Deep Learning-Based Optimization for Maritime Relay Networks

Abstract

The complexity and uncertainty of the marine environment pose significant challenges to the stability and coverage of communication links. Due to the limited coverage range of traditional onshore base stations (BSs) in marine environments, relay technology has become an essential approach to extending communication coverage. However, the rapid variations in marine wireless channels and the complexity of hydrological conditions make it extremely difficult to obtain accurate channel state information (CSI). In particular, dynamic environmental factors such as waves, tides and wind speed cause channel parameters to fluctuate significantly over time. To address these challenges, this paper proposes a cooperative communication strategy based on ships and designs a novel channel modeling method to accurately capture the characteristics of marine wireless channels. Furthermore, a deep learning-based optimization scheme is proposed, which formulates the relay selection problem as a spatiotemporal classification task. By integrating the spatial positions of candidate relays and their communication conditions, the proposed scheme enables real-time selection of the optimal relay while evaluating link connectivity probabilities under hydrological influences. Simulation results confirm that the proposed method achieves high accuracy even in rapidly changing marine environments.

1. Introduction

The last two decades have witnessed great progress in terrestrial wireless communications to support various high-rate low-latency applications. However, the development of maritime communications is still far behind. In recent years, providing broadband communications for marine users has attracted attention due to rapidly increasing oceanic activities, including oil exploitation, sea farming, scientific expedition, tourism, etc.

Currently, obtaining access to onshore base stations (BSs) and terrestrial cellular networks is the most appropriate method to provide high-rate and low-cost network service. Hence, techniques to extend the coverage of onshore BSs have been investigated, among which relaying is a mature and effective one. Based on this technology, researchers have continuously explored new applications to enhance long-distance communication capabilities. For example, a multi-hop solution utilizing balloons to achieve broadband communication in remote maritime areas was proposed in [1], providing a novel perspective on addressing maritime communication challenges. Additionally, the work in [2] explored the design of relay networks for multi-domain maritime operations (land, sea, air and space), highlighting the critical need for connectivity and efficient network performance across these diverse environments. Moreover, the flexibility and mobility of unmanned aerial vehicles (UAVs) have garnered significant attention as a means to improve the coverage of maritime communication networks [3]. To address the challenges posed by the lack of stable connections between maritime communication nodes, researchers have proposed integrating UAVs into the topology of sixth-generation maritime communication networks [4]. In scenarios where direct connections to terrestrial base stations are limited, UAV swarms can significantly improve wireless communication performance. However, due to the mobility of UAVs, the state of communication links can easily be affected during operations. To address this issue, Ref. [5] proposed a channel relay selection method based on UAV position prediction to improve the stability of communication links. Additionally, Ref. [6] introduced a Q-learning-based relay selection scheme using the signal-to-noise ratio (SNR) to identify the optimal relay for multi-hop transmissions. Ref. [7] proposed an optimal relay selection scheme based on deep neural networks (DNNs), where the DNN is used to predict the outage probability and perform optimal relay selection, thereby improving network performance. Ref. [8] introduced the optimization of relay selection to improve the performance of Device-to-Device (D2D) communication systems in complex wireless propagation environments.

Broadband reliable wireless communications depend on accurate and instantaneous knowledge of channel information. However, in highly dynamic oceanic environments, channel information changes with hydrological and meteorological parameters, including wind, wave, moisture and temperature, to name a few. The fluctuations of these parameters present significant challenges for estimating channel information in maritime communications, which makes link analysis a challenging task. Ref. [9] proposed a novel opportunistic routing scheme aimed at improving the efficiency of nautical wireless ad hoc networks, particularly in situations where nodes frequently switch between network types. The scheme enhances packet delivery performance by optimizing relay node selection and utilizing prediction mechanisms. Ref. [10] made an initial attempt to leverage the advantages of machine learning to solve the relay selection problem in two-hop networks. In [11], a solution based on Q-learning was proposed to optimize relay selection in scenarios where channel state information (CSI) is unavailable. Furthermore, in reference [12], the relay selection problem was formulated as a multi-class classification problem, and a decision tree-based approach was used to address the challenges of estimating and collecting accurate eavesdropper CSI. Additionally, Ref. [13] proposed a deep learning-based channel estimation algorithm that models the rapid variations in communication channels as two-dimensional images, using known pilot positions to predict unknown channel responses. Deep learning can extract implicit features from large amounts of history data and therefore is regarded as a promising solution for connectivity analysis without full channel information.

Maritime communications are significantly influenced by unique environmental factors, such as seawater evaporation and tidal movements, which render terrestrial communication channel models inadequate for oceanic applications. Current maritime channel models often rely on well-established terrestrial frameworks but fail to capture the complex and dynamic propagation characteristics of marine environments [14]. This highlights the critical need for wireless channel models specifically tailored to maritime conditions. For instance, Ref. [15] analyzed wireless channel models in oceanic environments and identified key challenges, including highly dynamic network topologies and sparse user distributions, which lead to communication instability and high operational costs [16]. In addition, traditional modeling methods demonstrate clear limitations in rapidly changing oceanic conditions. Their lack of real-time adaptability and comprehensiveness prevents them from effectively characterizing the dynamic behavior of wireless channels in maritime environments.

