Hybrid deep unrolling and graph neural networks for super-resolution direction-of-arrival estimation in physics-informed antenna arrays

1 Introduction

The rapid evolution of high-speed wireless communications, radar imaging, and industrial sensing—particularly within the framework of Industry 4.0 and decentralized pharmaceutical manufacturing—has intensified the demand for high-fidelity Direction-of-Arrival (DoA) estimation (Qin and Braatz, 2021). Accurate spatial localization of electromagnetic (EM) sources is fundamental to beamforming, array calibration, and process monitoring. However, as antenna systems scale in complexity and operate in increasingly dense EM environments, traditional computational electromagnetics (CEM) and classical signal processing frameworks encounter significant bottleneck constraints.

Before the advent of deep learning, Direction-of-Arrival (DoA) estimation saw significant progress through Sparse Signal Processing (SSP) and Compressed Sensing (CS). Methods such as ℓ1-SVD and iterative shrinkage algorithms offered high resolution by treating the spatial spectrum as a sparse reconstruction problem. However, these classical CS methods often suffer from high computational complexity due to their iterative nature and require manual tuning of regularization parameters. Our proposed model bridges this gap by ‘unrolling’ these iterative processes into a neural network, combining the mathematical rigor of CS with the real-time inference speed of deep learning.

For decades, subspace-based methods such as Multiple Signal Classification (MUSIC) have served as the industry standard for DoA estimation (Schmidt, 1986). While these methods offer asymptotic consistency, their performance in practical scenarios is often compromised by two primary factors: “sample starvation” and physical array impairments. Mathematically, the resolution of subspace methods is strictly bounded by the number of available snapshots (T); when T is low, the sample covariance matrix (SCM) becomes an unstable representation of the signal environment, leading to the collapse of the noise subspace (Pillai and Kwon, 1989). Furthermore, real-world arrays are subject to mutual coupling effects, where the EM interaction between adjacent elements distorts the steering vectors. Failure to rigorously account for the mutual coupling matrix (C) leads to severe bias and the emergence of “ghost” peaks in the spatial spectrum (Gupta and Weiner, 1983; Friedlander and Weiss, 1991). Mutual coupling introduces non-linearities into the signal model. Standard black-box networks struggle to generalize these effects without immense datasets.

To address these limitations, Deep Learning (DL) has emerged as a transformative paradigm for EM modeling. Early DL approaches utilized Convolutional Neural Networks (CNNs) and Recurrent Neural Networks (RNNs) to map raw signal inputs to angular outputs (Huang et al., 2018). While these “black box” models demonstrate high accuracy in specific training regimes, they often lack the mathematical transparency and physical consistency required for critical antenna engineering applications. This has led to the emergence of “Model-Based Deep Learning,” which seeks to integrate domain-specific knowledge into neural architectures (Shlezinger et al., 2020).

One of the most promising avenues in this domain is Algorithm Unrolling (or Unrolling), which transforms iterative optimization algorithms into layer-wise neural network structures (Monga et al., 2021). By unrolling the Iterative Soft-Thresholding Algorithm (ISTA), researchers have developed Learned ISTA (LISTA) networks that achieve sparse signal reconstruction with significantly lower computational latency than traditional solvers (Gregor and LeCun, 2010). Concurrently, Geometric Deep Learning and Graph Neural Networks (GNNs) have provided a robust mathematical framework for modeling non-Euclidean spatial dependencies in antenna arrays, where the elements can be represented as nodes in a graph (Bronstein et al., 2021; Jiang et al., 2022).

