Graph Neural Network-Based Beamforming Optimization for Multi-BS RIS-Aided Communication Systems

Abstract

The optimization of beamforming in multi-base station (multi-BS) reconfigurable intelligent surface (RIS)-aided systems is a challenging task due to its high computational complexity. This paper first demonstrates that an optimized multi-BS system exhibits superior communication performance compared to a centralized large-scale single-BS system. To efficiently solve the complex beamforming problem in the multi-BS environment, this paper proposes a novel optimization solver based on a graph neural network (GNN) that models the physical structure of the system. Experimental results show that the proposed GNN solver finds solutions of higher quality, achieving a 42% performance increase with 45% less total computational complexity compared to a conventional iterative optimization method. Furthermore, when compared to other complex AI models such as transformer and Bi-LSTM, the proposed GNN achieves similar state-of-the-art performance while having less than 1% of the parameters and a fraction of the computational cost. These findings demonstrate that the GNN is a powerful, efficient, and practical solution for beamforming optimization in multi-BS RIS-aided systems, satisfying the demands for performance, computational efficiency, and model compactness.

1. Introduction

The advent of beyond-5G (B5G) and 6G communication systems has spurred research into innovative technologies to meet the ever-increasing demands for higher data rates, broader coverage, and improved energy efficiency. Reconfigurable intelligent surfaces (RISs) have emerged as a promising technology capable of intelligently manipulating the wireless propagation environment by controlling the phase shifts of a large number of passive reflecting elements [1,2,3]. An RIS-assisted system can create favorable propagation paths, mitigate interference, and extend coverage, all with minimal energy consumption thanks to the passive nature of its elements.

While initial research has predominantly focused on single-base station (single-BS) scenarios, this paradigm faces fundamental limitations in practical large-scale deployments. Single-BS systems, even those with large numbers of antennas, suffer from coverage holes at the cell edges and are vulnerable to link blockages by obstacles. In contrast, a distributed multi-base station (multi-BS) architecture offers significant advantages. By leveraging macroscopic diversity, multi-BS architectures can provide more robust and uniform coverage while mitigating the impact of shadowing and fading [4]. However, this transition also presents a formidable challenge in the form of joint active and passive beamforming optimization. The RIS must now coherently combine signals originating from multiple geographically distributed sources, creating a large-scale coupled optimization problem that is highly sensitive to inter-cell interference. The computational complexity of finding an optimal solution for this coordinated beamforming problem increases dramatically compared to the single-BS case [5].

1.1. Related Work

Traditional approaches to this problem often rely on iterative optimization techniques such as semidefinite programming (SDP) or alternating optimization (AO) [6,7,8]. While capable of finding high-quality solutions, these methods suffer from prohibitively high computational complexity, often scaling polynomially with the number of RIS elements or antennas. This renders them unsuitable for real-time applications where channel conditions change dynamically.

To overcome this complexity, artificial intelligence (AI) has been explored as a powerful alternative [9,10]. While the multi-BS system in this paper is a form of distributed network, it differs from other prominent architectures. Compared to cell-free massive MIMO [11], which uses a massive number of simple access points, the approach in this paper focuses on a scalable number of powerful base stations. Unlike reconfigurable distributed antenna and reflecting surface (RDARS) [12] or distributed RIS [13] approaches, which distribute reconfigurable or passive surfaces, this paper focuses on the challenge of coordinating distributed active base stations with a centralized RIS.

Despite these architectural differences, all distributed systems share the fundamental challenge of high-complexity joint optimization. This paper focuses on tackling this common computational bottleneck by proposing a novel GNN-based solver. Therefore, the contribution is not a competing architecture but rather an efficient optimization tool that could enhance the practicality of these systems and serve as a foundation for more complex future architectures such as multi-BS, multi-RIS/RDARS systems. Deep neural networks (DNNs), convolutional neural networks (CNNs), and long short-term memory (LSTM) networks have been applied to various wireless communication problems, demonstrating the ability to perform near-instantaneous inference after an offline training phase [14]. However, these architectures often treat the channel information as a generic vector or image, failing to exploit the underlying physical structure of the communication network. This leads to large and complex "black-box" models that require a massive number of parameters and may struggle to adapt to changes in network topology.

Graph neural networks (GNNs) offer a compelling solution to these inefficiencies [15]. A wireless network can be naturally represented as a graph, with its base stations, users, and RIS elements as nodes and the channels as edges. GNNs are specifically designed to operate on such graph-structured data, allowing them to explicitly model the network topology and the relationships between its components. This structural inductive bias makes GNNs a highly promising yet less explored candidate for the multi-BS RIS beamforming problem, with the potential to learn the complex channel interactions more efficiently and with significantly fewer parameters [16,17].

