Optimized Bi-LSTM Model for Short-Term Predicting of Ship State with Definitions of Surf-Riding and Broaching

Abstract

This paper introduces a hybrid prediction method that combines the Bi-LSTM neural network with definitions of surf-riding, wave-blocking and broaching to enhance the safety and stability of ship navigation. The hybrid method can accurately predict ships’ attitude motions and states by recognizing ship states and encoding them into one-hot representations. The Bi-LSTM model’s bidirectional learning capability captures significant temporal dependencies, enabling precise and timely predictions of complex maritime events across various conditions. Additionally, the direct output approach of state features improves prediction accuracy by eliminating intermediate steps, allowing for the better anticipation of and response to critical events. Validated with ship navigation data from autopilot simulations in wave conditions, the hybrid method outperforms conventional methods based on LSTM and Bi-LSTM models, demonstrating strong generalization capabilities and significantly contributing to safer and more stable ship navigation.

1. Introduction

The prediction of a ship’s attitude motions in sea waves is crucial for ensuring maritime safety. Surf-riding occurs when a ship is accelerated by a following wave, matching its speed. Based on the state of surf-riding and broaching, capsizing will happen when the velocity of the ship exceeds the velocity of the wave [1]. It is difficult for fishing boats, high-speed ferries and naval vessels, which are mainly small vessels, to maintain stability in such conditions. Understanding these phenomena involves analyzing wave patterns, ship dynamics and environmental factors. The phenomenon of surf-riding/broaching has been listed as one of the stable failure modes by the International Maritime Organization (IMO) on the second generation intact stability criteria for ships in 2020 (IMO, 2020) [2]. Thus, advanced computational models and simulations should be employed to predict these behaviors, enabling the design of safer and more efficient ships capable of withstanding challenging sea conditions.

The research on the prediction of ships’ motion has evolved significantly over the decades. Early studies primarily relied on empirical observations and basic linear mathematical models [3,4]. These initial models focused on the fundamental principles of ship hydrodynamics and stability, providing a rudimentary understanding of ship motions in regular and irregular waves. Based on these studies, analytical methods began to incorporate more complex nonlinear effects, addressing the limitations of earlier linear models [5]. Researchers started to use the strip theory, which divides the ship into cross-sectional strips, allowing for a more detailed analysis of ship motion responses [6,7]. Despite these advancements, the computational resources available at the time constrained the complexity and accuracy of predictions.

With the advancement of computer technology, Computational Fluid Dynamics (CFD) emerged as a powerful tool, enabling detailed simulations of fluid–structure interactions [8]. Many researchers have employed CFD methods to study the hydrodynamic performance of ships in different motion states, and the numerical simulation and experimental data are verified and compared [9,10]. Carrica et al. (2008) used CFD-based URANS to simulate broaching in irregular waves, highlighting the role of coupled motions and autopilot dynamics [11]. Later, using CFD, Carrica et al. (2012) validated broaching simulations in regular waves, showing that optimizing the autopilot can prevent broaching and expand the safe operating range [12]. These studies emphasize CFD’s key contribution to understanding and mitigating broaching. Gong et al. (2019) verified the effectiveness of numerical methods based on CFD for the simulation of different ships by comparing grid independence tests and self-propulsion experiments and highlighted the significance of accounting for ship attitude in simulations [13]. Jiao et al. (2021) employed a solver based on CFD utilizing an approach combining the URANS equations with a Volume of Fluid (VOF) model, predicting ship motion responses in bi-directional cross waves [14]. Furthermore, CFD models provided insights into the nonlinear behaviors of ships, including the prediction of the ships’ attitude motions. For example, Begovic et al. (2020) investigated the occurrence of surf-riding in ships using three degrees of freedom (DOF) CFD simulations [15]. They aimed to establish surf-riding limitations for ship operability by analyzing wave-induced forces and nonlinear effects, comparing these to calm water conditions. Gong et al. (2022) simulated the ship’s navigation in oblique stern waves using a hybrid fully nonlinear potential theory (FNPT) and CFD method to study its surf-riding and broaching characteristics. Although the accuracy of the CFD solution has been very high, its computational complexity and time cost are still significant disadvantages [16]. However, with the development of artificial intelligence technology, AI has gradually entered the field of vision of scholars, providing a new way to improve simulation efficiency and prediction accuracy.

