
(L) we summarize each channel by extracting specific trend and g = Π HLLM,cont . (43) periodicity descriptors from the current look-back window. We compute the slope as   ψbt+1:t+Hpred |t , θbt+1:t+Hpred |t , ϕbt+1:t+Hpred |t =
PLwin et [i, ℓ] − X̄i   (ℓ − ℓ̄) X gyaw (Γ(g)), gpitch (Γ(g)), groll (Γ(g)) , (44) , i = 1, . . . , Nvar , (37) κi = ℓ=1 PLwin 2 ℓ=1 (ℓ − ℓ̄) + εκ where each axis-wise head outputs an Hpred -dimensional where the variable κi approximates the signed temporal trend sequence, and Γ(·) denotes an axis router/gating module. Each per channel and εκ > 0 is a small constant for numerical head g , g yaw pitch , groll maps the context vector to an Hpred stability. dimensional sequence, a direct multi-horizon forecasting that To capture periodicity, we estimate a dominant period from avoids error accumulation across steps. the peak of the one-sided FFT power spectrum while excluding The forecast attitudes are converted into the analog beamthe zero-frequency component. This is achieved through the former sequence for the target-slot window according to the following formulations geometric and steering relations in Section II-A, while the
e digital beamformer is refined online using the instantaneous Fi [f ] = rFFT Xt [i, 1:Lwin ] [f ], effective channel. Lwin fi⋆ = arg max |Fi [f ]|2 , Pbi ≈ ⋆ , (38) f ≥1 fi E. Training and Offline Calibration where Fi [f ] denotes the one-sided FFT coefficient at frequencyThe forecasting model is trained for direct multi-horizon bin index f for channel i. We additionally include dispersion forecasting so that all future forecast steps are produced in summaries as v one shot, which is consistent with the delay-aware target-slot u win u 1 LX
beamformer framework. Given a length-Lwin history ending 2 e t [i, ℓ] − X̄i . X (39) at t, the model forecasts an H ςi = t pred -step attitude sequence Lwin TABLE II R EPRESENTATIVE PROMPT STRUCTURE FOR THE TEXT MODULE

ℓ=1

Let Ω denote fixed domain tokens (HAP/beam task), I the instruction tokens, and Pt the textualization of Nvar var b Nvar {{κi }N i=1 , {Pi }i=1 , {X̄i , ςi }i=1 }. A tokenizer and embedding map Tok(·) converts this structured information into vectors according to
Etext = Tok [Ω; I; Pt ] ∈ RTtext ×dLLM . (40) D. Multimodal Fusion and Output Heads The multimodal fusion stage combines task-conditioned text tokens, CVA tokens, and vision tokens into a unified sequence for direct multi-horizon attitude forecasting. We form the multimodal input sequence to the frozen LLM as   c ; TruncN (Vt ) , Zt = TruncTmax (Etext ) ; M (41) max where TruncL (·) keeps the first L tokens along the token dimension. The resulting multimodal sequence Zt is fed to the frozen LLM, and the output representations are further processed by a lightweight mixer and axis-wise heads for multi-horizon forecasting. This concatenation preserves the complementary roles of the three modules where the text tokens provide task priors, the CVA tokens summarize cross-channel dependencies, and the vision tokens encode rendered telemetry patterns.

bt+s|t ≜ (ψbt+s|t , θbt+s|t , ϕbt+s|t ), a

s = 1, . . . , Hpred .

(45)

Let Yt+1:t+Hpred ∈ RHpred ×C denote the ground-truth b t+1:t+H attitude sequence and Y ∈ RHpred ×C denote the pred corresponding forecast sequence, where C is the number of attitude channels. The dataset standardizes the targets using scalers fitted on the training split, and we invert the scaling inside the loss so that errors are measured in the original physical units. We minimize a weighted Huber regression loss PHpred PC ws wc Huberδ (∆s,c ) , (46) Lreg = s=1 PHc=1 PC pred s=1 c=1 ws wc where δ is the Huber threshold, wc are axis/channel weights, and ws are horizon weights. The per-step error is ∆s,c = b t+1:t+H [s, c] − Yt+1:t+H [s, c]. For yaw, we use the Y pred pred sine-cosine representation (Ysin , Ycos ) and evaluate the angular error via Lyaw using wrapπ (·) below. We further add smoothness and geometry-aware terms and define the overall forecasting loss as Lpred = Lreg +λvel Lvel +λacc Lacc +λyaw Lyaw +λ◦ L◦ , (47) where λvel , λacc , λyaw , λ◦ ≥ 0 are loss weights. Define the channel-wise error es,c ≜ Ybs,c − Ys,c with Ybs,c ≜ b t+1:t+H [s, c] and Ys,c ≜ Yt+1:t+H [s, c]. Let ∆es,c ≜ Y pred pred

9

es,c − es−1,c (s ≥ 2) and ∆2 es,c ≜ ∆es,c − ∆es−1,c (s ≥ 3). Then Lvel =

Hpred C X X 1 Huberδ (∆es,c ) , (Hpred − 1)C s=2 c=1

(48a)

Lacc =

Hpred C X X
1 Huberδ ∆2 es,c . (Hpred − 2)C s=3 c=1

(48b)

For yaw, we compute ψt+s = atan2(Ysin,t+s , Ycos,t+s ) and ψbt+s|t = atan2(Ybsin,t+s|t , Ybcos,t+s|t ) and set Lyaw =

1

Hpred

X

Hpred s=1


 Huberδ wrapπ ψbt+s|t − ψt+s , (49)

where wrapπ (·) maps angles to (−π, π]. Finally, L◦ enforces (Ybsin )2 + (Ybcos )2 ≈ 1. The LLM backbone is kept frozen, and only lightweight task adapters are trained. The certificates in Lemma 1 and (26) require an uncertainty description of the target-slot pointing mismatch ∆ω τ defined in (15). For each forecast time t and horizon h, the corresponding future slot is τ = t + h. We perform a one-time offline calibration on a separate validation set by collecting ∆ω τ over the target-slot horizons h ∈ {d+1, . . . , Hpred }, which are the horizons used to program the analog beamformer sequence over Tt . This offline calibration yields fixed uncertainty parameters that are reused during online operation and do not add online computation. The use of a separate calibration set and empirical quantiles follows standard forecasting practice [54] and is related to recent conformal methods for multi-step time-series forecasting [48]. Deterministic calibration. On an independent calibration split, we slide the same windowing protocol and, for each forecast time t, compute the target-window maximum Zt ≜

max

h∈{d+1,...,Hpred }

∆ω t+h|t 2 ,

(50)