Despite these advances, a gap remains in creating architectures that simultaneously leverage the spatial reasoning of GNNs and the interpretable sparsity of unrolled LISTA networks for superresolution DoA estimation. In this paper, we propose a Hybrid GNN-LISTA architecture specifically designed for physics-informed antenna arrays. Our contributions are three-fold:

• Architectural Innovation: We develop a hybrid model that utilizes a shared spatial Smatrix to maintain physical consistency across layers, modulated by learned layer-wise scalars (α and λ) to provide a supervised denoising curriculum.
• Ghost-Killer Optimization: We introduce a specialized loss function that combines weighted Mean Squared Error (MSE) with a quartic penalty term to eliminate spurious “ghost” peaks, a common failure mode in deep-learning-based DoA estimation.
• Performance Breakthroughs: We demonstrate that the proposed model achieves a 98.0% success rate and resolves sources at a 3.0° separation with 70% accuracy, significantly surpassing the capabilities of SS-MUSIC (36%). Moreover, we prove the model’s robustness to sample starvation, maintaining a 78.0% success rate at T = 50 snapshots, a +14% improvement over classical subspace baselines.

The unique novelty of this work distinguishes it from both standard deep unrolling models and standalone GNN signal processing pipelines. Conventional deep unrolling techniques apply physics-informed constraints but struggle when the input array measurements are warped by nonlinear phenomena such as mutual coupling. Conversely, typical GNN signal architectures excel at extracting topological relationships from array layouts but drop structural constraints within their deep inference layers, acting as black boxes. This framework uniquely resolves this dichotomy by employing a GNN to map raw, coupling-warped covariance manifolds into structured inputs, which are then processed by an unrolled layer sequence strictly bound by a physically consistent Shared S-matrix.

2 System model and data generation

To evaluate the performance of the proposed Hybrid GNN-LISTA architecture, we consider a physics-informed signal model that rigorously accounts for both the geometric properties of the antenna array and the electromagnetic impairments inherent in real-world systems.

To ensure accessibility, we define core concepts used in this work:

• Deep Unrolling: The process of mapping an iterative algorithm (like ISTA) onto a layer-wise neural network architecture.
• Physical Consistency: Ensuring the network layers mimic the steps of a physical signal recovery model rather than acting as a black box.
• Non-Euclidean Spatial Dependency: The irregular relationship between antenna elements modeled as a graph, rather than a simple grid.
• Denoising Curriculum: The strategy where the network learns to progressively refine the signal estimate across layers, modeled by learned parameters γ and λ.

2.1 Antenna array geometry and steering vector

We consider a Uniform Linear Array (ULA) consisting of N = 12 identical omnidirectional antenna elements. The elements are arranged along the x-axis with a uniform inter-element spacing of d = λ/2, where λ represents the wavelength of the incident narrowband signals.

For a signal impinging from a far-field direction Θ, the ideal steering vector a(θ)∈CN×1 represents the relative phase shifts across the array and is defined as (Schmidt, 1986):

a θ = 1 , e − j 2 π λ d ⁡ sin θ , … , e − j 2 π N − 1 λ d ⁡ sin θ T

Where (⋅)T denotes the transpose operator. In the presence of K uncorrelated sources located at angles Θ = θ 1 , θ 2 , … , θ K, the composite steering matrix is defined as A Θ = a θ 1 , a θ 2 , … , a θ K.

2.2 Mutual coupling formulation

In practical antenna systems, the electromagnetic interaction between adjacent elements—known as mutual coupling—distorts the received signal manifold. We model this effect using a Mutual Coupling Matrix (MCM), denoted as C∈CN × N. For a ULA of identical elements, C is characterized by a complex symmetric Toeplitz structure, where the coupling coefficient between two elements depends primarily on their physical separation (Gupta and Weiner, 1983).

The MCM is defined as:

C = 1 c 1 ⋮ C N − 1 c 1 c 2 … 1 c 1 … ⋮ C N − 2 ⋮ C N − 3 ⋮ … C N − 1 C N − 2 ⋮ 1

where cm represents the coupling coefficient between elements separated by m inter-element spacings. Following the principle that mutual coupling decays inversely with distance, we implement a banded model where cm = 0 for m > P, where P is the coupling range (Friedlander and Weiss, 1991). This rigorous modeling ensures that the neural network learns to compensate for “ghost” peaks and manifold deviations caused by electromagnetic interference.