In recent years, the wide employment of machine learning and artificial intelligence has further advanced the prediction capabilities of ship attitude motions. Machine learning algorithms, trained on vast ship motion and wave pattern datasets, can now predict ship behaviors with remarkable precision. Gao et al. (2023) proposed a novel prediction model for ship motion attitude by integrating the Adaptive Discrete Wavelet Transform (ADWT) with a space-time residual recurrent neural network (RRNN) [17]. Their approach significantly enhances long-term prediction accuracy, particularly under challenging conditions such as rough seas. However, the surf-riding and broaching need to be judged by several conditions based on the ship’s attitude motion. In other words, to accurately predict the occurrence of broaching based on multiple motion features, it is crucial to maintain the precision of these predictions of features concurrently. Zhang et al. (2023) applied support vector machines to analyze vessel motion and quantify incident risk during hurricanes, demonstrating its potential for enhancing safety through risk-aware routing and monitoring [18]. Xu et al. (2024) proposed, for the first time, using a state vector to assist the LSTM neural network in predicting ship state, which averted the occurrence of broaching that is indirectly obtained from motion features [19]. However, when broaching occurs, the ship’s state in both the preceding and succeeding moments often exhibits significant characteristics, such as rudder gradient and value changes. A Bi-LSTM, which can learn dependencies in both directions, is more effective than a unidirectional LSTM for capturing these bidirectional dependencies and improving model performance.

This study presents a method for the short-term forecasting of ship motions and states, focusing on a rapid response to unexpected broaching events during navigation. The vessel type used in this paper is a trimaran, which is selected as the study object for the analysis. The key innovation lies in the real-time classification of ship states (such as normal sailing, surf-riding and broaching) at each time step. These states are represented as a state vector using one-hot encoding, which is then combined with motion features and fed into a neural network for learning. The Bi-LSTM model is employed and optimized to predict ship attitude motions and states in wave conditions. The performance of the Bi-LSTM model, incorporating state features, is compared with conventional LSTM models, highlighting the importance of motion features, alongside state features, for enhanced predictive accuracy. The study aims to optimize the Bi-LSTM structure and determine the most effective input feature components, ultimately enabling timely predictions and responses to critical ship behavior, such as broaching, during navigation. The overall configuration of the proposed Bi-LSTM model for predicting surf-riding, wave-blocking and broaching is shown in Figure 1.

2. Principles and Methods

2.1. The Simulation of Ship Autopilot in Waves

The training dataset in this paper comes from the simulation of the autopilot of the ship in waves. The validity of the data was verified in a previous paper [13,16]. The dataset used in this study comprises multiple variables that describe the dynamic response of the ship under various flow and wave conditions. These variables include the time (simulation time steps in seconds), rudder angle (in radians or degrees) representing the control input for ship heading and displacements along the X, Y and Z axes (DisX, DisY and DisZ) indicating the vessel’s linear movements. Additionally, the rotational angles (AngX, AngY and AngZ) describe the roll, pitch and yaw motions, while the linear velocities (Ux, Uy and Uz) and angular velocities (OmegaX, OmegaY and OmegaZ) capture the ship’s translational and rotational speeds, respectively. The dataset also includes hydrodynamic forces (Fx, Fy and Fz) acting on the hull and moments (Mx, My and Mz) influencing the ship’s rotational behavior. Together, these variables provide a comprehensive description of the ship’s motion and interaction with the surrounding fluid environment.

This section is just a brief summary of the simulation of ship autopilot in waves. Nonlinear effects, such as side hull emergence, bow diving and transient draft variation, are crucial for accurately simulating a trimaran’s surf-riding and broaching. The viscous flow method is employed to solve the flow around the trimaran within its internal domain. Assuming an incompressible fluid with constant density, the flow field adheres to principles of mass and momentum conservation while neglecting heat conduction and water exchange. The incompressible URANS equations are expressed as follows:

$${\displaystyle \frac{\partial (\rho \mathit{U})}{\partial \mathit{t}}}=0$$

(1)

$$▽·\left[\rho (\mathit{U}-{\mathit{U}}_{\mathit{g}})\mathit{U}\right]+{\displaystyle \frac{\partial (\rho \mathit{U})}{\partial \mathit{t}}}=▽·({\mu }_{\mathit{eff}}▽\mathit{U})+(▽\mathit{U})·▽{\mu }_{\mathit{eff}}-▽{\mathit{p}}_{\mathit{d}}-\mathit{g}·\mathit{x}▽\rho +{\mathit{f}}_{\mathsf{\sigma }}+{\mathit{f}}_{\mathit{s}}$$

(2)

where $\rho$ represents density, $\mathit{t}$ denotes time, $\mathit{g}$ represents the gravitational acceleration, $\mathit{U}$ and ${\mathit{U}}_{\mathit{g}}$ are velocity field and the moving velocity of the grid moving, ${\mathit{p}}_{\mathit{d}}$ denotes the dynamic pressure field, ${\mu }_{\mathit{eff}}$ represents the effective dynamic viscosity, and ${\mathit{f}}_{\mathsf{\sigma }}$ and ${\mathit{f}}_{\mathit{s}}$ denote the surface tension term and the source term, respectively. The Volume of Fluid (VOF) method and artificial compression techniques in OpenFOAM are utilized to simulate a Eulerian two-phase flow, ensuring that fluid volume fractions are non-zero solely at the air–water interface [20,21].