We then set
δω ≜ Quantile1−ρ {Zt } ,

(51)

offline and then treated as fixed parameters during online operation. In contrast, Jξ,k in (16) is evaluated online at the operating point associated with the analog beamformer applied in the corresponding target-slot. These offline estimates provide the uncertainty descriptors required by the deterministic and statistical certificates in the subsequent robust beamforming formulation without introducing additional online calibration cost. IV. L EARNING -A SSISTED Q O S- DRIVEN D OWNLINK B EAMFORMING With proactive analog beam steering in Sec. III, the analog beamformer used in the target-slot τ is set as Aτ ≜ Aτ |t⋆ (τ ) and treated as fixed during online operation. The remaining online task is QoS-driven downlink digital beamforming with admission control under instantaneous QoS and transmitpower constraints. This task is challenging because the digital beamforming and admission decisions must be computed online within each slot. Recent model-based iterative solvers, such as the WMMSE and SCA, can be accurate but typically require multiple updates and repeated numerical solves [26]– [30]. Learning-based forecaster are fast, yet they may output infeasible admissions and beamformers without explicit safeguards [31], [32]. To bridge this gap, we develop a learningassisted framework that produces per-slot digital beamforming and admission decisions with strict feasibility guarantees. Robustness against forecast-induced pointing errors is handled offline through calibrated analog-pointing certificates, while the online stage focuses on satisfying instantaneous QoS and transmit-power constraints via a lightweight repair routine. We adopt the bound-based pointing certificate and use it as a deterministic robustness certificate. In each target-slot, Aτ ∈ Apred is produced by VL-LLM from the HAP attitude forecasts, and the corresponding pointing loss is controlled by the offline-calibrated parameters in Lemma 1. Accordingly, this section focuses on the remaining online variables (α, D) given Aτ . Our goal is to avoid running a full iterative solver in every slot. To maintain feasibility without high complexity, we employ a lightweight repair step.

where {Zt } collects the target-window maxima computed over all forecast times t in the calibration split. This calibrated radius A. Target-Slot QoS-driven Beamforming Formulation Consider a snapshot at a target-slot τ and omit the slot index directly interfaces with Lemma 1, which demonstrates that the analog-beam main-lobe gain loss is uniformly bounded over for brevity. When serving all users is infeasible under Pmax , all target-slot horizons by L2k δω2 for user k under the small- we introduce admission variables αk ∈ {0, 1}. For fixed Aτ , detuning approximation and the target-window event. Therefore, define the per-user QoS violation choosing δω through the empirical (1 − ρ)-quantile provides a vk (D, αk ) ≜ αk rkmin − Rk (Aτ , D), k ∈ K. (52) direct way to translate observed target-window forecast errors PK 1 The QAR objective is K k=1 αk . Feasibility additionally into a high-confidence analog-pointing robustness margin. A requires ∥Aτ D∥2F ≤ Pmax . Once Aτ is fixed, the pointing larger confidence level (1 − ρ) yields a larger calibrated radius δω and leads to a more conservative but more robust analog certificate is enforced through offline-calibrated constants and steering certificate, an effect that is later quantified in the a simple pre-screening rule, and it does not introduce any calibration-sensitivity experiments. Together with the steering- additional online optimization variables. For efficient evaluation, H K×NRF and angle event in Assumption A2, the resulting target-window define Heff ≜ H Aτ ∈ C K×K guarantee holds with probability at least 1−(ρ+ρs ), as detailed G ≜ Heff D ∈ C , Gk,j = hH (53) k Aτ dj , in the proof of Lemma 1 in the supplementary material. 2 |Gk,k | Statistical calibration: We estimate (µω , Σω ) as the sample so that SINRk = Pj̸=k |Gk,j |2 +σ2 and Rk = B log2 (1 + Hpred NRF mean and covariance of {∆ω t+h }h=d+1 on the validation set SINRk ). Let heff,k ≜ (Heff )H denote the effective k,: ∈ C and use them in (26). Both δω and (µω , Σω ) are computed channel vector for user k.

10

B. Solver-Guided Labels and KKT Targets

C. KKT-Guided Reconstruction With Strict Feasibility Repair

We design an offline reference solver to generate supervision by computing feasible high-quality solutions to (28) and (29) using a lightweight iterative solver. The offline reference solver follows a two-stage routine and stores the resulting admission decisions, beamformers, and auxiliary variables as distillation labels. Since Lk and δω are fixed constants calibrated offline, the pointing certificate can be enforced by a simple prescreening rule. Define the certified user set

Next, we design a compact online forecaster that outputs soft b from b ∈ [0, 1]K and auxiliary variables ζ admission scores α snapshot information. The inputs include the instantaneous effective CSI Heff , the QoS targets {rkmin }, and pointing2 uncertainty proxies such as σ bξ,k in (19). Admission decisions are binarized as αk = I{b αk ≥ η} using a fixed threshold η ∈ (0, 1), where I{·} denotes the indicator function. Instead of directly regressing the digital beamformer D, we reconstruct it using a WMMSE-inspired closed-form mapping that is consistent with the corresponding KKT conditions. b include per-user scalars (b Specifically, let ζ uk , w bk ) and an initial estimate of the power-dual variable νb ≥ 0. Define

 Kcert ≜ k ∈ K : L2k δω2 ≤ ϵ .

(54)

Only users in Kcert are eligible for admission in the targetK slot optimization, and we set αk = 0 for all k ∈ / Kcert . X Accordingly, the admission and strict-repair procedures operate C(ν) ≜ αk w bk |b uk |2 heff,k hH eff,k + νI, on Kcert without introducing additional online variables. k=1   Stage I. Feasibility-driven admission. Starting from the dk (ν) ≜ C(ν)−1 αk w bk u b∗k heff,k , (56) certified candidate set Kcand = Kcert , the offline reference solver iteratively removes users until the snapshot becomes where D(ν) = [d1 (ν), . . . , dK (ν)]. We select ν by a few feasible under Pmax . To decide which user to be removed, the bisection steps, since ∥Aτ D(ν)∥2F is non-increasing in ν for offline reference solver uses a low-cost required-power proxy this WMMSE-type mapping. πk computed from the snapshot. For reproducibility and low To guarantee the instantaneous power constraint, we further online complexity, we define πk as apply the scaling projection ( s ) 2 P min γ σ max k , (57) D ← D · min 1, γk ≜ 2rk /B − 1, πk ≜ , (55) 2 ∥Aτ D∥2F + εp hH Aτ fk + επ k