1.2. Contribution and Novelty of This Paper

This paper addresses the limitations of existing methods by proposing a novel and efficient optimization solver based on a GNN. To establish a clear motivation for this research, the paper first experimentally validates the superiority of an optimized multi-BS RIS system over a centralized large-scale single-BS system. The paper then demonstrates, using the GNN architecture itself, that using preprocessed SVD-based channel features is a more effective input strategy for AI solvers than using the full raw channel information. The primary contribution is the proposed GNN-based solver, which is novel in its application to the multi-BS RIS beamforming problem. Through comprehensive simulations, this solver is shown to not only outperform a conventional iterative optimization technique in both solution quality and total computational cost but also to achieve state-of-the-art performance with unparalleled efficiency compared to other AI architectures.

2. System Model and Problem Formulation

This section details the mathematical framework for the communication systems under investigation. First, the conventional single-BS architecture is presented in order to establish a baseline and highlight its inherent limitations. Then, the multi-BS architecture is introduced as a solution, followed by a detailed comparison of two distinct modeling approaches for this more complex system. Finally, based on the selected model, the signal transmission process and the beamforming optimization problem are formally defined.

2.1. Single-BS RIS-Aided System: Model and Limitations

The analysis begins with a conventional single-BS RIS-aided system, as depicted in Figure 1. In this setup, a single BS equipped with $N_t$ antennas serves a group of $N_u$ users via a shared RIS composed of $N_{\mathrm{ris}}$ elements.

The channel from the BS to the RIS is denoted by ${\mathbf{H}}_1\in {\mathbb{C}}^{N_{\mathrm{ris}}\times N_t}$, while the channel from the RIS to the users is ${\mathbf{H}}_r\in {\mathbb{C}}^{N_u\times N_{\mathrm{ris}}}$. The RIS applies a phase shift matrix, $\mathsf{\Theta }=\mathrm{diag}(e^{j{\theta }_1},\dots ,e^{j{\theta }_{N_{\mathrm{ris}}}})$ to the incident signal. The end-to-end effective channel is then formulated as

$${\mathbf{H}}_{\mathrm{eff},\mathrm{single}}\left(\mathsf{\Theta }\right)={\mathbf{H}}_r\mathsf{\Theta }{\mathbf{H}}_1.$$

(1)

Let the signal transmitted by the BS be $\mathbf{x}\in {\mathbb{C}}^{N_t\times 1}$ and the signal received by the users be $\mathbf{y}\in {\mathbb{C}}^{N_u\times 1}$. The input–output relationship, including the additive white Gaussian noise (AWGN) vector $\mathbf{n}$, is provided by

$$\left(\begin{array}{c}{y}_1\\ ⋮\\ y_{N_u}\end{array}\right)={\mathbf{H}}_{\mathrm{eff},\mathrm{single}}\left(\begin{array}{c}{x}_1\\ ⋮\\ x_{N_t}\end{array}\right)+\left(\begin{array}{c}{n}_1\\ ⋮\\ n_{N_u}\end{array}\right).$$

(2)

While this centralized architecture is straightforward to model, it suffers from fundamental limitations. The system’s performance is highly dependent on a single link, making it vulnerable to deep fading or physical blockages and often creating coverage holes at the cell edge. These limitations motivate the transition to a more distributed architecture.

2.2. Multi-BS RIS-Aided System: Architecture and Advantages

To overcome the limitations of the single-BS setup, this paper focuses on a distributed multi-BS system, as shown in Figure 2. In this architecture, K coordinated BSs, each with $N_t$ antennas, jointly serve the same user group through the shared RIS.

This distributed approach offers significant advantages over its centralized counterpart. By leveraging signals from multiple geographically dispersed locations, the system benefits from macroscopic diversity. This mitigates the impact of large-scale fading and shadowing, providing more robust and uniform coverage across the entire service area. However, the core challenge now lies in jointly optimizing the RIS phase shifts in order to coherently combine signals from these multiple sources, which requires a more sophisticated system model.

2.3. Channel Modeling Approaches for the Multi-BS System

To analyze the complex multi-BS system, the individual channels from all K BSs must be integrated into a unified multiple-input multiple-output (MIMO) model. Let ${\mathbf{H}}_{1,k}\in {\mathbb{C}}^{N_{\mathrm{ris}}\times N_t}$ be the channel from the k-th BS to the RIS and ${\mathbf{H}}_r\in {\mathbb{C}}^{N_u\times N_{\mathrm{ris}}}$ be the channel from the RIS to the user group. Two primary and mathematically equivalent approaches for this integration are presented below.