In recent years, deep learning has shown significant potential in various fields. For example, Ref. [17] attempted to combine machine learning techniques with wireless communications. A relay selection scheme in millimeter-wave D2D communications within fifth-generation cellular networks is presented in [18]. The authors in [19] introduced a deep learning-assisted cooperative framework called Predictive Relay Selection (PRS), which enhances the quality of CSI by accurately predicting fading channels. Building upon this progress, the researchers in [20] explored and attempted to apply machine learning in maritime communications. In [21], the authors proposed a deep learning-based approach that analyzes channel and navigation information to predict link availability, providing a foundation for intelligent maritime communication networks.

However, existing strategies often rely on accurate CSI. Measuring and integrating such critical data pose extreme challenges in the dynamic and unpredictable oceanic environment. To overcome these challenges, this paper proposes a deep learning-based optimization strategy for maritime relay networks that eliminates the need for real-time CSI estimation. By enabling multi-hop transmission and prioritizing link connectivity, the proposed approach ensures efficient data transmission and enhances communication reliability. Moreover, link connectivity serves as a critical metric, offering intelligent node selection to optimize overall network performance in maritime scenarios.

In this paper, a channel modeling method is proposed to approximate the oceanic environment, and a deep learning-based scheme is developed to analyze relay networks. The main contributions of this work are as follows:

To capture the highly dynamic characteristics of wireless environments over the ocean, a channel modeling method is established, allowing channel parameters to vary randomly.

A deep learning-based optimization scheme is proposed, transforming the relay selection problem into a spatiotemporal classification and prediction task, effectively addressing the challenge of obtaining accurate channel state information (CSI) in dynamic oceanic conditions.

By incorporating hydrological factors into the evaluation of connectivity probability, this approach systematically analyzes communication systems in complex marine environments. The results demonstrate that optimizing relay selection strategies significantly enhances the system’s adaptability to dynamic conditions and improves data transmission reliability.

2. System Model

In the maritime communication environment, a dual-hop network model is considered, which comprises a source node ship (S), a destination node ship (D) and K relay node ships. Each ship travels at varying speeds (v) and in diverse directions ($\theta$), thereby generating a dynamic network environment (depicted in Figure 1). At two distinct time points, $t_1$ and $t_2$, the positions of the ships and their communication links vary, reflecting the instability engendered by the movement of ships within maritime communication networks. In this scenario, data transmission is facilitated via relay nodes that employ the Decode-and-Forward (DF) protocol.

The source node ship must transmit data to the destination node ship; however, the long distance between them makes direct communication impractical. To overcome this limitation, relay node ships serve as intermediaries to facilitate data transmission.

To further expand, a multi-hop network model is considered, where data are required to pass through multiple relay clusters to reach the destination node (D). The network is composed of a source node (S), a destination node (D) and M relay clusters. These relay clusters are labeled $C_1$, $C_2$, …, $C_M$, with each cluster containing multiple relay nodes denoted as $R_{m1}$, $R_{m2}$, …, $R_{m{K}_m}$, where $m=1,...,M$. S and D cannot communicate directly, and except for the nodes in $C_M$, other relay nodes also cannot establish direct connections with D.

The transmission process begins with S, broadcasting data to the first relay cluster $C_1$. Subsequently, a relay node from cluster $C_1$ is selected to receive the data and forward them to the next cluster, $C_2$. This process continues sequentially across the relay clusters until D successfully receives the data. The complete transmission path is formed by the relay nodes selected from each cluster, with the data being forwarded hop-by-hop along the path to ensure their successful delivery to the destination node.

2.1. Marine Communication Environment

In maritime environments, the highly dynamic and variable propagation characteristics of oceanic wireless channels render traditional modeling methods inadequate. To address this issue, a novel channel modeling approach is proposed, which integrates both large-scale and small-scale fading effects through the combination of statistical distribution models with time-series analysis. This approach is designed to provide a more accurate representation of wireless channel characteristics within oceanic environments. For simplicity, it is assumed that the statistical distribution of the channel follows a Rayleigh distribution, and the SNR is modeled as a random variable following an exponential distribution. Additionally, the temporal correlation of channel parameters, including the impacts of wave height and wind speed variations on channel characteristics, is taken into account. This facilitates a more accurate representation of the dynamic propagation characteristics of wireless channels in maritime environments. The channel model is formulated as follows:

$$H\left(t\right)=g\left(t\right)·\alpha \left(t\right),$$

(1)

where $g\left(t\right)={\displaystyle \frac{1}{L^n}}$ represents large-scale fading, L denotes the distance, n is the path loss exponent and $\alpha \left(t\right)$ represents small-scale fading.

Furthermore, accurately obtaining instantaneous CSI in maritime environments is challenging. In traditional terrestrial communication systems, where complete CSI is available for each transmission link, the relay selected must satisfy the following condition:

$$K^{*}=arg\underset{1\le k\le K}{max}{R}_k=arg\underset{1\le k\le K}{max}{\displaystyle \frac{1}{2}}log\left(1+{\displaystyle \frac{P{g}_k}{{\sigma }^2}}\right),$$

(2)

where $R_k$ represents the end-to-end rate when the k-th relay is cooperating. $K^{*}$ denotes the selected relay, while K is the total number of relay nodes.