2 where fk is a low-complexity reference beam, e.g., the matched- which enforces ∥Aτ D∥F ≤ Pmax by construction and εp > 0 is a small constant. filter fk = heff,k /∥heff,k ∥2 , and επ > 0 is a small constant. Even with the above reconstruction, QoS constraints may This proxy approximates the transmit power required to still be violated due to model mismatch, finite-precision effects, meet the SINR target under a noise-limited approximation and or an overly aggressive admission decision. We therefore apply is used only for ranking and removal. The admission priority a deterministic strict-repair routine based on worst-first user is determined by the snapshot-wise required-power proxy πk , removal followed by an add-back step. The routine terminates in which jointly reflects effective channel quality, QoS target, and at most K removals because the admitted set strictly decreases current analog steering state. Users with larger πk are more and the empty set is always feasible. Moreover, the power costly to satisfy and are removed first in Stage I. In each projection in (57) enforces the instantaneous transmit-power iteration, the offline reference solver removes the user with the constraint by design. The removal loop iteratively reduces largest πk from Kcand and repeats until feasibility is reached the admitted set and terminates once a feasible solution is or Kcand = ∅, yielding an admission label α⋆ . We denote the found. Consequently, the repaired output satisfies both the ⋆ resulting admitted set by Kadm = {k : αk⋆ = 1}. instantaneous power constraint and the QoS requirements of Stage II. Utility refinement over the admitted set. Given all retained users. ⋆ the admitted set Kadm , the offline reference solver refines the Let Kadm = {k : αk
= 1} and compute the QoS gap Stage-II utility or an equivalent sum-rate surrogate using a small gk = rkmin − Rk (Aτ , D) + for k ∈ Kadm . If all gk = 0, the number of iterations of a local solver. A typical choice is the current solution is feasible. Otherwise, we remove the user WMMSE method or SCA on Heff [26]–[30]. The refinement with the largest normalized violation, ⋆ maintains Rk ≥ rkmin for all k ∈ Kadm and ∥Aτ D∥2F ≤ Pmax . gk The offline reference solver outputs a feasible beamformer D⋆ k † = arg max , (58) k∈K π + επ adm k and auxiliary variables that are consistent with the local KKT conditions of the refinement solver. For WMMSE-type updates, where πk is the required-power proxy in (55). we store per-user scalars u⋆k and wk⋆ and the power-dual scalar We set αk† ← 0, update Kadm ← Kadm \ {k † }, and ⋆ ⋆ ν , and denote the collected auxiliary labels by ζ . These reconstruct D again. We repeat until feasibility is reached auxiliary variables are aligned with the stationarity structure or Kadm = ∅. After feasibility is achieved, we add back of WMMSE-type KKT conditions and are used to supervise removed users in ascending πk order and keep a user only if all the subsequent closed-form beamformer reconstruction. constraints remain satisfied. Finally, we run a few iterations of For each snapshot, the offline reference solver provides α⋆ , a local refinement initialized from the repaired D, and accept D⋆ , the proxy vector π computed by (55), and KKT-related an iteration only if it preserves Rk ≥ rkmin for all admitted labels ζ ⋆ , such as {u⋆k , wk⋆ } and ν ⋆ . users and satisfies (57).

11

Algorithm 1 QoS-driven downlink beamforming with KKTguided reconstruction and strict repair

removal rounds in the strict-repair loop, where 0 ≤ Ndrop ≤ K. Let Nref be the number of capped refinement iterations. INPUT: Snapshot (Hτ , Aτ ), QoS targets {rkmin }, Pmax Forming Heff = HH τ Aτ costs O(KM NRF ) in general, and 1: Heff ← HH τ Aτ this step can be skipped if Heff is directly available. For a 2: Extract snapshot features from CSI Heff , QoS targets, and uncertainty 2 fixed ν, constructing C(ν) costs O(KNRF ). Solving the linear proxies b b ζ) and binarize αk = I{b 3: The online forecaster outputs (α, αk ≥ η} system for all users can be implemented by one matrix factor3 4: Compute πk via (55) and set Kadm ← {k : αk = 1} ization with cost O(NRF ) followed by K back-substitutions 5: repeat 2 with cost O(KNRF ). Therefore, the reconstruction cost per ν 6: Reconstruct D via (56) 3 2 7: Choose ν by bisection and apply (57)
is O(NRF + KNRF ). Evaluating G = Heff D and the resulting 8: Compute gk = rkmin − Rk (Aτ , D) + for k ∈ Kadm SINR/rates costs O(K 2 NRF ). The online forecaster network 9: if all gk = 0 then inference can be written as O(CNN ) and is typically dominated 10: break by the above matrix operations. 11: Set k† by (58), update αk† ← 0, and update Kadm ← Kadm \ {k† }
3 2 12: until Kadm = ∅ Overall, the bisection contributes O Nbis (NRF + KNRF ) 13: Attempt to add back removed users in ascending πk order and keep a per reconstruction. The strict repair repeats one reconstruction user if feasibility is preserved and one rate evaluation for at most Ndrop rounds. The refine14: QoS-safe refinement iterations e e D) 15: Output repaired (α, ment stage runs for Nref iterations with the same order of matrix operations and rate evaluation. As a result, the total per-snapshot  3 D. Distillation-Based Training Objective online complexity is O KM NRF + (Ndrop + 1) Nbis (NRF +  The online forecaster is trained to distill the admission deci- KN 2 )+K 2 N
+N (N 3 +KN 2 +K 2 N )+C RF ref RF NN . RF RF RF sions generated by the offline reference solver and KKT-related In hybrid beamforming, NRF is typically small and we auxiliary variables, while penalizing QoS violations. For admiscap (Nbis , Ndrop , Nref ) by small constants. This keeps the sion P distillation, we use the binary cross-entropy loss
 Ladm = online computation bounded per slot. Therefore, the proposed K  ⋆ 1 ⋆ − K k=1 αk log(σ(b αk )) + (1 − αk ) log 1 − σ(b αk ) , online stage avoids expensive iterative optimization, allowing where σ(·) is the sigmoid. For the utility-related term ⋆ 2 e − D ∥ , or admission and digital beamforming decisions to be computed Lutil , a typical choice is the regression loss ∥D F e The overall within strict per-slot latency constraints. The offline reference e D). a negative-utility surrogate evaluated on (α, solver cost is incurred only during training and is amortized beamforming distillation objective is defined as through distillation. ⋆ 2 ⋆ b b α ) + λζ ∥ζ − ζ ∥2 Lbf = λadm Ladm (α, K   V. S IMULATION R ESULTS X e + λqos ϖk ϕβ α ek rkmin − Rk (Aτ , D) We present two complementary sets of simulations. First, k=1 we evaluate VL-LLM on multi-horizon attitude forecasting K 1 X e D⋆ ), using HAP flight telemetry and quantify its effect on proactive − λqar · α ek + λutil Lutil (D, K analog beamforming over the target-slot window. Second, we k=1 2 ϖk = 1 + cω σ bξ,k . (59) benchmark the proposed QoS-driven beamforming solver on synthetic multiuser snapshots under strict instantaneous QoS where ϕβ (x) = β1 log(1 + exp(βx)) is a smooth penalty for and transmit-power constraints. e denotes the final outputs e D) QoS violations. The pair (α, after KKT-guided reconstruction and the power projection A. Comparison Algorithms and Simulation Parameters e is the binary admission decision after in (57). The vector α For attitude forecasting, we compare the proposed VLthresholding and strict feasibility repair. The uncertainty-aware 2 weight ϖk = 1 + cω σ bξ,k assigns a larger QoS-violation cost LLM with TimeLLM [36], PatchTST [55], TimesNet [40]. We also evaluate prompt ablations, input modes. Robustness is to users with higher pointing uncertainty. We first warm up the online forecaster using admission and further tested against inertial measurement unit (IMU) noise, auxiliary distillation, then enable the feasibility-aware term. The calibration confidence, and inference latency. Performance is strict repair is always applied at inference. It can also be enabled measured by mean absolute error (MAE) and root mean square during training-time evaluation to reduce train-test mismatch. error (RMSE) in degrees, along with the horizon-Hpred mean Overall, the main steps of the proposed target-slot QoS-driven MAE in the target-slot window. For calibration, we use the scalar mismatch magnitude ∥∆ω∥2 in (50). beamforming method are summarized in Algorithm 1. For QoS-driven beamforming, we compare the proposed learning-assisted solver with proximal policy optimization E. Algorithm and Complexity (PPO), PPO-λ [56], [57], and a denoising diffusion probaWe report the per-snapshot computational complexity of the bilistic model (DDPM)-based generative baseline [58], [59]. proposed online procedure. Let K be the number of users, To evaluate attitude compensation impact, we consider no NRF the number of RF chains, and M the number of antennas. compensation, reactive, forecast-based, and ideal modes. We Let Nbis be the number of bisection steps for selecting the also test under different channel conditions, user distributions, power-dual variable ν. Let Ndrop be the number of worst-first and admission priorities.