2.3.1. The Computational Model (Horizontal Concatenation)

This model prioritizes computational efficiency by creating compact dense matrices. The aggregate BS-to-RIS channel ${\mathbf{H}}_{1,\mathrm{agg}}$ is formed by concatenating the individual channel matrices horizontally:

$${\mathbf{H}}_{1,\mathrm{agg}}=[{\mathbf{H}}_{1,1}{\mathbf{H}}_{1,2}\dots {\mathbf{H}}_{1,K}].$$

(3)

In this compact representation, the RIS-to-user channel is simply ${\mathbf{H}}_r$, while the RIS phase matrix is the original $\mathsf{\Theta }$. The effective channel is then derived by the sequential multiplication of these three matrices:

$$\begin{array}{cc}\hfill {\mathbf{H}}_{\mathrm{eff}}& ={\mathbf{H}}_r\mathsf{\Theta }{\mathbf{H}}_{1,\mathrm{agg}}\hfill \\ \hfill & =[{\mathbf{H}}_r\mathsf{\Theta }{\mathbf{H}}_{1,1}{\mathbf{H}}_r\mathsf{\Theta }{\mathbf{H}}_{1,2}\dots {\mathbf{H}}_r\mathsf{\Theta }{\mathbf{H}}_{1,K}].\hfill \end{array}$$

(4)

2.3.2. The Structural Model (Block-Diagonal)

This model preserves the modularity of the distributed system by creating large matrices that maintain the separation of each signal path. The BS-to-RIS channel ${\mathbf{H}}_1$ is constructed as a large block-diagonal matrix:

$${\mathbf{H}}_1=\left(\begin{array}{cccc}{\mathbf{H}}_{1,1}& \mathbf{0}& \dots & \mathbf{0}\\ \mathbf{0}& {\mathbf{H}}_{1,2}& \dots & \mathbf{0}\\ ⋮& ⋮& \ddots & ⋮\\ \mathbf{0}& \mathbf{0}& \dots & {\mathbf{H}}_{1,K}\end{array}\right).$$

(5)

To align with this structure, the other matrices are expanded. The RIS-to-user channel ${\mathbf{H}}_2$ is formed by repeating ${\mathbf{H}}_r$ horizontally, while the RIS phase matrix ${\mathbf{T}}_{\mathrm{eff}}$ is formed by using the Kronecker product to apply $\mathsf{\Theta }$ to each parallel path.

$$\begin{array}{cc}\hfill {\mathbf{H}}_2& =[{\mathbf{H}}_r{\mathbf{H}}_r\dots {\mathbf{H}}_r]\hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill {\mathbf{T}}_{\mathrm{eff}}& ={\mathbf{I}}_K\otimes \mathsf{\Theta }\hfill \end{array}$$

(7)

The effective channel is then calculated as ${\mathbf{H}}_{\mathrm{eff}}={\mathbf{H}}_2{\mathbf{T}}_{\mathrm{eff}}{\mathbf{H}}_1$. The derivation below shows how the expanded matrices interact to form the final effective channel, which is identical to the one derived using the computational model.

$$\begin{array}{cc}\hfill {\mathbf{H}}_{\mathrm{eff}}& =[{\mathbf{H}}_r\dots {\mathbf{H}}_r]\left(\begin{array}{ccc}\mathsf{\Theta }& & \\ & \ddots & \\ & & \mathsf{\Theta }\end{array}\right)\left(\begin{array}{ccc}{\mathbf{H}}_{1,1}& & \\ & \ddots & \\ & & {\mathbf{H}}_{1,K}\end{array}\right)\hfill \\ \hfill & =[{\mathbf{H}}_r\mathsf{\Theta }{\mathbf{H}}_{1,1}{\mathbf{H}}_r\mathsf{\Theta }{\mathbf{H}}_{1,2}\dots {\mathbf{H}}_r\mathsf{\Theta }{\mathbf{H}}_{1,K}]\hfill \end{array}$$

(8)

2.3.3. Model Selection

As the above derivations demonstrate, both models produce the identical mathematical result for the effective channel. Therefore, the choice between them is not about correctness but rather utility for the specific task. The objective of this paper is to design a multi-stage optimization algorithm that requires analyzing each BS–RIS link as a distinct subproblem. For this purpose, the block-diagonal model is adopted for several compelling reasons.

First, the proposed solution methodologies fundamentally rely on processing the individual channel matrices ${\mathbf{H}}_{1,k}$. The block-diagonal model provides a natural framework for this decomposition, whereas the monolithic representation of the computational model would require an artificial and counterintuitive slicing process.

Furthermore, this structural separation offers significant computational advantages in terms of scalability. A prime example is matrix inversion, which is common in advanced beamforming techniques. Inverting a large and dense $(K{N}_t\times K{N}_t)$ matrix has a complexity of $\mathcal{O}\left({\left(K{N}_t\right)}^3\right)$. In contrast, inverting the corresponding block-diagonal matrix only requires inverting K smaller $(N_t\times N_t)$ matrices, resulting in a much lower complexity of $\mathcal{O}\left(K{N}_t^3\right)$. This implies that for achieving a large antenna array, it is far more computationally efficient to increase the number of distributed BSs (K) rather than concentrating a massive number of antennas ($N_t$) at a single location. This aligns perfectly with the trend towards distributed and cell-free architectures. Therefore, for its superior logical alignment with the proposed modular optimization strategy and its critical advantages in computational scalability, the block-diagonal model is adopted throughout this paper.