2.3 Signal model and snapshot estimation

The observed signal vector at the array output at time t is given by:where s(t)∈CK×1 is the vector of incident signal waveforms and n(t)∼CN(0,σ2I) is the additive white Gaussian noise (AWGN) with variance σ2.

x t = CA Θ s t + n t

In practical scenarios, the true covariance matrix is unavailable, and we must rely on the Sample Covariance Matrix (SCM), R^, estimated from T finite snapshots (Pillai and Kwon, 1989):where (⋅)H denotes the conjugate transpose. The performance of the architecture is specifically evaluated under “sample starvation” conditions (e.g., T = 50), where R^ is a noisy and potentially rank-deficient estimate of the spatial environment.

R ^ = 1 T ∑ t = 1 T x t x H t

2.4 Physics-informed dataset generation

The dataset for training and validation was generated using a Monte Carlo approach to ensure statistical diversity. We generated 20,000 signal realizations with the following parameters:

• Source Configuration: Randomly distributed sources with an angular separation Δθ ranging from 1.0° to 10.0°.
• Coupling Range: P = 3 with complex coefficients to simulate realistic EM interactions.
• SNR Regime: Signal-to-Noise Ratios (SNR) varied from −10 dB to 20 db.
• Snapshot Scenarios: Training conducted at high T, with testing performed across a range of T in [50, 1000] to verify the 14% efficiency gain observed in low-sample regimes.

3 Proposed methodology

The proposed framework bridges the gap between geometric deep learning and iterative optimization unrolling. By mapping the antenna array manifold to a graph structure and unrolling the sparse reconstruction solver, the model leverages both the spatial reasoning of Graph Neural Networks (GNNs) and the convergence guarantees of the Learned Iterative Soft-Thresholding Algorithm (LISTA) (Gregor and LeCun, 2010; Monga et al., 2021).

3.1 Graph construction from covariance manifolds

To capture the spatial dependencies and the underlying geometric structure of the Uniform Linear Array (ULA), we represent the Sample Covariance Matrix (SCM) as a graph G=(V,E). Following the principles of Geometric Deep Learning (Bronstein et al., 2021), we apply a forward-backward spatial smoothing technique to the SCM to mitigate source correlation effects (Pillai and Kwon, 1989).

The smoothed SCM is then decomposed into its eigen structure. The node feature matrix X∈R L × 2L is constructed by concatenating the real and imaginary components of the signal eigenvectors, ensuring the network is cognizant of the signal subspace. The edge set E is defined as a fully connected graph of the smoothed subarrays, allowing the GNN to learn the complex electromagnetic interactions and mutual coupling residuals across the array manifold (Jiang et al., 2022).

3.2 Hybrid GNN-LISTA architecture

The core of the architecture is a deep Unrolling network that mimics the behavior of a modelbased iterative solver (Shlezinger et al., 2020). The network consists of a GNN-based front-end followed by K unrolled layers.

The GNN front-end utilizes Graph Convolutional Network (GCN) layers to extract high-level spatial features from the eigen structure. These features are projected into the angular domain to form an initial estimate z (0). The subsequent LISTA layers perform iterative refinements based on the following update rule:where z(k) is the estimated spatial spectrum at layer k, and η λ is the soft-thresholding operator defined as soft (x, λ) = sign(x)max (∣x∣−λ,0).

z k + 1 = η λ k z 0 + γ k Sz k

A critical innovation in this architecture is the Shared S-matrix. Unlike standard LISTA, which learns independent weight matrices for every layer, our model utilizes a shared spatial kernel S∈R181 × 181 that acts as a learned dictionary of spatial interactions. This ensures that the physical constraints of the array remain consistent throughout the unrolling process.

A key advantage of this architecture is the implementation of a Shared S-matrix across all K iterations. In a standard non-shared unrolled network, memory requirements scale linearly with the number of layers (K × grid size2). By sharing the spatial kernel, we reduce the parameter footprint by 90% for a 10-layer model, decreasing memory overhead from O (K⋅N ²) to O (N²).