12

TABLE III M AIN EXPERIMENTAL SETTINGS FOR FORECASTING AND COMMUNICATION EVALUATION . Category

Setting

Forecasting data & timing

Real-flight telemetry, fs = 10 Hz, train/validation/test split = 70%/10%/20%, Lwin = 192, Llabel = 24, Hpred = 12, decision delay d = 6, target-slot window h ∈ {7, . . . , 12} TimeLLM, PatchTST, TimesNet task-only, task+physics, visual-only, numeric-only; IMU-noise, calibration-confidence, and latency tests DeepSeek-R1-7B, ViLT, AdamW, early stopping patience = 15, BF16 mixed precision (λadm , λζ , λqos , λqar , λutil , cω ) = (2, 1, 20, 0.1, 2, 0), (λvel , λacc , λyaw , λ◦ ) = (0.02, 0, 0, 0) DualFormer (proposed), PPO, PPO-λ, DDPM none, reactive, forecast-based, ideal, Rician strong, Rician weak; uniform, clustered, edge-biased, forecastedQoS-difficulty, channel-gain, random; K = 10 K = 10 users; 12×12 UPA; validation: seed 2025, 2048 samples; test: seed 2026, 2048 samples Admission gate with kmin = 8, deterministic strict repair, QoS-safe WMMSE refinement with 10 iterations, DDPM uses best-of-32 sampling and two WMMSE warm-start iterations NVIDIA RTX 5090 GPU, 32 GB memory

Forecasting baselines Ablations and robustness Forecasting model & training Loss weights Communication baselines Evaluation modes Digital beamforming solver setting Common post-processing Hardware

(a) Yaw trajectory.

(b) Pitch trajectory.

(c) Roll trajectory.

(d) Yaw deviation.

(e) Pitch deviation.

(f) Roll deviation.

Fig. 4. Attitude trajectories and forecasting deviations for yaw, pitch, and roll.

Table III summarizes the main experimental settings. These include the real-flight telemetry source, the delay-aware forecasting setup, the backbone and training configuration, and the communication evaluation settings.

TABLE IV F ORECASTING PERFORMANCE , ABLATION , ROBUSTNESS , AND LATENCY ANALYSIS . Setting

MAE

RMSE

Main baseline comparison

B. Performance Evaluation

VL-LLM (Proposed)

0.3722

0.6188

TimeLLM [36] 0.5880 0.8990 1) VL-LLM for Delay-Aware Attitude Forecasting: We first PatchTST [55] 0.5890 0.8770 evaluate whether the proposed VL-LLM can provide sufficiently TimesNet [40] 0.9830 1.9430 accurate short-term attitude forecasts for delay-aware proactive Prompt ablation beam steering. We compute MAE and RMSE in degrees and Task-only 0.3872 0.6295 evaluate yaw errors using wrap-to-(−180◦ , 180◦ ]. We focus on Task+physics 0.3851 0.6328 the target-slot windoww h ∈ {d + 1, . . . , Hpred }. We report Input-mode comparison horizon-Hpred metrics to reflect forecast-based beam-steering Visual-only 0.4735 0.7673 quality under the modeled decision delay. Numeric-only 0.4299 0.7072 Fig. 4 shows that the forecast yaw, pitch, and roll trajecIMU-noise robustness tories closely follow the ground truth over representative test Noise std = 0.005 0.3799 0.6305 segments. Analog pointing is compensated over the actionable Noise std = 0.010 0.3799 0.6302 Noise std = 0.020 0.3800 0.6306 target-slot window defined in Sec. II-A1. Therefore, horizonLatency profiling aligned errors are more relevant than one-step accuracy for assessing proactive beam steering quality. Total mean (ms) 36.24 Total p99 (ms) 40.13 Figs. 4(a), 4(b), and 4(c) show that VL-LLM closely follows the ground truth for yaw, pitch, and roll over representative test trajectories. The corresponding deviation plots in Figs. 4(d), 1.99◦ , and 1.98◦ . These horizon-aligned tail errors directly 4(e), and 4(f) show that the errors remain concentrated around support the offline calibration of δω in (51), thereby fixing zero and stay controlled at the forecast horizon. Specifically, at the uncertainty parameters used for robust beamforming. h = 12, the absolute errors are within ±4◦ for 97.10%, 98.93%, To quantify calibration-confidence sensitivity, we vary the and 99.52% of yaw, pitch, and roll samples, respectively. confidence level (1−ρ) from 0.80 to 0.99. The interval coverage The corresponding 95th-percentile absolute errors at 3.29◦ , increases from 0.9877 to 0.9968. Meanwhile, the MAE and