2.2. Data Acquisition

In maritime communication networks, the communication quality among ships is influenced by multiple factors, including ship speed (v), direction ($\theta$), distance (d) and wave height (h). At a specific time point t, ships may be elevated to different sea surface heights due to the influence of ocean waves. Let $h_1$, $h_2$ and $h_3$ denote the wave heights of the ships in their respective sea area. Assuming that wave height is a random variable with a Gaussian distribution, its fluctuations can significantly affect communication quality. Furthermore, particular attention is given to the wave height differences, denoted as $\Delta h_1=|h_{1t}-h_{2t}|$ and $\Delta h_2=|h_{2t}-h_{3t}|$, as these differences directly affects the stability of signal transmission. To determine the optimal relay node ship, two critical distance parameters are introduced, which can be expressed as:

$$L_{1t}=\sqrt{{\left(d_{1t}\right)}^2+{|h_{1t}-h_{2t}|}^2},$$

(3)

$$L_{2t}=\sqrt{{\left(d_{2t}\right)}^2+{|h_{2t}-h_{3t}|}^2},$$

(4)

where, at time t, $L_{1t}$ represents the distance between the source node ship and the relay ship, while $L_{2t}$ represents the distance between the relay ship and the destination node ship. Additionally, $d_{1t}$ and $d_{2t}$ capture the horizontal spatial distances between adjacent ships at this time, t.

$v_{1t}$, $v_{2t}$ and $v_{3t}$ denote the scalar speed of ships at the time point t and are given by:

$$v_{1t}=V_{1t}+{\Delta }_{1t},$$

(5)

$$v_{2t}=V_{2t}+{\Delta }_{2t},$$

(6)

$$v_{3t}=V_{3t}+{\Delta }_{3t},$$

(7)

where $V_{1t}$, $V_{2t}$ and $V_{3t}$ are the speeds measured by ship logs and ${\Delta }_{1t}$, ${\Delta }_{2t}$ and ${\Delta }_{3t}$ are the random deviations. ${\theta }_{1t}$, ${\theta }_{2t}$ and ${\theta }_{3t}$ denote the angles of cruising directions deviating from the nominal direction at the time point t.

In a Rayleigh fading channel, the fading coefficient $\alpha$ is a random variable, and the instantaneous SNR ($\gamma$) of the channel is also a random variable. ${\gamma }_{S,R_k}$ and ${\gamma }_{R_k,D}$ are independent exponential random variables representing the instantaneous SNR of the small-scale fading from S to R and from R to D, respectively. In the considered system, both small-scale fading and large-scale fading effects are taken into account. Consequently, the overall average end-to-end SNR reflects the combined influence of both fading types. The expression for the overall average SNR is presented as follows:

$$\overline{\gamma }={\displaystyle \frac{{P\left|\alpha \right|}^2}{W{L}^n}},$$

(8)

where P represents the transmission power, and W represents the noise power.

In the DF protocol, the parameters ${\gamma }_{R_k,1}$ and ${\gamma }_{R_k,2}$ denote the end-to-end SNRs of the first and second hops on the k-th relay’s communication link, respectively. The equivalent SNR for this link is determined as the minimum value between ${\gamma }_{R_k,1}$ and ${\gamma }_{R_k,2}$. According to the maximum function criterion, the relay with the highest equivalent SNR is selected as the optimal relay. The expressions for ${\gamma }_{R_k}$ and ${\gamma }^{*}$ are given as follows:

$$\begin{array}{cc}\hfill {\gamma }_{R_k}& =min\{{\overline{\gamma }}_{R_k,1},{\overline{\gamma }}_{R_k,2}\}\hfill \\ \hfill & =min\left\{{\displaystyle \frac{P_t{\left|{\alpha }_{R_k,1}\right|}^2}{W{L}_1^n}},{\displaystyle \frac{P_t{\left|{\alpha }_{R_k,2}\right|}^2}{W{L}_2^n}}\right\},\hfill \end{array}$$

(9)

$${\gamma }^{*}=\underset{1\le k\le K}{max}{\gamma }_{R_k}.$$

(10)

In maritime communications systems, link quality is a critical determinant of communication performance and reliability. Thus, assessing the connectivity probability of the optimal link is essential to ensuring stable and robust communication. In a dual-hop network, the communication process involves the source node transmitting data to the destination node through an intermediate relay node. For a single relay node, the connectivity probability is defined as the probability that the SNR of the relay node exceeds the outage threshold.