The internal domain moves with the ship within the external domain, where the fluid is regarded as ideal. The velocity potential in the external domain satisfies the Laplace equation, with a wavemaker on one side and a self-adaptive wavemaker for wave absorption on the other. Both kinematic and dynamic boundaries are maintained on the free surface. The hybrid method, applied for various fluid–structure interaction (FSI) issues, connects the two domains via interfaces and a transition zone.

For the autopilot trimaran, a global coordinate system ($\mathit{a}$, $\mathit{b}$, $\mathit{c}$) solves the flow field, whereas a local coordinate system ($\mathit{a}$′, $\mathit{b}$′, $\mathit{c}$′), centered on the trimaran’s center of gravity, is employed to address its motion, thereby avoiding the need for force and moment corrections. The ship’s 6DOF motion equations are as follows [22]:

$$\begin{gathered} \mathit{m}({\dot{\mathit{u}}}_{\mathit{a}’}-{\mathit{u}}_{\mathit{b}’}{\mathit{r}}_{\mathit{c}’}+{\mathit{u}}_{\mathit{c}’}{\mathit{r}}_{\mathit{b}’})={\mathit{F}}_{\mathit{Ha}’}+{\mathit{F}}_{\mathit{WJa}’} \\ \mathit{m}({\dot{\mathit{u}}}_{\mathit{b}’}-{\mathit{u}}_{\mathit{c}’}{\mathit{r}}_{\mathit{a}’}+{\mathit{u}}_{\mathit{a}’}{\mathit{r}}_{\mathit{c}’})={\mathit{F}}_{\mathit{Hb}’}+{\mathit{F}}_{\mathit{WJb}’} \\ \mathit{m}({\dot{\mathit{u}}}_{\mathit{b}’}-{\mathit{u}}_{\mathit{a}’}{\mathit{r}}_{\mathit{b}’}+{\mathit{u}}_{\mathit{b}’}{\mathit{r}}_{\mathit{a}’})={\mathit{F}}_{\mathit{H}\mathit{c}’}+{\mathit{F}}_{\mathit{WJ}\mathit{c}’} \\ {\mathit{I}}_{\mathit{a}’}{\mathit{r}}_{\mathit{a}’}+({\mathit{I}}_{\mathit{c}’}-{\mathit{I}}_{\mathit{b}’}){\mathit{r}}_{\mathit{b}’}{\mathit{r}}_{\mathit{c}’}={\mathit{M}}_{\mathit{Ha}’}+{\mathit{M}}_{\mathit{WJa}’} \\ {\mathit{I}}_{\mathit{b}’}{\mathit{r}}_{\mathit{b}’}+({\mathit{I}}_{\mathit{a}’}-{\mathit{I}}_{\mathit{c}’}){\mathit{r}}_{\mathit{a}’}{\mathit{r}}_{\mathit{c}’}={\mathit{M}}_{\mathit{Hb}’}+{\mathit{M}}_{\mathit{WJb}’} \\ {\mathit{I}}_{\mathit{c}’}{\mathit{r}}_{\mathit{c}’}+({\mathit{I}}_{\mathit{b}’}-{\mathit{I}}_{\mathit{a}’}){\mathit{r}}_{\mathit{a}’}{\mathit{r}}_{\mathit{b}’}={\mathit{M}}_{\mathit{Hc}’}+{\mathit{M}}_{\mathit{WJc}’} \end{gathered}$$