13

TABLE V C OMMUNICATION PERFORMANCE UNDER DIFFERENT COMPENSATION AND SYSTEM SETTINGS . Setting

QAR

Sum-rate

Compensation mode No compensation Reactive forecast-based Ideal

0.1984 0.1986 0.1994 0.2000

25.6294 25.7547 25.8342 28.0534

Channel condition Rician weak Rician strong

0.1910 0.1994

24.1283 25.8342

User distribution Uniform Clustered Edge-biased

0.1994 0.1980 0.1986

25.8342 25.5361 25.7268

Admission priority forecasted-QoS-difficulty Channel-gain Random

0.1994 0.1978 0.1990

25.8342 25.6584 25.7638

RMSE remain unchanged. This confirms that higher calibration confidence mainly renders the offline robustness margin more conservative, without affecting the point-forecast accuracy of VL-LLM. Table IV shows that VL-LLM achieves the best aggregate forecasting accuracy among the compared baselines. Compared with the strongest learning baseline PatchTST, VL-LLM reduces the RMSE from 0.8770◦ to 0.6188◦ , corresponding to a 29.4% improvement. It also reduces the MAE from 0.5890◦ to 0.3722◦ . These results confirm that the proposed VL-LLM provides a substantially more accurate basis for delay-aware proactive beam steering. Removing prompt components or replacing the multimodal input with single-modality input degrades forecasting accuracy. This confirms that both structured task priors and multimodal telemetry representations are beneficial. VL-LLM also remains stable under moderate IMU perturbations. From the reference model to the noisiest case, the RMSE increases from 0.6188◦ to 0.6306◦ , an increase of less than 2%. 2) Communication Performance After Attitude Compensation: We then evaluate whether improved attitude forecasting translates into actual communication gains after compensation and beamforming refinement. To this end, we compare different compensation modes. We also test the method under different channel conditions, user distributions, and admission priorities. Table V summarizes the end-to-end communication results after attitude compensation under different compensation modes and system settings. Forecast-based compensation consistently outperforms both no compensation and reactive modes, while the ideal mode remains the upper bound. Compared with the reactive baseline, forecast-based compensation increases the QAR from 0.1986 to 0.1994 and the sum-rate from 25.7547 to 25.8342 under the same settings. Although the gain is moderate, it is consistent and confirms that the proposed VLLLM provides communication benefits. The same table further shows that the proposed method remains effective under non-LoS (NLoS) uncertainty and different system settings. The two Rician settings lead to

TABLE VI B ENCHMARK OF THE Q O S- DRIVEN DIGITAL BEAMFORMING SOLVER . Method

Feasible

QAR

Sum-rate

DualFormer (Proposed) PPO PPO-λ DDPM

1.0000 0.9980 1.0000 1.0000

0.9944 0.8100 0.6810 0.8142

77.3820 68.5984 68.8129 66.2873

different communication performance, with the strong-Rician case achieving higher QAR and sum-rate. The clustered and edge-biased user layouts are slightly more challenging than the uniform layout. This reflects less balanced QoS difficulty under the same power budget. Among the admission strategies, the forecasted-QoS-difficulty priority achieves the best overall tradeoff in both QAR and sum-rate. This is consistent with its design objective of ranking users according to their instantaneous QoS difficulty under the current analog pointing and power budget. 3) Learning-Assisted QoS-driven Downlink Beamforming Solver: Finally, we benchmark the Stage-II digital beamforming solver itself using static multiuser LoS scenarios. This experiment uses identical settings for user admission, deterministic repair, and power budget. Table VI shows that DualFormer achieves the best sum-rate and the highest QAR while maintaining near-unity feasibility under the same user admission, repair, and power budget. Compared with the best-QAR baseline DDPM, DualFormer improves the QAR from 0.8142 to 0.9944. Compared with the best-sum-rate baseline PPO-λ, DualFormer improves the sumrate from 68.8129 to 77.3820, a gain of 12.5%. These results support the design in Sec. IV, where KKT-guided reconstruction reduces approximation error and deterministic repair prevents QoS violations.

VI. C ONCLUSION This paper investigated HAP downlink mmWave communications. It addressed platform attitude-induced beam misalignment and stringent per-slot online decision requirements. We developed a multimodal LLM-enabled beamforming framework for robust HAP downlink communications. Specifically, we designed a VL-LLM that learned from multivariate real-flight telemetry to forecast short-term HAP attitudes under platform shaking. We also introduced an offline forecast-error calibration procedure. This yielded reliable upper bounds on the residual errors. Based on the compensated analog beamformer, we further developed a learning-assisted digital beamforming and admission scheme. It incorporated lightweight feasibility enforcement to satisfy instantaneous transmit-power and QoS constraints. Using real-flight telemetry sampled at 10 Hz, the proposed framework achieved accurate short-term attitude forecasting with p99 latency of 40.13 ms. This enabled delayaware proactive beam steering and consistent communication gains over reactive and no compensation baselines across diverse settings. Future work will consider stronger NLoS effects, nonstationary sensing uncertainty, cooperative aerialnetwork settings.

14

A PPENDIX A P ROOF OF L EMMA 1

TABLE VII C OMMUNICATION PERFORMANCE UNDER DIFFERENT USER - NUMBER SETTINGS .

Proof. Fix a user k and consider the target-window horizons h ∈ {d+1, . . . , Hpred } programmed at time t. Under Assumption A2, the forecasted steering angles satisfy (ϑk , φk ) ∈ Sk on an event with probability at least 1 − ρs . Under the smalldetuning regime where the quadratic approximation in Eq. (23) of the main manuscript holds, the normalized main-lobe gain loss can be expressed as ∆GUPA,k ≈ ∆ω ⊤ t+h|t Qk (ϑk , φk )∆ω t+h|t .