The mathematical expression for the connectivity probability of a single relay node is presented in Equation (11) as follows:

$$\begin{array}{cc}\hfill P_r({\gamma }_{R_k}>{\gamma }_0)& =P_r\left(min\left\{{\gamma }_{R_k,1},{\gamma }_{R_k,2}\right\}>{\gamma }_0\right)\hfill \\ \hfill & =P_r\left({\gamma }_{R_k,1}>{\gamma }_0,{\gamma }_{R_k,2}>{\gamma }_0\right)\hfill \\ \hfill & =P_r\left({\gamma }_{R_k,1}>{\gamma }_0\right)·P_r\left({\gamma }_{R_k,2}>{\gamma }_0\right)\hfill \\ \hfill & ={\int }_a^b{\int }_{{\gamma }_0}^{\infty }{f}_1\left(x\right)f_{\lambda }\left(y\right)dxdy+{\int }_a^b{\int }_{{\gamma }_0}^{\infty }{f}_2\left(x\right)f_{\lambda }\left(y\right)dxdy\hfill \\ \hfill & ={\int }_a^b{\left[1-F_{\gamma }\left({\gamma }_0\right)\right]}^2{f}_{\lambda }\left(y\right)dy\hfill \\ \hfill & ={\int }_a^b{e}^{-2{\gamma }_0\lambda }·{\displaystyle \frac{1}{b-a}}dy\hfill \\ \hfill & ={\displaystyle \frac{1}{2{\gamma }_0(b-a)}}\left(e^{-2a{\gamma }_0}-e^{-2b{\gamma }_0}\right).\hfill \end{array}$$

(11)

For a system with multiple relay nodes, the connectivity probability must consider the scenario where the maximum SNR among all relay nodes exceeds the threshold (${\gamma }_0$). Accordingly, the connectivity probability of the entire system is expressed as:

$$P_r({\gamma }^{*}>{\gamma }_0)=1-P_r({\gamma }^{*}\le {\gamma }_0).$$

(12)

Since ${\gamma }^{*}$ represents the maximum SNR among all relay nodes, the connectivity probability can be further expressed as:

$$\begin{array}{cc}\hfill P_r({\gamma }^{*}>{\gamma }_0)& =1-\prod _{k=1}^K{P}_r({\gamma }_{R_k}\le {\gamma }_0)\hfill \\ \hfill & =1-\prod _{k=1}^K\left(1-P_r({\gamma }_{R_k}>{\gamma }_0)\right)\hfill \\ \hfill & =1-\prod _{k=1}^K\left(1-{\displaystyle \frac{1}{2{\gamma }_0(b-a)}}\left(e^{-2a{\gamma }_0}-e^{-2b{\gamma }_0}\right)\right),\hfill \end{array}$$

(13)

where K denotes the total number of relay nodes.

In a multi-hop network, data transmission is performed through hop-by-hop forwarding across multiple relay clusters until the destination node is reached. The success rate of each hop is determined by the selection of the relay node in the preceding hop and its corresponding communication performance.

The connection probability of the first hop is defined as the probability that the source node (S), through broadcasting, communicates with the K relay nodes in the first cluster $C_1$ and selects the relay node with the highest SNR. The probability that the SNR of a single relay node exceeds ${\gamma }_0$ is calculated as:

$$\begin{array}{cc}\hfill P_r({\gamma }_{R_k,1}>{\gamma }_0)& ={\int }_a^b\left[{\int }_{{\gamma }_0}^{\infty }\lambda e^{-\lambda x}dx\right]{\displaystyle \frac{1}{b-a}}d\lambda \hfill \\ \hfill & ={\int }_a^b{e}^{-\lambda {\gamma }_0}{\displaystyle \frac{1}{b-a}}d\lambda \hfill \\ \hfill & ={\displaystyle \frac{1}{{\gamma }_0(b-a)}}\left(e^{-a{\gamma }_0}-e^{-b{\gamma }_0}\right).\hfill \end{array}$$

(14)

For the K independent relay nodes in cluster $C_1$, the probability that the maximum SNR among these nodes exceeds the threshold is expressed as:

$$\begin{array}{cc}\hfill P_r({\gamma }_{C_1}^{*}>{\gamma }_0)& =1-\prod _{k=1}^K{P}_r({\gamma }_{R_k,1}\le {\gamma }_0)\hfill \\ \hfill & =1-\prod _{k=1}^K\left(1-P_r({\gamma }_{R_k,1}>{\gamma }_0)\right)\hfill \\ \hfill & =1-\prod _{k=1}^K\left(1-{\displaystyle \frac{1}{{\gamma }_0(b-a)}}\left(e^{-a{\gamma }_0}-e^{-b{\gamma }_0}\right)\right).\hfill \end{array}$$

(15)

For the connection probability of the intermediate $M-1$ hops, each hop is influenced by the SNR distribution of the relay node selected in the preceding hop. Let the connection probability of the m-th cluster be denoted as $P_r({\gamma }_{C_m}^{*}>{\gamma }_0\mid {\gamma }_{C_{m-1}}^{*})$. The SNR of the current relay node follows an exponential distribution, which is expressed as:

$$f_{{\gamma }_{R_k,m}}(x\mid {\gamma }_{C_{m-1}}^{*})={\lambda }_{m,k}\left({\gamma }_{C_{m-1}}^{*}\right)e^{-{\lambda }_{m,k}\left({\gamma }_{C_{m-1}}^{*}\right)x},$$

(16)

where ${\lambda }_{m,k}$ represents the channel parameter of the k-th relay node in the m-th cluster, which is related to the SNR of the previous hop.