This efficiency is vital for real-time deployment on edge-based signal processing hardware. The overall workflow is illustrated in Figure 1.

3.3 Layer-wise learned parameters

To achieve a “denoising curriculum,” the model learns layer-specific scalar modulators:

• Step-size γ(k): Controls the influence of the spatial dictionary at each iteration, allowing the network to adapt the convergence speed.
• Threshold λ(k): Dynamically adjusts the sparsity level, effectively filtering out noise and mutual coupling artifacts as the signal propagates through the layers.

By plotting these parameters, the model provides physics-informed interpretability, demonstrating how the network gradually suppresses the noise floor to reveal the true signal peaks.

3.4 Ghost-Killer Loss optimization

Standard Mean Squared Error (MSE) loss often fails in DoA estimation because it does not distinguish between a missed source and a spurious “ghost” peak caused by sidelobes or coupling errors. We introduce a multi-objective Ghost-Killer Loss function:where:

L = MSE w e i g h t e d + β L g h o s t

• MSEweighted: Assigns higher weights (e.g., w = 20) to the true source locations to prioritize resolution.
• Lghost: A quartic penalty term, E [(z⋅ M) 4], where M is a mask for the noise zone (target <0.01). The fourth-order power ensures that even small spurious peaks are aggressively penalized, leading to a “clean” spatial spectrum.
• While standard Mean Squared Error (MSE) is effective for peak alignment, it often fails in DoA estimation because it treats every angular bin with equal weight, leading to ‘blurred’ peaks and a lack of sparsity. Furthermore, MSE does not aggressively penalize low-power spurious artifacts. The quartic penalty term (P^(θ)4) addresses this by providing a significantly steeper gradient for non-zero power in indices outside the ground truth. This forces the optimizer to prioritize a clean noise floor, effectively suppressing ‘ghost peaks’ that typically bypass quadratic-based loss functions.
• The mathematical selection of a quartic (x⁴) penalty over a standard quadratic (L₂) penalty is rooted in gradient dynamics. Consider an isolated spurious peak with an out-of-bounds power amplitude x. Under a traditional quadratic loss, the gradient vector scales as ∂/∂ x (x²) = 2x, which provides weak, linear correction forces for small-scale artifacts. Under our quartic ‘Ghost-Killer’ formulation, the error gradient scales as ∂/∂ x (x⁴) = 4x³. This cubic gradient growth ensures that even minor false peaks trigger a massive, disproportionate penalty, forcing the network parameters to decisively drive false spectral artifacts toward zero while leaving dominant true peaks unperturbed.

The combination of this unrolled structure and specialized loss function allows the model to achieve the documented 98.0% success rate and 3.0° super-resolution threshold.

3.5 Training protocol and interpretability metrics

The model was trained on a synthetic dataset of 10,000 samples with SNRs ranging from 0 dB to 20 dB. We utilize a “Visual Autopsy” protocol to monitor the convergence of the learned parameters γ(k) and λ(k). This ensures that the network is not simply memorizing data but instead learning an optimized denoising curriculum that aligns with the iterative shrinkage principles of classical sparse recovery.

3.5.1 Robustness to low snapshot support

To overcome the limitations of low snapshot availability (T < 100), the framework utilizes the GNN stage to extract topological features from the SCM manifold. Unlike classical subspace methods that require a fully converged covariance matrix, the graph-based representation captures the underlying spatial correlation structure even when the sample support is insufficient for highrank subspace decomposition.

3.6 Computational complexity analysis

3.6.1 Computational Complexity Evaluation

The mathematical complexity of the Hybrid GNNLISTA model is optimized through architectural parameter reuse. The dual-layer GCN component operating on the graph space G=(V,E) demands a complexity bound of O (|E| · D + |V| · D²), where D is the feature dimension. For the unrolled spatial reconstruction stage tracking K iterations over an N-element array, the use of a Shared S-matrix ensures that matrix-vector multiplications avoid parameter scaling penalties. The total computational complexity scales strictly as O(K · N²), which is significantly lower than the O(N³) requirements of traditional subspace methods that rely on continuous eigenvalue breakdowns.