2.4. Signal Transmission and Problem Formulation

Using the selected block-diagonal model, the end-to-end effective channel is denoted by ${\mathbf{H}}_{\mathrm{eff}}\left(\mathsf{\Theta }\right)$. Let the aggregate signal vector transmitted from all K base stations be $\mathbf{x}$ and the received signal vector at the $N_u$ users be $\mathbf{y}$. The fundamental input–output relationship of the wireless channel, including the AWGN vector $\mathbf{n}$, is provided by

$$\mathbf{y}={\mathbf{H}}_{\mathrm{eff}}\left(\mathsf{\Theta }\right)\mathbf{x}+\mathbf{n}.$$

(9)

The following block matrix equation explicitly shows how the signal ${\mathbf{x}}_k$ from each BS is shaped by its unique channel path and subsequently combined, demonstrating that the final received signal is a superposition of signals from all K paths:

$$\left(\begin{array}{c}{y}_1\\ ⋮\\ y_{N_u}\end{array}\right)=[{\mathbf{H}}_r\dots {\mathbf{H}}_r]\left(\begin{array}{ccc}\mathsf{\Theta }& & \\ & \ddots & \\ & & \mathsf{\Theta }\end{array}\right)\left(\begin{array}{ccc}{\mathbf{H}}_{1,1}& & \\ & \ddots & \\ & & {\mathbf{H}}_{1,K}\end{array}\right)\left(\begin{array}{c}{\mathbf{x}}_1\\ ⋮\\ {\mathbf{x}}_K\end{array}\right)+\left(\begin{array}{c}{n}_1\\ ⋮\\ n_{N_u}\end{array}\right).$$

(10)

To transmit independent data streams, the signal $\mathbf{x}$ is generated by applying a linear precoding matrix $\mathbf{P}$ to a vector of data symbols $\mathbf{s}$ such that $\mathbf{x}=\mathbf{P}\mathbf{s}$. Substituting this into the channel equation yields the complete end-to-end signal model:

$$\mathbf{y}={\mathbf{H}}_{\mathrm{eff}}\left(\mathsf{\Theta }\right)\mathbf{P}\left(\mathsf{\Theta }\right)\mathbf{s}+\mathbf{n}.$$

(11)

To mitigate multi-user interference, this paper employs zero-forcing (ZF) precoding, where the precoder is centrally computed using the pseudo-inverse of the effective channel:

$$\mathbf{P}\left(\mathsf{\Theta }\right)=\beta {\mathbf{H}}_{\mathrm{eff}}^H\left(\mathsf{\Theta }\right){\left({\mathbf{H}}_{\mathrm{eff}}\left(\mathsf{\Theta }\right){\mathbf{H}}_{\mathrm{eff}}^H\left(\mathsf{\Theta }\right)\right)}^{-1}$$

(12)

where $\beta$ is a normalization factor to satisfy the total transmit power constraint.

The primary objective is to find the optimal RIS phase shift matrix $\mathsf{\Theta }$ that maximizes the overall system performance. Under the ZF scheme, the postprocessing signal-to-noise ratio (SNR) is directly related to the condition of the effective channel. A higher channel gain (as captured by the Frobenius norm) leads to a better-conditioned channel matrix, which in turn results in a higher SNR. This directly improves system-level metrics such as the bit error rate (BER) and spectral efficiency (SE). Therefore, the optimization problem is formulated as maximizing the squared Frobenius norm of the effective channel:

$$\underset{\mathsf{\Theta }}{max}\parallel {\mathbf{H}}_{\mathrm{eff}}{\left(\mathsf{\Theta }\right)\parallel }_F^2\mathrm{subject}\mathrm{to}\left|{\mathsf{\Theta }}_{n,n}\right|=1,\forall n=1,\dots ,N_{\mathrm{ris}}.$$

(13)

This is a non-convex optimization problem with a unit-modulus constraint, which is generally difficult to solve optimally.

3. Beamforming Optimization Techniques

This section presents the different techniques used to solve the optimization problem defined in (13). First, several baseline heuristic methods are described to serve as benchmarks. Then, the proposed GNN-based solver is detailed.

3.1. Baseline Heuristic Techniques

Before presenting the specific multi-BS heuristics, this section first defines a powerful method based on singular value decomposition (SVD) for optimizing the RIS phase shifts for a single BS–RIS link. This method serves as a foundational building block for the more advanced heuristics. The specific implementation is formally described in Algorithm 1.

Algorithm 1: SVD-Based RIS Phase Shift Optimization for a Single Link

Input: $H_1\in {\mathbb{C}}^{N_{\mathrm{ris}}\times N_t}$ (BS→RIS),   $H_2\in {\mathbb{C}}^{N_u\times N_{\mathrm{ris}}}$ (RIS→User) Perform singular-value decomposition on the RIS–BS covariance: $$H_1{H}_1^H=U\Sigma V^H,$$    then extract the right singular matrix V Select the first $N_u$ columns: $$V_{\mathrm{opt}}=V(:,1:N_u)$$ Form the phase-mapping matrix and extract its diagonal entries: $$M_{\mathrm{phase}}=V_{\mathrm{opt}}{H}_2,v_{\mathrm{phase}}=diag\left(M_{\mathrm{phase}}\right)$$ Build the phase-shift matrix: $${\mathsf{\Theta }}_{\mathrm{opt}}=diag\left(e^{-j\angle \left(v_{\mathrm{phase}}\right)}\right)$$ Output: ${\mathsf{\Theta }}_{\mathrm{opt}}$ (optimized RIS phase matrix)

With this foundational single-link optimization method established, three baseline heuristic techniques for the multi-BS problem are now introduced.