The cumulative distribution function (CDF) of the maximum SNR in the current cluster is given as:

$$\begin{array}{cc}\hfill P_r({\gamma }_{C_m}^{*}\le x\mid {\gamma }_{C_{m-1}}^{*})& =\prod _{k=1}^K{P}_r({\gamma }_{R_k,m}\le x\mid {\gamma }_{C_{m-1}}^{*})\hfill \\ \hfill & =\prod _{k=1}^K{F}_{{\gamma }_{R_k,m}}(x\mid {\gamma }_{C_{m-1}}^{*}).\hfill \end{array}$$

(17)

Therefore, the connection probability in the current cluster can be expressed as:

$$\begin{array}{cc}\hfill P_r({\gamma }_{C_m}^{*}>{\gamma }_0\mid {\gamma }_{C_{m-1}}^{*})& =1-P_r({\gamma }_{C_m}^{*}\le {\gamma }_0\mid {\gamma }_{C_{m-1}}^{*})\hfill \\ \hfill & =1-\prod _{k=1}^K\left(1-e^{-{\lambda }_{m,k}\left({\gamma }_{C_{m-1}}^{*}\right){\gamma }_0}\right),m=2,...,M.\hfill \end{array}$$

(18)

For the M-th cluster, the connection probability of the selected relay node to D is given by:

$$\begin{array}{cc}\hfill P_r({\gamma }_D>{\gamma }_0\mid {\lambda }_M)& ={\int }_{{\gamma }_0}^{\infty }{\lambda }_M{e}^{-{\lambda }_{M}x}dx\hfill \\ \hfill & =e^{-{\lambda }_M{\gamma }_0}.\hfill \end{array}$$

(19)

Consequently, the overall connection probability is expressed as:

$$\begin{array}{cc}\hfill P_r({\gamma }_D>{\gamma }_0)& ={\int }_a^b{e}^{-\lambda {\gamma }_0}{\displaystyle \frac{1}{b-a}}d\lambda \hfill \\ \hfill & ={\displaystyle \frac{1}{b-a}}·{\displaystyle \frac{e^{-a{\gamma }_0}-e^{-b{\gamma }_0}}{{\gamma }_0}}.\hfill \end{array}$$

(20)

The connectivity probability of the entire system must account for the probabilities of all individual hops. It can be expressed as:

$$P_{\mathrm{total}}=P_r({\gamma }_{C_1}^{*}>{\gamma }_0)·\prod _{m=2}^M{P}_r({\gamma }_{C_m}^{*}>{\gamma }_0\mid {\gamma }_{C_{m-1}}^{*})·P_r({\gamma }_D>{\gamma }_0\mid {\gamma }_{C_M}^{*}).$$

(21)

To comprehensively understand the changing trends in the sustainable propagation link states, extensive data were collected while considering multiple factors that influence ship-to-ship communication. By integrating historical data from maritime networks, a thorough analysis of link state variations was performed, resulting in the development of representative training samples.

3. Relay Network Based on Deep Learning

The proposed algorithm utilizes a CNN-GRU architecture to optimize relay link prediction and evaluate link connectivity, thereby enhancing communication quality and link reliability in maritime environments while enabling longer successful communication cycles. The model efficiently extracts features from input data through Convolutional Neural Network (CNN) layers and passes the processed data to Gated Recurrent Unit (GRU) layers for prediction and adaptive adjustments. Similar to Long Short-Term Memory (LSTM) networks, GRUs effectively handle time-series data, extracting useful temporal information and identifying long-term dependencies. The core structure of the algorithm, as shown in Figure 2, consists of two main components: a feature extractor and an estimator.

The dynamic nature of the maritime environment and the motion of ships pose significant challenges to predicting the optimal relay link. The proposed solution addresses these challenges by utilizing features extracted from time-series data. It incorporates several factors that influence communication quality between ships, including their speed and direction of movement, the distance between ships, signal quality (SNR) and the presence of K relays. By collectively analyzing these factors, the optimal relay is selected, and the connectivity probability of the optimal relay link is determined.

The input features are structured as parameters over a series of time points, represented as samples X with multiple attributes. These samples are fed into the convolutional layer (CL) of the CNN model and are defined as:

$$X=\{x_1,x_2,\dots ,x_t\},$$

(22)

$$x_t=\left[v_{1t},v_{2t},v_{3t},{\theta }_{1t},{\theta }_{2t},{\theta }_{3t},L_{1t},L_{2t},h_{1t},h_{2t},h_{3t},{\gamma }_{1t},{\gamma }_{2t}\right].$$

(23)

The output data correspond to the prediction label and connectivity probability of training samples, which can be expressed as:

$$Y=\{({\widehat{K}}_1^{*},P_1),({\widehat{K}}_2^{*},P_2),\dots ,({\widehat{K}}_n^{*},P_n)\},$$

(24)

where ${\widehat{K}}_i^{*}$ represents the predicted label of the i-th time slot, and $P_i$ denotes the corresponding connectivity probability.

In maritime networks, huge amounts of historical data are produced during navigation and can be utilized as training samples. In this work, it is neither economical nor practical to collect these data within a short time. To validate the proposed algorithm and to reduce the cost of experiment, we designed a software-aided data generating method. This method implements the data acquisition presented in Section 2.2 and also generates both navigation and hydrological parameters. Input data and output labels are produced out of these data and packed into training and prediction samples. The Algorithm 1 of method is summarized as follows.

Algorithm 1: Data Generating Method.