(60)

Setting

QAR

Sum-rate

K=8 K = 10 K = 12

0.2299 0.1994 0.1687

23.0020 25.8342 26.2766

TABLE VIII F ORECASTING SENSITIVITY TO CALIBRATION CONFIDENCE . Calibration confidence (1 − ρ)

Interval coverage

MAE

RMSE

0.80

0.9877

0.3722

0.6188

0.85 0.9900 0.3722 0.6188 Since cx , cy > 0 and Qk (ϑ, φ) = 0.90 0.9921 0.3722 0.6188 ⊤ Jξ,k (ϑ, φ) diag(cx , cy )Jξ,k (ϑ, φ), we have Qk (ϑ, φ) ⪰ 0. 0.95 0.9943 0.3722 0.6188 0.99 0.9968 0.3722 0.6188 Therefore, the Rayleigh quotient bound gives
2 ∆ω ⊤ ∆ω t+h|t 2 .A. User-Number Variation t+h|t Qk (ϑk , φk )∆ω t+h|t ≤ λmax Qk (ϑk , φk ) (61) Table VII provides the communication performance results By the definition of L2k in Eq. (24) of the main manuscript,
under different user-number settings compared to the main for all (ϑ, φ) ∈ Sk , λmax Qk (ϑ, φ) ≤ L2k . Hence, whenever manuscript. As the number of users increases, the admission (ϑk , φk ) ∈ Sk , difficulty naturally becomes higher, which reduces the QoS 2

∆GUPA,k ≤ L2k ∆ω t+h|t 2 .

(62)

Next, under Assumption A1, the target-window event max

h∈{d+1,...,Hpred }

∆ω t+h|t 2 ≤ δω

holds with probability at least 1 − ρ. On this event, (62) implies that for every h {d+1, . . . , Hpred }, ∆GUPA,k ≤ L2k δω2 .

(63)

∈ B. Calibration Sensitivity (64)

If L2k δω2 ≤ ϵ, then ∆GUPA,k ≤ ϵ holds uniformly over all target-window horizons programmed at time t. Finally, letting E1 denote the event in Assumption A1 and E2 denote the event {(ϑk , φk ) ∈ Sk } in Assumption A2, we have P(E1 ∩ E2 ) ≥ 1 − P(E1c ) − P(E2c ) ≥ 1 − (ρ + ρs ),

admission ratio (QAR). Meanwhile, the sum-rate varies more mildly across the tested range. This is because admitting more users does not necessarily improve feasibility, but the admitted set can still be efficiently refined under the same beamforming method. Overall, these results provide evidence for the scalability of the algorithm proposed in the main manuscript.

(65)

which completes the proof. A PPENDIX B C OMMUNICATION P ERFORMANCE U NDER D IFFERENT U SER N UMBERS AND C ALIBRATION S ETTINGS To complement the main manuscript evaluation, we provide additional communication performance under different usernumber settings and calibration settings. In addition to the default case K = 10 reported in the main manuscript, we further test K = 8 and K = 12 to assess whether the proposed delay-aware forecast-then-beamform method remains effective when the users decreases or increases around the default operating point. The simulation protocol follows the same setting as in the main manuscript, and only the user number or calibration setting is changed.

Table VIII reports the sensitivity of the offline calibration step to the confidence level (1−ρ). As (1−ρ) increases, the interval coverage increases monotonically, confirming that a larger confidence level yields a more conservative calibrated bound. By contrast, the macro MAE and RMSE remain unchanged across the sweep, since the calibration step only affects the post-hoc robustness margin and does not modify the point forecaster itself. These results support the main-text discussion that calibration confidence mainly controls the conservativeness of the offline robustness certificate rather than the point-forecast accuracy. R EFERENCES [1] X. Cao, X. Su, P. Yang, Y. Gao, D. Oliver Wu, and T. Q. S. Quek, “Survey on near-space information networks: Channel modeling, transmission, and networking perspectives,” IEEE Commun. Surveys Tuts., vol. 28, pp. 672–715, 2026. [2] K. Mashiko, Y. Kawamoto, N. Kato, K. Yoshida, and M. Ariyoshi, “Combined control of coverage area and HAPS deployment in hybrid FSO/RF SAGIN,” IEEE Trans. Veh. Technol., vol. 74, no. 7, pp. 10 819– 10 828, 2025. [3] T. Xu, J. Zhu, Y. Xu, P. Peng, X. Chen, Q. Wu, and D. Niyato, “NOMAoriented spectrum sensing for joint HAP and HEO nonterrestrial uplink communications,” IEEE Internet Things J., vol. 12, no. 16, pp. 34 603– 34 606, 2025. [4] A. M. Benaya, M. S. Hassan, M. H. Ismail, and T. Landolsi, “Aerial ISAC: A HAPS-assisted integrated sensing, communications and computing framework for enhanced coverage and security,” IEEE Trans. Green Commun. Networking, vol. 9, no. 4, pp. 2101–2114, 2025.