Technique 1 (Random Phase) serves as a performance lower bound. In this approach, the phase shifts for all $N_{\mathrm{ris}}$ elements are chosen independently and uniformly at random from the interval $[0,2\pi )$.

Technique 2 (SVD-Based Phase Averaging) extends the SVD principle to the multi-BS problem. For each subsystem k, an optimal phase vector ${\mathbf{t}}_k$ is computed using Algorithm 1. These K individually optimized phase vectors are then combined by calculating the vector sum of all ${\mathbf{t}}_k$ and setting the final RIS phase to the angle of this resultant vector.

Technique 3 (Optimized Weighted Combination) improves upon simple averaging. This method also uses the SVD-based phase vectors ${\mathbf{t}}_k$ as a starting point, but introduces a set of complex weights $w_k$ to create a weighted combination $\sum w_k{\mathbf{t}}_k$. A conventional iterative solver is then used to find the optimal weights ${\left\{w_k\right\}}^{*}$ that maximize the final objective function.

However, these baseline techniques possess fundamental limitations. Simple heuristics such as random phase and averaging cannot guarantee a coherent superposition of signals. Even the more sophisticated iterative method faces two critical challenges: it is prone to becoming trapped in local optima in the non-convex landscape, and its high computational cost makes it unsuitable for real-time environments. These limitations create a clear need for a new solution paradigm, motivating the shift towards data-driven AI approaches.

3.2. Proposed GNN-Based Solver

To overcome the limitations of conventional methods, this paper proposes a solver based on a GNN. This approach is designed not merely to find a solution, but to learn how to intelligently synthesize a globally optimal RIS configuration from a set of individually optimized yet mutually conflicting sub-solutions.

3.2.1. Overall Architecture

The core idea is to treat the GNN as an “information synthesis engine.” The architecture of the proposed solver (illustrated in Figure 3) is designed around this principle. The process begins not with raw channel data but with a set of high-quality SVD-based features (“Per-BS Optimal Phases”). These features are then fed into a pipeline of two GCN layers, a mean pooling layer, and a decoder to synthesize the final, coherent RIS phase vector θ. To validate this feature-based approach, a “GNN (Direct)” variant, which uses the identical pipeline but takes raw channel data as input, was also evaluated for comparison.

3.2.2. Core Components and Mechanism

The solver’s effectiveness comes from the careful design of its core components, which are the static graph structure, the dynamic input features, and the mechanism that processes them. The GNN operates on a graph that models the physical topology of the RIS panel, where the $N_{\mathrm{ris}}$ elements are the graph’s nodes. The connectivity between these nodes is defined by physical proximity.

This connection rule is translated into a 64 × 64 adjacency matrix $\mathbf{A}$ and its version with self-loops, $\tilde{\mathbf{A}}=\mathbf{A}+\mathbf{I}$. The elements of $\tilde{\mathbf{A}}$ are formally defined based on the grid coordinates of the RIS elements. Let $(r_k,c_k)$ be the row and column coordinates of the k-th element on the $\sqrt{N_{\mathrm{ris}}}\times \sqrt{N_{\mathrm{ris}}}$ grid. Then, each element of the matrix is provided by

$${\left(\tilde{\mathbf{A}}\right)}_{ij}=\left\{\begin{array}{cc}1,\hfill & \mathrm{if}max\left(\right|r_i-r_j|,|c_i-c_j\left|\right)\le 1\hfill \\ 0,\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

(14)

This rule states that a connection exists only if two elements are immediate neighbors, including diagonally. As visualized in Figure 4, the resulting matrix is static, sparse, and channel-independent, serving as a fixed `blueprint’ that dictates how information flows between RIS elements.

While the graph structure is static and channel-independent, the information that flows across this fixed blueprint is dynamic. This information is captured in the GNN’s node feature matrix, denoted as $\mathbf{F}$. The process begins with an initial feature matrix ${\mathbf{F}}^{\left(0\right)}$, where each node’s feature vector is the collection of optimal phase shifts suggested by each of the K base stations as derived via the SVD-based method. Each of these suggestions represents a locally optimal solution tailored to a single BS–RIS link, meaning that it is designed to maximize signal strength for its corresponding path while disregarding the impact on others.