Step 1: Generate input data       Set up parameters.    1.1 Generate distances       Generate ${\mathit{d}}_{\mathbf{1}}$ and ${\mathit{d}}_{\mathbf{2}}$ with uniform distribution ${\mathit{d}}_{\mathit{i}}\sim U(d_a,d_b)$, which reflects the spatial randomness of node deployment within a defined operational area.    1.2 Generate wave heights       Generate ${\mathit{h}}_{\mathbf{1}}$, ${\mathit{h}}_{\mathbf{2}}$ and ${\mathit{h}}_{\mathbf{3}}$ with Gaussian distribution ${\mathit{h}}_{\mathit{i}}\sim N(\mu ,{\sigma }^2)$, which is widely used in oceanography to approximate the variability of wave heights under diverse sea conditions.    1.3 Generate speed and direction       Generate ${\mathit{v}}_{\mathbf{1}},{\mathit{v}}_{\mathbf{2}}$, and ${\mathit{v}}_{\mathbf{3}}$, ${\mathit{\theta }}_{\mathbf{1}},{\mathit{\theta }}_{\mathbf{2}}$, and ${\mathit{\theta }}_{\mathbf{3}}$ follow the uniform distribution ${\mathit{v}}_{\mathit{i}}\sim U(v_a,v_b)$, ${\mathit{\theta }}_{\mathit{i}}\sim U({\theta }_a,{\theta }_b)$. These distributions capture the random nature of node movement while remaining within environmental and operational constraints.    1.4 Generate SNR       Assuming a Rayleigh channel, the generated channel parameter $\lambda$ follows a uniform distribution $\lambda \sim U({\lambda }_a,{\lambda }_b)$, which models the random yet bounded variability in channel conditions.       The SNR vectors ${\gamma }_1$ and ${\gamma }_2$ follow the exponential distribution ${\gamma }_i\sim Exp\left(\lambda \right)$, which is consistent with the theoretical Rayleigh fading model.    1.5 Pack data       Repeat steps 1.1–1.4 for $2K$ times.       Re-organize all data and pack them into $\mathit{X}=[{\mathit{X}}_{\mathit{t}},{\mathit{X}}_{\mathit{p}}]$. Step 2: Generate output data    2.1 ${\widehat{K}}^{*}$ is the selection result of the optimal strategy.    2.2 Monte Carlo method is used to calculate the connectivity probability of the optimal result.       Pack the results into $\mathit{Y}=[{\mathit{Y}}_{\mathit{t}},{\mathit{Y}}_{\mathit{p}}]$. Step 3: Predict relay selection    3.1 Pack the training samples $[{\mathit{X}}_{\mathit{t}},{\mathit{Y}}_{\mathit{t}}]$ and train the network.    3.2 Feed ${\mathit{X}}_{\mathit{p}}$ into the network and obtain the results ${\mathit{Y}}_{\mathit{p}}^{\mathbf{\prime }}$.    3.3 Compute the mean squared error (MSE) by comparing ${\mathit{Y}}_{\mathit{p}}$ and ${\mathit{Y}}_{\mathit{p}}^{\mathbf{\prime }}$.

3.1. Data Features Extractor

The feature extractor is specifically designed for highly mobile nodes in dynamic networks, with a focus on efficiently capturing essential patterns from fluctuating input data. Initially, CNN layers are utilized to extract latent features from raw data. These layers transform low-level input variations into high-level representations, which are subsequently passed to fully connected layers for further processing. This hierarchical structure significantly enhances the model’s capacity to recognize complex relationships, such as those between communication links. CNNs, as multilayer neural networks within a deep supervised learning framework, are designed to process diverse data types, including time-series and spatially structured data. Although traditionally applied to two-dimensional image processing tasks, the same principles can be effectively extended to handle one-dimensional sequential data [22]. In the proposed scheme, the CNN architecture incorporates convolutional and pooling layers, where each convolutional layer applies a set of kernels for feature extraction. The resulting features are passed through a Rectified Linear Unit (ReLU) activation function, which introduces non-linearity into the model. Pooling layers, particularly max-pooling, play a critical role in reducing the feature dimensions while retaining essential information, thereby improving computational efficiency and mitigating overfitting risks. The integration of convolutional and pooling layers not only reduces the complexity of the feature space but also ensures robustness to variations in the input data. By preserving the invariance of feature mappings, the model adapts effectively to dynamic input conditions. The max-pooling operation can be represented by (25) as:

$$y_t^l=maxpooling(x_t^{l-1},s_{\mathrm{scale}},s_{\mathrm{stride}}),$$

(25)

the pooling window size $s_{\mathrm{scale}}$ is set to 2 × 2, and the stride $s_{\mathrm{stride}}$ is set to 2. These settings are chosen to balance computational efficiency and feature preservation, ensuring effective downsampling of the input feature maps while maintaining critical data characteristics.

The CNN extracts data features through layer-by-layer convolution and pooling operations. The filter’s window size and stride can be flexibly set based on the input data dimensions and the complexity of the features to be captured. The convolutional layers of the CNN are capable of identifying spatial features relevant to relay node selection, such as wave height, wind speed and ocean current direction. The extraction of these features helps enhance the predictive performance of the model. The calculation formula is shown in Equation (26).

$$l_t=tanh(x_t\ast k_t+b_t),$$

(26)

where $l_t$ represents the output value after convolution, tanh is the activation function, $x_t$ is the input vector, $k_t$ is the weight of the convolution kernel and $b_t$ is the bias of the convolution kernel.