15

[5] F. Wang, H. Yang, S. Zhang, J. Shi, Z. Li, and T. Q. S. Quek, “Mobilityaware multicast orchestration for low-altitude UAVs with integrated terrestrial and non-terrestrial networks,” IEEE Trans. Commun., vol. 73, no. 11, pp. 12 854–12 868, 2025. [6] P. An, S. Yang, J. Sun, J. Li, P. Yang, and X. Cao, “Survey on HA2G and A2A channel measurement and modeling for airborne maneuvering networks,” Chin. J. Aeronaut., p. 104036, early access, Dec. 24, 2025, DOI:10.1016/j.cja.2025.104036. [7] P. Kanani, M. Javad Omidi, M. Modarres-Hashemi, and H. Yanikomeroglu, “Optimizing network performance and resource allocation in HAPS-UAV integrated sensing and communication systems for 6G,” IEEE Trans. Wireless Commun., vol. 25, pp. 4098–4112, 2026. [8] F. Wang, S. Zhang, J. Shi, Z. Li, and T. Q. S. Quek, “Sustainable UAV mobility support in integrated terrestrial and non-terrestrial networks,” IEEE Trans. Wireless Commun., vol. 23, no. 11, pp. 17 115–17 128, 2024. [9] M. Wu, Z. Gao, Z. Wang, D. Niyato, G. K. Karagiannidis, and S. Chen, “Deep joint semantic coding and beamforming for near-space airshipborne massive MIMO network,” IEEE J. Sel. Areas Commun., vol. 43, no. 1, pp. 260–278, Jan. 2025. [10] O. E. Ayach, S. Rajagopal, S. Abu-Surra, Z. Pi, and R. W. Heath, “Spatially sparse precoding in millimeter wave MIMO systems,” IEEE Trans. Wireless Commun., vol. 13, no. 3, pp. 1499–1513, 2014. [11] F. Sohrabi and W. Yu, “Hybrid digital and analog beamforming design for large-scale antenna arrays,” IEEE J. Sel. Top. Signal Process., vol. 10, no. 3, pp. 501–513, 2016. [12] H.-J. Moon, C.-B. Chae, K.-K. Wong, and M.-S. Alouini, “A generalized pointing error model for FSO links with fixed-wing UAVs for 6G: Analysis and trajectory optimization,” IEEE Trans. Wireless Commun., vol. 24, no. 7, pp. 5723–5737, 2025. [13] K. Lee, D. You, H. Noh, and C. Lee, “Robust beamforming for UAV communication with jittering effects,” IEEE Wireless Commun. Lett., vol. 14, no. 1, pp. 48–52, 2025. [14] Q. Xue, C. Ji, S. Ma, J. Guo, Y. Xu, Q. Chen, and W. Zhang, “A survey of beam management for mmwave and THz communications towards 6G,” IEEE Commun. Surv. Tutorials, vol. 26, no. 3, 2024. [15] M. Ghanbari, M. T. Dabiri, O. S. Badarneh, M. Hasna, Y. H. Al-Badarneh, M. K. Alshawaqfeh, and K. Qaraqe, “When future communications shift toward narrow beams: A forward looking survey on pointing errors and alignment limits,” 2025, arXiv:2512.20874. [16] A. P. Salmeno and I. Zakia, “Reliable beam tracking on high-altitude platform for millimeter wave high-speed railway,” IEEE Access, vol. 12, pp. 71 997–72 012, 2024. [17] V. Ardianto Nugroho and B. Moo Lee, “GPS-aided deep learning for beam prediction and tracking in UAV mmwave communication,” IEEE Access, vol. 13, pp. 117 065–117 077, 2025. [18] Y. Ke, W. Xu, W. Ni, X. Yuan, and D. Niyato, “Combating beam misalignment in mmwave UAV networks: An attitude compensationbased method,” IEEE Trans. Veh. Technol., vol. 74, no. 2, pp. 3521–3526, 2025. [19] J. Shao, J. Tong, Q. Wu, W. Guo, Z. Li, Z. Lin, and J. Zhang, “WirelessLLM: Empowering large language models towards wireless intelligence,” J. Commun. Inf. Networks, vol. 9, no. 2, pp. 99–112, 2024. [20] R. Nikbakht, M. Benzaghta, and G. Geraci, “TSpec-LLM: An opensource dataset for LLM understanding of 3GPP specifications,” 2024, arXiv:2406.01768. [21] N. Piovesan, A. D. Domenico, and F. Ayed, “Telecom language models: Must they be large?” 2024, arXiv:2403.04666. [22] A.-L. Bornea, F. Ayed, A. D. Domenico, N. Piovesan, and A. Maatouk, “Telco-RAG: Navigating the challenges of retrieval-augmented language models for telecommunications,” 2024, arXiv:2404.15939. [23] R. Zhang, H. Du, Y. Liu, D. Niyato, J. Kang, Z. Xiong, A. Jamalipour, and D. In Kim, “Generative AI agents with large language model for satellite networks via a mixture of experts transmission,” IEEE J. Sel. Areas Commun., vol. 42, no. 12, pp. 3581–3596, 2024. [24] S. Zhang, Q. Liu, K. Chen, B. Di, H. Zhang, W. Yang, D. Niyato, Z. Han, and H. V. Poor, “Large models for aerial edges: An edge-cloud model evolution and communication paradigm,” IEEE J. Sel. Areas Commun., vol. 43, no. 1, pp. 21–35, 2025. [25] X. Cheng, R. Meng, X. Xu, H. Gao, P. Zhang, and D. Niyato, “APEG: Adaptive physical layer authentication with channel extrapolation and generative AI,” IEEE Trans. Inf. Forensics Secur., vol. 21, pp. 1257–1272, 2026. [26] X. Pan, Z. Zheng, X. Huang, and Z. Fei, “WMMSE-based joint transceiver design for multi-RIS-assisted cell-free networks using hybrid CSI,” IEEE Trans. Wireless Commun., vol. 24, no. 9, pp. 7654–7669, 2025.