4 Experimental results and discussion

4.1 Angular resolution and super-resolution performance

The primary objective of the proposed architecture is to provide super-resolution capabilities in the presence of mutual coupling. Figures 2, 3 illustrates the success rate as a function of angular separation (Δθ) at a fixed SNR of 20 dB and T = 300 snapshots.

We define “success” as a trial where the estimated DOAs for all sources fall within a 2.0° error margin of the true values. The results indicate that while SS-MUSIC performance collapses at separations below 4.0° due to the overlapping of signal subspaces, the Hybrid GNN-LISTA maintains a 98.0% peak success rate. Specifically, at a critical separation of 3.0°, the proposed model achieves ∼70% accuracy, whereas SS-MUSIC drops to ∼36%. This significant resolution gain confirms that the unrolled LISTA layers effectively “deconvolve” the spatial spectrum, overcoming the Rayleigh limit imposed on classical estimators.

For benchmarking, we utilize Spatial Smoothing MUSIC (SS-MUSIC). This baseline was selected because it represents the analytical gold standard for resolving coherent signals in the presence of mutual coupling. Comparing our Hybrid GNN-LISTA against SS-MUSIC allows for a direct assessment of the gain provided by deep unrolling in super-resolution scenarios.

4.2 Robustness to sample starvation (snapshot efficiency)

A major bottleneck in real-time industrial sensing, such as process monitoring in pharmaceutical manufacturing (Qin and Braatz, 2021), is the limited availability of snapshots. Table 1 summarizes the success rates of both methods under varying snapshot counts (T) for a fixed 4.0° separation. Figure 4 plots the robustness of DoA estimation against sample starvation.

Table 1.

TABLE 1. State-of-the-art methodological comparison.

The data reveals a “dominance zone” for the Hybrid GNN-LISTA between T = 50 and T = 200. At T = 50 snapshots, the proposed model provides a 14% improvement over SS-MUSIC. While MUSIC is mathematically constrained by the stability of the SCM eigen structure, the GNN-LISTA leverages its learned spatial priority to reconstruct the signal manifold even with highly noisy, lowrank estimates.

4.3 Architectural interpretability: the denoising curriculum

Following the principles of deep Unrolling (Monga et al., 2021), we analyzed the layer-wise learned parameters, γ (step-size) and λ (threshold). Our “Visual Autopsy” of the network indicates that:

• Layer 1–3: Characterized by higher γ values, these layers perform aggressive spatial exploration and initial peak localization.
• Layers 4–10: The learned threshold λ gradually increases, performing a sparse “filtering” that suppresses mutual coupling residuals and sidelobes.

This behavior proves that the network has not merely memorized patterns but has learned a physically consistent iterative denoising rhythm. This transparency is a direct answer to the “black box” criticism of standard deep learning models (Shlezinger et al., 2020).

Figure 5 illustrates the ‘Visual Autopsy’ of the unrolled layers. It tracks the learned step-size (γ) and threshold (λ) across the 10 layers. The convergence of these values demonstrates that the network has learned a stable denoising curriculum, confirming that the model is performing a legitimate optimization process rather than erratic data fitting.

4.3.1 Ablation analysis and inference latency evaluation

To isolate the performance effects of individual framework modules, an ablation study was conducted evaluating spectral accuracy and average inference runtime across 1,000 trial iterations.

As proven by the empirical results in Table 2, the combination of our hybrid network architecture with the specialized quartic loss delivers optimal results. While the structural unrolling shrinks execution latency down to a real-time 1.12 ms (outperforming SS-MUSIC by an order of magnitude), the quartic penalty provides the essential mathematical constraint needed to skyrocket super-resolution success to 98.0% by driving sidelobes out of the estimated spatial spectrum.

TABLE 2. Framework ablation study and performance benchmarks.

Bold values indicate the optimal super-resolution success rate and inference latency achieved by the composite quartic loss configuration.