However, these locally optimal solutions are mutually conflicting when considered for the global multi-BS system. Therefore, the fundamental task of the GNN is to learn the complex and nonlinear process of reconciling these conflicting inputs into a single coherent phase vector that is globally optimal. The GNN must learn to weigh the importance of each suggestion across different parts of the RIS, effectively finding a sophisticated compromise. This initial feature matrix, which artfully encapsulates the inherent challenge of the multi-BS problem, serves as the input to the first GCN layer. The core mechanism inside each GCN layer, known as neighborhood aggregation, then begins the iterative process of refining this information.

As conceptually visualized in Figure 5, the neighborhood aggregation process allows each node to update its state by gathering feature vectors from its physical neighbors. This intuitive process is efficiently implemented for all nodes simultaneously using the GCN propagation rule, where ${\mathbf{F}}^{\left(l\right)}$ is the matrix of all node features at layer l and ${\mathbf{W}}^{\left(l\right)}$ is a trainable weight matrix for that layer.

$${\mathbf{F}}^{(l+1)}=\sigma \left(\widehat{\mathbf{A}}{\mathbf{F}}^{\left(l\right)}{\mathbf{W}}^{\left(l\right)}\right)$$

(15)

The matrix multiplication $\widehat{\mathbf{A}}{\mathbf{F}}^{\left(l\right)}$ is the mathematical embodiment of the parallelized neighborhood aggregation. The full architecture applies this mechanism sequentially. The initial features are passed through two GCN layers. The first layer performs a local reconciliation of the conflicting suggestions by aggregating information from immediate physical neighbors. The second layer then establishes a broader panel-wide consensus by allowing this refined information to propagate across a two-hop neighborhood. This two-layer design was chosen to balance the need for capturing sufficient spatial correlation against the risk of ‘over-smoothing’, which is a common issue where node features become indistinguishable in deeper GNNs. After this feature refinement, the resulting node features are summarized by a mean pooling layer, which distills the rich node-level information into a single global consensus vector. Finally, this global vector is mapped by the decoder to the final unified phase vector θ.

3.2.3. Cost Definition for AI Solvers

The efficiency of the AI solvers is evaluated using two key metrics based on floating-point operations (FLOPs), which provides a hardware-independent measure of computational complexity.

First, the total computational cost is defined as the total FLOPs consumed during the entire training process to find an optimized solution for the given problem instance. This is approximated as follows: Total FLOPs ≈ (Inference FLOPs) × 3 × (Number of Epochs), reflecting the computational load of backpropagation. This metric serves as a strong proxy for the total energy consumption required by the solver.

Second, the inference cost is the number of FLOPs required for a single forward pass of the model. This metric reflects the operational cost (latency and energy) if the model were deployed as a pretrained real-time solver.

4. Performance Evaluation

4.1. Experimental Setup

The simulation framework is defined by the system parameters in

Table 1. System simulation parameters.

Parameter	Value
Number of BSs (K)	4
Antennas per BS ($N_t$)	8
Number of Users ($N_u$)	8
Number of RIS elements ($N_{\mathrm{ris}}$)	64
Channel Model	Rayleigh Fading
Modulation Scheme	16-QAM
Transmit Power (P)	0 to 20 dB
Noise Model	AWGN

and the AI model architectures detailed in

Table 2. Core architectural parameters of evaluated AI models.

Model	Layer/Input Details	Shape
GNN (Proposed)	Input Features (SVD-based)	($N_{\mathrm{ris}},K\times 2$)
	GCN Layer 1 (32 units)	($N_{\mathrm{ris}}$, 32)
	GCN Layer 2 (64 units)	($N_{\mathrm{ris}}$, 64)
	Linear Decoder	($N_{\mathrm{ris}}$)
GNN (Direct)	Input Features (Raw Channel)	($N_{\mathrm{ris}},(K{N}_t+N_u)\times 2$)
	GCN Layer 1 (128 units)	($N_{\mathrm{ris}}$, 128)
	GCN Layer 2 (256 units)	($N_{\mathrm{ris}}$, 256)
	Linear Decoder	($N_{\mathrm{ris}}$)
CNN	Input Image (SVD-based)	(8, 8, $K\times 2$)
	Conv2D Layer 1 (16 filters, 3 × 3 kernel)	(8, 8, 16)
	Conv2D Layer 2 (32 filters, 3 × 3 kernel)	(8, 8, 32)
	Linear Layer (64 units)	(64)
	Linear Decoder	($N_{\mathrm{ris}}$)
Bi-LSTM	Input Sequence (SVD-based)	($N_{\mathrm{ris}},K\times 2$)
	2-layer Bi-LSTM (128 hidden units)	($N_{\mathrm{ris}}$, 256)
	Linear Decoder	($N_{\mathrm{ris}}$)
Transformer	Input Sequence (SVD-based)	($N_{\mathrm{ris}},K\times 2$)
	Input Projection (Linear, 64 units)	($N_{\mathrm{ris}}$, 64)
	Self-Attention Encoder (3 layers, 4 heads/layer)	($N_{\mathrm{ris}}$, 64)
	Linear Decoder	($N_{\mathrm{ris}}$)
Common Training Hyperparameters
Optimizer	Adam
Learning Rate	$1\times {10}^{-4}$ for GNN (Direct), $1\times {10}^{-3}$ for all other models
Scheduler	ReduceLROnPlateau (patience = 20, factor = 0.5)
Training Epochs	200