[27] N. Zhou, Z. Wang, Q. Shi, W. Xu, and Y. Huang, “Model-driven distributed WMMSE for downlink massive MIMO systems,” IEEE Wireless Commun. Lett., vol. 14, no. 11, pp. 3675–3679, 2025. [28] Z. Zhao, X. Guan, F. Xu, and Q. Shi, “Enhancing WMMSE with EVM feedback through bayesian optimization,” IEEE Trans. Veh. Technol., vol. 74, no. 10, pp. 16 579–16 583, 2025. [29] Z. Xinyu, Z. Lijuan, L. Shidang, Z. Yanping, C. Jianbin, and L. Chunguo, “Joint robust beamforming design for RIS-assisted ISAC systems with hardware impairments,” AEU Int. J. Electron. Commun., vol. 206, p. 156209, 2026. [30] Z. Bowen, Z. Pan, P. Zihao, Z. Wenming, Y. Ning, and G. Daoxing, “Joint optimization of multi-beam configuration and resource allocation for low-earth orbit satellites,” Phys. Commun., vol. 75, p. 103001, 2026. [31] L. Ma, N. Cheng, X. Wang, Z. Yin, H. Zhou, and W. Quan, “Distilling knowledge from resource management algorithms to neural networks: A unified training assistance approach,” in Proc. IEEE 98th Veh. Technol. Conf. (VTC2023-Fall), Dec. 2023, pp. 1–5. [32] X. Xu, Y. Liu, X. Mu, D. Gan, and A. Nallanathan, “Beamforming for pinching antenna systems (PASS): KKT-guided large model learning,” in Proc. IEEE/CIC Int. Conf. Commun. China (ICCC), Aug. 2025, pp. 1–6. [33] X. Wang, X. Zhao, and N. Cheng, “Differentiable projection-based learn to optimize in wireless network-part I: Convex constrained (non-)convex programming,” 2025, arXiv:2502.00053. [34] K. Deng, H. Zhang, J. Lu, and H. Sun, “HoP: Homeomorphic polar learning for hard constrained optimization,” 2025, arXiv:2502.00304. [35] X. Liu, J. Hu, Y. Li, S. Diao, Y. Liang, B. Hooi, and R. Zimmermann, “Unitime: A language-empowered unified model for cross-domain time series forecasting,” CoRR, vol. abs/2310.09751, 2023. [36] M. Jin, S. Wang, L. Ma, Z. Chu, J. Y. Zhang, X. Shi, P.-Y. Chen, Y. Liang, Y.-F. Li, S. Pan, and Q. Wen, “Time-LLM: Time series forecasting by reprogramming large language models,” in Proc. 12th Int. Conf. Learn. Represent. (ICLR), May 2024, pp. 1–24. [37] Y. Sheng, K. Huang, L. Liang, P. Liu, S. Jin, and G. Ye Li, “Beam prediction based on large language models,” IEEE Wireless Commun. Lett., vol. 14, no. 5, pp. 1406–1410, 2025. [38] X. Liu, J. Hu, Y. Li, S. Diao, Y. Liang, B. Hooi, and R. Zimmermann, “UniTime: A language-empowered unified model for cross-domain time series forecasting,” in Proc. ACM Web Conf. Association for Computing Machinery, May 2024, p. 4095–4106. [39] S. Wang, J. LI, X. Shi, Z. Ye, B. Mo, W. Lin, J. Shengtong, Z. Chu, and M. Jin, “Timemixer++: A general time series pattern machine for universal predictive analysis,” in Proc. 13th Int. Conf. Learn. Represent. (ICLR), Apr. 2025, pp. 1–37. [40] H. Wu, T. Hu, Y. Liu, H. Zhou, J. Wang, and M. Long, “Timesnet: Temporal 2d-variation modeling for general time series analysis,” in Proc. 11th Int. Conf. Learn. Represent. (ICLR), Apr. 2023, pp. 1–23. [41] A. Dosovitskiy, L. Beyer, A. Kolesnikov et al., “An image is worth 16×16 words: Transformers for image recognition at scale,” in Proc. 9th Int. Conf. Learn. Represent. (ICLR), 2021. [42] N. Chen, S. Zhang, X. Zhou, X. Huang, D. O. Wu, and T. Qiu, “ CuIoT: Advancing Network Connectivity with Motif Knowledge-Centric for Robust Topology ,” IEEE Trans. Knowl. Data Eng., no. 01, pp. 1–14, Mar. 5555. [43] N. Chen, T. Qiu, X. Zhou, S. Zhang, W. Si, and X. Wang, “LEGOMotif: Enhancing IoT topology robustness with evolutionary motif-based generation,” IEEE Trans. Syst. Man Cybern.: Syst., pp. 1–14, 2026. [44] N. Chen, S. Zhang, X. Zhou, S. Cao, and T. Qiu, “Fast robustness enhancement for dynamic IIoT topology with adaptive bayesian learning,” IEEE Trans. Mob. Comput., vol. 24, no. 10, pp. 10 886–10 899, 2025. [45] N. Chen, T. Qiu, X. Zhou, S. Zhang, W. Si, and D. Oliver Wu, “A distributed co-evolutionary optimization method with motif for largescale IoT robustness,” IEEE/ACM Trans. Networking, vol. 32, no. 5, pp. 4085–4098, 2024. [46] M. Khemiri, M. Erdem, A. Saeed, O. Gurbuz, and M.-S. Alouini, “Robust beam control for terahertz drone networks,” IEEE Trans. Aerosp. Electron. Syst., vol. 61, no. 5, 2025. [47] A. R. Geist, J. Frey, M. Zhobro, A. Levina, and G. Martius, “Learning with 3D rotations, a hitchhiker’s guide to SO(3),” in Proc. 41th Int. Conf. Machine Learning (ICML), vol. 235, Jul 2024, pp. 15 331–15 350. [48] M. Sousa, A. M. Tomé, and J. Moreira, “A general framework for multistep ahead adaptive conformal heteroscedastic time series forecasting,” Neurocomputing, vol. 608, p. 128434, 2024. [49] W. Chen, C. Liu, W. Wang, M. Peng, and W. Zhang, “Adaptive hybrid beamforming for UAV mmwave communications against asymmetric jitter,” IEEE Trans. Wireless Commun., vol. 23, no. 8, pp. 9432–9445, Feb. 2024.

16

[50] K. H. Irani, Y. Huang, and S. A. Vorobyov, “SINR maximizing distributionally robust adaptive beamforming,” IEEE Trans. Signal Process., vol. 73, pp. 2542–2557, 2025. [51] L. F. Abanto-Leon, A. Asadi, A. Garcia-Saavedra, G. H. Sim, and M. Hollick, “Radiorchestra: Proactive management of millimeter-wave self-backhauled small cells via joint optimization of beamforming, user association, rate selection, and admission control,” IEEE Trans. Wireless Commun., vol. 22, no. 1, pp. 153–173, 2023. [52] L. F. Abanto-Leon, A. Krishnamoorthy, A. Garcia-Saavedra, G. H. Sim, R. Schober, and M. Hollick, “Radio resource management design for RSMA: Optimization of beamforming, user admission, and discrete/continuous rates with imperfect SIC,” IEEE Trans. Mob. Comput., vol. 23, no. 12, pp. 11 498–11 518, 2024. [53] T. Kim, J. Kim, Y. Tae, C. Park, J.-H. Choi, and J. Choo, “Reversible instance normalization for accurate time-series forecasting against distribution shift,” in Proc. 10th Int. Conf. Learn. Represent. (ICLR), 2022. [54] Y. Romano, E. Patterson, and E. Candes, “Conformalized quantile regression,” in Proc. 33rd Adv. Neural Inf. Process. Syst. (NeurIPS), Jul 2019. [55] Y. Nie, N. H. Nguyen, P. Sinthong, and J. Kalagnanam, “A time series is worth 64 words: Long-term forecasting with transformers,” in Proc. 11th Int. Conf. Learn. Represent. (ICLR), Apr. 2023, pp. 1–23. [56] M. Cheng, C. Zhu, M. Lin, and W.-P. Zhu, “A MAPPO based scheme for joint resource allocation in UAV assisted MEC networks,” in Proc. IEEE/CIC Int. Conf. Commun. China (ICCC), Aug. 2024, pp. 42–47. [57] A. B. Adam, E. M. Diallo, M. S. A. Muthanna, R. I. Alkanhel, A. Muthanna, and M. Hammoudeh, “Generative AI-driven reinforcement learning for beamforming and scheduling in multi-cell MIMO-NOMA systems,” Phys. Commun., vol. 72, p. 102771, 2025. [58] J. Zhang, S. Xu, Z. Zhang, C. Li, and L. Yang, “A denoising diffusion probabilistic model-based digital twinning of ISAC MIMO channel,” IEEE Internet Things J., vol. 12, no. 15, pp. 29 121–29 134, 2025. [59] C. Zhao, J. Wang, R. Zhang, D. Niyato, G. Sun, H. Du, D. I. Kim, and A. Jamalipour, “Generative AI-enabled wireless communications for robust low-altitude economy networking,” IEEE Wireless Commun., pp. 1–9, 2025.