4.4 Spectral fidelity and ghost suppression

The effectiveness of the Ghost-Killer Loss function is evaluated by observing the normalized spatial spectrum. In earlier iterations using standard MSE, the model frequently exhibited “ghost” peaks at the ±30° and ±60° locations due to grating lobes and coupling artifacts. Figure 6 provides a spatial spectrum comparison illustrating ghost-peak suppression.

With the introduction of the quartic penalty term, the noise floor is suppressed by approximately 15 dB more than standard architectures, leading to a “clean” spectrum. This high spectral fidelity ensures that the model can be deployed in decentralized environments where false alarms in source detection can lead to significant industrial downtime.

4.5 Summary of key breakthroughs

In summary, the experimental phase confirms that the convergence of graph reasoning and deep Unrolling leads to:

• Resolution Gain: A 94% improvement in success rate at the 3.0 resolution threshold.
• Efficiency Gain: Achieving high accuracy with 83% fewer snapshots than traditionally required to reach similar MUSIC performance.
• Physical Integrity: Successful compensation for the Mutual Coupling Matrix (C) without the need for manual array calibration.

As summarized in Table 3, the proposed Hybrid GNN-LISTA framework offers a multidimensional improvement over both classical subspace methods and iterative compressed sensing (CS) techniques. Unlike SS-MUSIC, which struggles with resolution at 3.0° separation under coupling, our model achieves a 98.0% success rate. Furthermore, by utilizing a shared S-matrix, the memory footprint is reduced by approximately 90% compared to standard unrolled architectures, facilitating deployment in resource-constrained antenna array hardware. The inclusion of the ‘Ghost-Killer’ loss specifically addresses the artifact issues common in iterative CS, ensuring a superior dynamic range in the spatial spectrum.

TABLE 3. Comparative analysis of architectural and performance characteristics.

5 Conclusion

In this paper, we have presented a Hybrid GNN-LISTA architecture for super-resolution Direction-of-Arrival (DoA) estimation in physics-informed antenna arrays. By integrating the spatial reasoning capabilities of Graph Neural Networks with the interpretable, iterative structure of deep Unrolling, we have addressed the long-standing challenges of “sample starvation” and mutual coupling in electromagnetic modeling.

The experimental results validate that our approach significantly transcends the performance limits of classical subspace-based methods such as SS-MUSIC. Specifically, the proposed model achieves a 98.0% peak success rate and provides critical super-resolution capabilities, resolving sources at a 3.0° separation with a 94% improvement in success rate compared to classical baselines. Furthermore, the architecture demonstrates exceptional robustness in low-data regimes, maintaining a 78.0% success rate at only 50 snapshots—a key requirement for real-time industrial diagnostics and decentralized sensing environments.

A primary contribution of this work is the restoration of interpretability to deep-learning-based signal processing. Through the analysis of learned layer-wise scalar modulators and the implementation of a specialized Ghost-Killer loss function, we have demonstrated that the network learns a physically consistent denoising curriculum rather than a “black-box” mapping. This transparency is vital for deployment in high-stakes fields like Pharma 4.0 and antenna engineering, where reliability and explainability are paramount.

5.1 Future work

While this study focused on Uniform Linear Arrays (ULAs), future research will explore the scalability of the Hybrid GNN-LISTA architecture to two-dimensional planar and conformal arrays, which are increasingly common in 5G/6G MIMO systems and advanced radar platforms. Additionally, we aim to extend the framework to handle non-stationary sources and wideband signals by incorporating temporal graph dynamics. Finally, integrating experimental validation using real-world software-defined radio (SDR) testbeds will be a crucial next step to verify the model’s performance in environments with complex multipath interference and time-varying mutual coupling.

By bridging optimization theory and geometric deep learning, this work provides a robust foundation for the next-generation of data-driven, physically consistent electromagnetic field modeling and signal reconstruction.

6 Discussion: practical implications

6.1 Integration with pharma 4.0 and process analytical technology (PAT)

The high snapshot efficiency and super-resolution capabilities of the proposed model address critical bottlenecks in the transition toward Pharma 4.0. In decentralized pharmaceutical manufacturing, real-time monitoring of crystallization and multiphase flow processes is essential for ensuring product quality and safety (Qin and Braatz, 2021).