The tanh activation function introduces non-linearity, which is essential for enabling the CNN to learn complex feature mappings. Additionally, the tanh function outputs values in the range of [−1, 1], ensuring smooth transitions between negative and positive inputs, which is beneficial for capturing variations in features such as wave height and navigation direction, thereby enhancing the robustness and stability of the network.

3.2. Sequence Learning Estimator

A training model consisting of a GRU layer and a Dropout layer is defined, where the GRU model contains only two gates, the reset gate and the update gate, and the specific structure is illustrated in Figure 3. Compared to LSTM, the GRU has a simpler architecture, fewer parameters and a faster training speed. The output data from the feature extractor are fed into the GRU, which performs temporal modeling and captures temporal dependencies within the sequential data.

The GRU layer processes sequential data using its reset and update gates. At each time step t, the update gate ($z_t$) determines how much of the previous hidden state ($h_{t-1}$) should be retained in the current hidden state, while the reset gate ($r_t$) decides how much of the past information to forget. The output of the GRU layer is a sequence of hidden states, which encode the temporal dynamics of the input data. The calculations are as follows:

$$z_t=\sigma \left(W_z·\left[h_{t-1},x_t\right]+b_z\right),$$

(27)

$$r_t=\sigma \left(W_r·\left[h_{t-1},x_r\right]+b_r\right),$$

(28)

$${\tilde{h}}_t=\tan \mathrm{h}\left(W·\left[r_t\ast h_{t-1},x_t\right]+b\right),$$

(29)

$$h_t=(1-z_t)\ast h_{t-1}+z_t\ast {\tilde{h}}_t,$$

(30)

where W and b are weights and bias, $\sigma$ is the sigmoid activation function and $\tilde{h}$ is the candidate hidden state, while $h_t$ is the final hidden state, respectively.

The hidden states produced by the GRU, specifically the hidden state of the last time step ($h_T$), are passed through a Dropout layer to prevent overfitting and improve generalization. Dropout randomly sets a fraction of the units in $h_T$ to zero during training, reducing the model’s reliance on specific neurons. This prevents the model from overfitting to particular features and encourages better generalization. The resulting feature vector, which encapsulates temporal dependencies from the entire input sequence, is then fed into a fully connected (FC) layer. Finally, accurate classification predictions are achieved through the Softmax classifier.

The integration of GRU layers into the model architecture not only accelerates the training process but also enhances the ability to process effective link features extracted by CNNs. By leveraging the spatial feature extraction capabilities of CNNs and the sequential modeling strengths of GRUs, the proposed algorithm captures both the spatial correlations present in the input data and the temporal dependencies across time steps. Specifically, spatial features are extracted through CNN layers and then sequentially passed to the GRU layers, where the temporal dynamics of the data are modeled. This combined architecture allows the model to effectively adapt to the dynamic environment of maritime communication networks. As a result, the algorithm achieves more accurate data modeling and prediction, addressing the challenges posed by real-time communication in complex oceanic environments.

4. Simulation Experiment and Analysis

In deep learning-based relay networks, the acquisition of training data plays a critical role in model training, directly impacting the effectiveness and accuracy of the experiments. MATLAB R2022a simulations are utilized to generate raw data by modeling key channel characteristics in maritime communication environments, including Rayleigh fading and large-scale path loss. These simulations are subsequently used to construct channel dataset samples for training. The key communication parameter settings for generating the dataset are summarized in

Table 1. Communication parameter.

Parameter	Value
Carrier frequency	5.8 GHz
Transmit antenna height	8 m
Receive antenna height	8 m
Number of relays	4
Transmission power	20∼25 dBm
Noise power	10∼11 dBm
Path loss	90∼200 dB
SNR	10∼30 dB
Speed	$0\le v\left(t\right)\le 30\mathrm{km}/\mathrm{h}$
Direction	$-\pi /6\le \theta \left(t\right)\le \pi /6$
Distance	20 km

.

Before training the model, the sample data must be preprocessed. As the deep learning-based maritime communication relay network optimization model follows a supervised learning paradigm, each set of relay link data must be labeled with the optimal relay and its corresponding connectivity probability. During the training process, the network architecture is iteratively adjusted, and parameter values are fine-tuned to maximize predictive performance.

Table 2. Model simulation parameters.

Parameter	Value
CNN convolutional filters	64
Convolution kernel size	$2\times 1$
Activation function	Relu
GRU hidden units	16
Dropout rate	0.2
Batch size	128
Epochs	100
Learning rate	0.01

provides the parameter settings for the CNN-GRU method used in this experiment.

To validate the consistency between the theoretical analysis and the simulation results for connectivity probability, Figure 4 presents the connectivity probability under different system parameters. Figure 4a,b demonstrate that an increase in the number of relay nodes K significantly enhances the connectivity probability, while higher SNR thresholds ${\gamma }_0$ lead to a reduction. Figure 4c illustrates a gradual decline in connectivity probability as the number of clusters M increases, emphasizing the trade-off inherent in multi-cluster systems. The high levels of consistency between the theoretical and simulation results validate the accuracy of the formulas and the reliability of the methods.