. To create a stringent and realistic testbed, this paper models a non-line-of-sight (NLOS) environment using independent Rayleigh fading, assuming that direct BS–user links are fully blocked. This setup reflects a key challenge for future B5G/6G systems in which higher-frequency bands suffer from poor diffraction, making RIS-assisted communication essential in scattering-rich NLOS environments. All experiments are conducted on a single fixed channel realization in order to ensure a fair comparison of the optimization capabilities of different solvers on an identical problem instance. The solvers are evaluated on two primary fronts: communication performance, measured by Bit Error Rate (BER) and Spectral Efficiency (SE), and cost efficiency, measured by total computational cost (Total MFLOPs) and number of trainable parameters.

4.2. Results and Analysis

4.2.1. System Architecture Comparison: Single-BS vs. Multi-BS

First, the motivation for focusing on the multi-BS scenario is validated by comparing the fundamental performance of a distributed multi-BS system against a centralized single-BS counterpart. The performance in terms of both BER and SE is presented in Figure 6. The results for the multi-BS system consistently outperform the single-BS system across both metrics. Specifically, the optimized multi-BS system achieves a significant power gain for the same BER target and demonstrates substantially higher SE. This result highlights the advantage of macroscopic diversity and confirms that investigating optimization techniques for multi-BS systems is a crucial research direction.

4.2.2. Input Strategy for GNN Solvers: Direct vs. Feature-Based

To validate the core design choice of using preprocessed SVD-based features, this paper compares the ‘GNN (Proposed)’ and ‘GNN (Direct)’ models with the iterative solver (“Optimized”) as a baseline in Figure 7. The figure first confirms that both GNN solvers find higher-quality solutions than the iterative baseline. While both GNN variants achieve a similar high level of communication performance, the critical advantage of the proposed feature-based approach is its exceptional cost efficiency, requiring only a fraction of the parameters and computational cost of the “GNN (Direct)” variant, as detailed in

Table 3. Comprehensive analysis of performance and cost efficiency for all solvers.

Solver Method	Objective	Total MFLOPs	Inf. MFLOPs	Params. (K)	Perf./Param.
Technique 3 (Optimized)	2.155 $\times {10}^4$	∼266	-	-	-
GNN (Proposed)	3.077 $\times {10}^4$	∼144	0.24	6.6	4.69
CNN	3.071 $\times {10}^4$	∼228	0.38	12.1	2.54
Transformer	3.071 $\times {10}^4$	∼30,282	50.47	848.2	0.04
Bi-LSTM	3.068 $\times {10}^4$	∼20,772	34.62	553.0	0.06
GNN (Direct)	2.958 $\times {10}^4$	∼1884	3.14	59.8	0.49

. This ability to deliver comparable performance with significantly greater efficiency validates the feature-based strategy as the superior solution, which is consequently used for all subsequent AI model comparisons.

4.2.3. Performance Comparison of Feature-Based AI Solvers

The final communication performance of all feature-based AI models is presented and compared against the conventional iterative solver (“Technique 3”), which serves as a strong baseline. Figure 8 shows the communication performance in terms of both BER and SE as a function of transmit power. A key observation is that all AI-based solvers consistently outperform the iterative optimization baseline, achieving lower BER and higher SE. This indicates their superior ability to navigate the high-dimensional and non-convex solution space to find higher-quality solutions. Among the AI models, the GNN, transformer, and Bi-LSTM models form a top-tier performance group, achieving the best results that are closely clustered together.

The learning dynamics shown in Figure 9 offer deeper insights that go beyond the final performance metrics. The plot visually confirms that all AI models converge to a significantly higher objective value than the iterative baseline, which directly explains their superior communication performance. However, the most critical insight comes from comparing the convergence behavior of the different AI architectures. The proposed GNN demonstrates remarkable training stability, a highly desirable property for practical systems. This stability is visually evident in two key aspects of its convergence curve.

First, the standard deviation band across multiple training runs, represented in the figure by the shaded area, is exceptionally narrow for the “GNN (Proposed)” model. This contrasts sharply with the wider bands of the transformer and Bi-LSTM models, which indicates that the GNN’s training process is highly consistent, repeatable, and less sensitive to random initialization. In other words, it reliably reaches a high-quality solution regardless of minor variations in the training setup. Second, its convergence curve is notably smooth, indicating a stable and predictable learning trajectory without significant oscillations.

This desirable stability is a direct result of GNNs’ strong structural inductive bias. Unlike models such as transformer or Bi-LSTM, which treat the input as a generic sequence and must learn all relationships from scratch, GNNs are explicitly designed to leverage the underlying physical topology of the RIS panel, which is provided as a graph structure. This inherent understanding of which RIS elements are neighbors acts as a powerful regularizer, guiding the optimization process more efficiently and preventing it from exploring less plausible regions of the solution space. Therefore, while the final performance of the top-tier models is nearly indistinguishable, the GNN’s superior training stability makes it a more robust and dependable choice for practical systems. This observation reinforces the idea that factors other than final accuracy, such as training stability and computational cost, become the primary differentiators in model selection.