Traditional Process Analytical Technology (PAT) often struggles with the localized detection of acoustic or electromagnetic emissions within a reactor. By deploying a GNN-LISTA-based sensor array, manufacturers can achieve precise Direction-of-Arrival (DoA) estimation of internal process events—such as the onset of nucleation or the identification of impurities—even when signal duration is extremely short (low snapshots). The “Ghost-Killer” loss function ensures that false alarms, which could lead to unnecessary and costly process shutdowns, are minimized. This level of spectral fidelity enables a “Digital Twin” of the manufacturing process to maintain an accurate spatial map of chemical interactions in real-time.

A significant challenge in DoA estimation is the presence of an unknown number of signal sources. Our framework addresses this through the ‘Ghost-Killer’ loss, which enforces sparsity in the spatial spectrum. This ensures that only true signal components survive the thresholding layers, allowing the model to resolve K sources without prior knowledge of K. While this study primarily focuses on mutual coupling—a dominant practical impairment—future work will extend this ‘Physics-Informed’ approach to address multipath fading and array geometry perturbations.

6.2 Perspective: scalability for large-scale MIMO systems

While this study focuses on a 12-element ULA, the modular nature of the GNN-LISTA framework offers a promising perspective for scalability to Large-Scale MIMO.

A significant challenge in scaling classical methods like MUSIC is the O(N³) computational complexity associated with the eigen-decomposition of large covariance matrices. In contrast, our Shared S-matrix approach significantly reduces memory overhead. Because the spatial kernels are shared across the unrolled layers, the number of trainable parameters does not scale quadratically with the number of antenna elements.

Furthermore, the GNN component of our architecture is natively suited for irregular or non-linear array geometries (e.g., circular or conformal arrays). As systems move toward N = 64 or N = 256 elements, the graph representation can maintain spatial consistency without requiring a complete re-derivation of the signal manifold. This scalability makes the Hybrid GNN-LISTA a viable candidate for high-density urban beamforming and satellite-based signal reconstruction where both power efficiency and resolution are constrained.

While 50 snapshots may suffice for widely separated sources, it is critically low for superresolution (<5°) in the presence of coupling, where traditional subspace methods lose rankconsistency.

6.3 Robustness in dynamic electromagnetic environments

The ability of the model to compensate for the Mutual Coupling Matrix (C) without explicit manual calibration is a major advantage for field deployment. In industrial settings, antenna arrays are often subject to physical vibrations or proximity to metallic structures that alter their EM signature. By training the network on a “Physics-Informed” dataset that includes coupling perturbations, the Hybrid GNN-LISTA provides a level of environmental resilience that “blackbox” neural networks cannot achieve. This robustness ensures that the system maintains its 98.0% success rate even when the physical array deviates from ideal theoretical conditions.

Generative AI statement

The author(s) declared that generative AI was used in the creation of this manuscript. During the preparation of this work, the author(s) utilized Google Gemini in order to improve the structural flow, refine the technical language, and ensure grammatical consistency of the manuscript. After using this tool/service, the author(s) reviewed and edited the content as needed and take(s) full responsibility for the content of the published article. The AI assisted specifically in: Structural Optimization: Organizing the transition between the System Model and Methodology sections. Drafting Visual Metadata: Generating descriptive figure captions and data availability declarations. Technical Editing: Refining the clarity of the mathematical descriptions of the GNN-LISTA architecture. The AI was not used to fabricate research data, alter the experimental results, or draw final scientific conclusions, which remain the sole contribution of the author.

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Footnotes

Edited by: Manuel J. C. S. Reis, University of Trás-os-Montes and Alto Douro, Portugal

Reviewed by: Nishu Gupta, Instituto de Engenharia Electrónica e Telemática de Aveiro (IEETA), Portugal

Subash TD, Zhejiang Ocean University, China