The accuracy of the model is evaluated by comparing the predicted labels of the test set with the corresponding optimal choices. To assess the link prediction performance of the CNN-GRU model, the mean squared error (MSE) is employed as the performance metric. The MSE is defined as

$$MSE={\displaystyle \frac{1}{N}}\sum _{i=1}^N{\left(p_i-{\widehat{p}}_i\right)}^2,$$

(31)

where N represents the number of predictions, $p_i$ denotes the true value and ${\widehat{p}}_i$ represents the predicted value. The smaller the MSE value, the better the prediction performance.

Figure 5 illustrates the variation in the model accuracy with the increase in the number of relays. It can be observed that the accuracy of the CNN-GRU-based relay selection scheme decreases slightly as K grows but remains consistently above 92%, which is within an acceptable range. This trend can be attributed to the fact that when the number of relays is small (e.g., two relays), the model can more effectively learn and memorize the dataset features, resulting in higher accuracy on the training set. However, as the number of relays increases, the model tends to overfit the training data, leading to a decrease in performance on the test data.

Figure 6 illustrates the prediction accuracy of the model’s labels over the entire time-series. It is evident that when comparing the predicted relay choices of the test set with corresponding optimal relay labels, the accuracy of the CNN-GRU-based model stabilizes at approximately 0.90 after 100 iterations, effectively meeting the expected performance. This result can be attributed to the model’s training process, where the CNN-GRU-based relay selection model maps the feedback data from each link to the corresponding optimal label. Through iterative training, the model establishes connections and retains the learned mappings. During the prediction phase, the model accurately classifies and predicts the data based on these established mappings.

To evaluate the effectiveness of the proposed scheme, a series of experiments were conducted to assess its performance. The comprehensive analysis and experimental results are as follows.

The MSE performance of the proposed scheme under different Rayleigh fading channel parameters is depicted in Figure 7. In such channels, the received SNR follows an exponential distribution, where the distribution parameter $\lambda$ was set to follow two different uniform distributions: $U(1,2)$ and $U(1,10)$. When $\lambda$ follows the distribution $U(1,2)$, the smaller fluctuation range of $\lambda$ results in a limited set of samples during training. This restricted range introduces a bias into the generated samples, as they fail to represent the diversity of real-world conditions. Consequently, the model trained on these biased samples demonstrates reduced performance, as reflected in the higher MSE values. Conversely, when $\lambda$ follows $U(1,10)$, the broader fluctuation range provides a more diverse and representative sample set. This increased diversity improves the model’s ability to generalize to different channel conditions, thereby reducing the MSE and enhancing performance.

Figure 8 analyzes the impact of different input sequence lengths on system performance. Longer input sequences provide more historical information, enabling the model to better capture the temporal correlations of signals and thereby enhancing its performance. In contrast, shorter input sequences lead to information loss and insufficient model complexity, which negatively affect system performance. When the input length is set to 56, the system demonstrates the poorest performance due to an imbalance between model complexity and information richness.

Maritime communication systems are significantly more affected by environmental conditions compared to terrestrial systems, with wave height being one of the critical influencing factors. Higher wave heights introduce greater signal fluctuation and instability due to harsher transmission conditions, leading to increased MSE. Conversely, lower wave heights provide a more stable environment, which improves signal quality and reduces MSE.

To analyze this impact, we evaluated the MSE performance of the proposed scheme under two different average wave heights ($\mu =1.5$ and $\mu =4.5$), as shown in Figure 9. The results indicate that as the wave height increases (i.e., the average wave height $\mu$ grows), the MSE also increases accordingly.

Compared to the impact of $\lambda$ parameters on the MSE, wave height primarily affects the physical stability of signals, whereas Rayleigh fading introduces statistical variations in channel conditions. Both factors have a significant impact on system performance, highlighting the necessity of considering environmental variability in the design of robust maritime communication systems.

5. Conclusions

In this paper, we explore the problem of relay-assisted maritime communication networks and propose a channel modeling method that closely represents real maritime environments. To effectively address the uncertainties associated with maritime conditions, we introduce a deep learning-based optimization scheme for maritime relay networks. This scheme predicts relay selection and calculates connectivity probabilities over multiple time points without relying on instantaneous CSI. Through simulation experiments and analysis, the proposed scheme demonstrates high accuracy and low MSE when compared to the optimal strategy. The model effectively captures the impact of system and environmental variations on performance, especially under different conditions of wave height and Rayleigh fading. Although the MSE fluctuates with changes in wave height and fading fluctuation range, the overall accuracy remains stable, validating the effectiveness and robustness of the method and demonstrating its reliability in complex maritime communication environments. However, the current model does not explore in depth the potential impact of extreme weather conditions (such as storms, fog, etc.) on performance. Future research can further investigate the performance under these extreme environmental conditions and enhance the model’s robustness in such scenarios. Furthermore, the real-world variability of the dataset may affect the model’s generalization ability, so future work will focus on optimizing the scheme in scenarios with higher data variability.