4.2.4. Comprehensive Cost Efficiency Analysis

To provide a holistic evaluation, the performance versus the computational and model costs is analyzed. Table 3 summarizes the key metrics for all solvers, including the final objective value, total computational cost, inference cost, and number of trainable parameters. The inference cost, measured in MFLOPs, serves as a hardware-agnostic proxy for crucial operational metrics such as inference latency and energy consumption.

The results presented in Table 3 reveal several key findings. The proposed GNN not only achieves a 42% higher objective value than the traditional solver but does so with 45% less total computational work, demonstrating the fundamental superiority of this AI-based approach.

Among the AI solvers, the proposed GNN stands out as the most efficient architecture. While achieving the highest objective value, it does so with orders of magnitude fewer parameters and less computational cost than complex models such as the transformer and Bi-LSTM. The “Performance per Parameter” metric, where the GNN scores an exceptional 4.69, quantitatively highlights this superior architectural efficiency. Furthermore, a cost–benefit analysis against the “GNN (Direct)” variant is telling; the proposed GNN achieves a higher objective value while using only 11% of the parameters and 8% of the inference FLOPs. This solidifies the proposed GNN as the most practical solution, since it delivers top-tier performance without the excessive overhead of larger or less efficient models, making it ideal for resource-constrained edge deployments.

5. Discussion

The results of this paper provide several key insights into the application of AI for RIS beamforming optimization.

First, the superiority of all AI-based solvers over the conventional iterative method suggests that data-driven approaches can more effectively navigate the complex non-convex solution space to find higher-quality solutions.

Second, the comparison among AI architectures reveals a clear tradeoff between model complexity and efficiency. While complex models such as the transformer and Bi-LSTM achieve high performance, they do so at a significant cost in terms of parameters and computations. This makes them less suitable for deployment in resource-constrained edge devices.

The proposed GNN strikes an optimal balance. Its remarkable efficiency stems from its inherent ability to exploit the underlying graph structure of the wireless system. This structural inductive bias allows the GNN to learn the essential relationships with unparalleled efficiency. This is quantitatively captured by the “Performance per Parameter” metric, where the GNN significantly outperforms all other models, as well as by its orders of magnitude lower total computational cost (Total MFLOPs).

This focus on computational efficiency is not merely a technical advantage but aligns with the broader goals of developing practical and sustainable AI solutions, often referred to as ‘Green Edge AI’ [18]. The total number of floating-point operations (FLOPs) serves as a strong and hardware-independent proxy for energy consumption. As such, the drastically lower computational cost of the GNN implies significant energy savings, making it a genuinely practical and sustainable solution for future networks.

Limitations and Future Work

This paper has several limitations that open up avenues for future research. The primary limitation is that the analysis is based on a single-channel realization. This approach was chosen in order to rigorously compare the optimization efficiency of different solvers on a fixed complex problem instance, rather than due to their generalization capabilities. Future work should focus on training the models on a large dataset of diverse channel realizations in order to evaluate and enhance their generalization capabilities across various user positions and mobility scenarios.

Furthermore, this paper assumes perfect channel state information (CSI). In practical systems, channel estimation errors are inevitable. Such inaccuracies would lead to a mismatch between the assumed channel and the actual channel, resulting in imprecise beamforming and residual inter-user interference, which would degrade the overall system performance in terms of BER and SE. Therefore, investigating the robustness of the proposed GNN under imperfect CSI is another critical direction for future work.

Finally, the GNN in this study was utilized as an optimization tool in a scenario where the training process itself finds the solution for a static problem. Adapting this efficient architecture into a pretrained general-purpose solver capable of near-instantaneous inference for dynamic channels remains a key objective. Such a pretrained model could potentially be updated for new network conditions without requiring full retraining, for instance through transfer learning or other fine-tuning techniques, which represents a promising direction for future investigation.

6. Conclusions

This paper has proposed a GNN-based solver for the complex reflective beamforming problem in multi-BS RIS-aided systems. The superiority of the multi-BS architecture over a centralized single-BS setup is first established. It is then demonstrated that AI-based solvers can find higher-quality solutions than conventional iterative optimization methods.

Through a comprehensive comparative analysis, this paper has shown that the proposed GNN is the most powerful and practical solution to the above problem. It surpasses the traditional optimization method in both performance and total computational efficiency. Furthermore, among various state-of-the-art AI architectures, the GNN delivers top-tier performance while also being exceptionally lightweight, requiring significantly fewer parameters and computational resources than complex models such as transformers and LSTMs. This remarkable efficiency stems from the inherent ability of GNNs to effectively model the graph structure of wireless communication systems. The findings in this paper suggest that GNNs are a highly promising paradigm for tackling complex optimization problems in future 6G networks.