(3)

where $\mathit{m}$ is the mass, (${\mathit{I}}_{\mathit{a}’}$, ${\mathit{I}}_{\mathit{b}’}$, ${\mathit{I}}_{\mathit{c}’}$) are inertia moments, (${\mathit{F}}_{\mathit{Ha}’}$, ${\mathit{F}}_{\mathit{Hb}’}$, ${\mathit{F}}_{\mathit{Hc}’}$) are total forces, and (${\mathit{M}}_{\mathit{Ha}’}$, ${\mathit{M}}_{\mathit{Hb}’}$, ${\mathit{M}}_{\mathit{Hc}’}$) are total moments, with subscripts $\mathit{WJ}$ indicating water jet forces and moments. The rotation is described by Euler angles (${\mathsf{\xi }}_4$, ${\mathsf{\xi }}_5$, ${\mathsf{\xi }}_6$):

$$\left[\begin{array}{ccc}1& \sin {\mathsf{\xi }}_4\tan {\mathsf{\xi }}_5& \cos {\mathsf{\xi }}_4\tan {\mathsf{\xi }}_5\\ 0& \cos {\mathsf{\xi }}_4& -\sin {\mathsf{\xi }}_4\\ 0& \sin {\mathsf{\xi }}_4/\cos {\mathsf{\xi }}_5& \cos {\mathsf{\xi }}_4/\cos {\mathsf{\xi }}_5\end{array}\right]\left[\begin{array}{c}{\mathit{r}}_{\mathit{x}’}\\ {\mathit{r}}_{\mathit{y}’}\\ {\mathit{r}}_{\mathit{z}’}\end{array}\right]$$

(4)

The autopilot utilizes a semi-empirical model for water jet thrust and steering moment [23,24], along with PD control, to achieve a target heading of 0°. The nozzle deflection angle $\delta$ is determined by $\delta ={\mathit{K}}_{\mathit{p}}{\mathsf{\xi }}_6+{\mathit{K}}_{\mathit{d}}{\dot{\mathsf{\xi }}}_6$, with ${\mathit{K}}_{\mathit{p}}$ = 9.5 and ${\mathit{K}}_{\mathit{d}}$ = 3.0, and ${\delta }_{\max }$ = 35$°$ and $\dot{\delta }$ = 10$°$/s.

The external domain grid is based on the QALE-FEM method, adjusted to the trimaran’s navigation range. The internal domain dimensions are 4.5 L × 3.0 L × 2.0 L, with the water above and below set to 0.5 L and 1.0 L, respectively. The grids are generated using OpenFOAM’s blockMesh and snappyHexMesh tools, following procedures in zigzag maneuver simulations. The internal domain uses a k−ω SST turbulence model and the PISO method to solve pressure–velocity coupling. Boundary conditions are coupled with the external domain, transferring velocity and volume fraction at each time step. Diagrams of the grid and boundary conditions illustrate the setup.

The hull characteristic dimensions used in the numerical simulation are presented in

Table 1. The specific dimensions of the trimaran.

Particulars	$\mathit{B}/\mathit{L}$	$\mathit{D}/\mathit{L}$	${\mathit{C}}_{\mathit{b}}$	${\mathit{B}}_1/{\mathit{L}}_1$	${\mathit{D}}_1/{\mathit{L}}_1$	${\mathit{C}}_{\mathit{b}1}$	$\mathit{p}/\mathit{L}$	$\mathit{t}/\mathit{L}$
Value	0.08	0.04	0.52	0.05	0.04	0.46	0.0	0.1

. $\mathit{L}$, $\mathit{B}$ and $\mathit{D}$ refer to the waterline length, waterline width, and draft of the center hull. Similarly, ${\mathit{L}}_1$, ${\mathit{B}}_1$ and ${\mathit{D}}_1$ denote the corresponding dimensions for the side hulls. The block coefficients for the center and side hulls are represented by ${\mathit{C}}_{\mathit{b}}$ and ${\mathit{C}}_{\mathit{b}1}$, respectively. The transverse and longitudinal positions of the side hulls are denoted by $\mathit{t}$ and $\mathit{p},$ as illustrated in Figure 2.

This image illustrates the behavior of a ship in motion, showing various key components related to its movement through water. This image illustrates the movement of a ship in water. The ship’s heading is indicated by the red arrow, showing the direction of travel. The bow waves are generated at the front of the ship as it moves, spreading outward in the direction of travel, as represented by the curved blue arrows on the left. The stern waves are produced at the rear of the ship and spread outward in the opposite direction to the bow waves, shown by the curved blue arrows on the right.

2.2. The Definition of Surf-Riding and Broaching

Surf-riding is commonly defined as a situation when a ship, initially traveling at a slower speed than the wave, is accelerated to the velocity of the wave. However, the forward speed and longitudinal position within the wave of the ship remain constant. The hull is captured by the following wave, typically on the front face of the wave crest that initially overtakes the ship. Marginal surf-riding is defined when the ship is accelerated to 90% or more of the wave celerity but does not reach the speed of the wave itself. It also means that the wave is still taking the ship gradually. So, the definition of surf-riding and marginal surf-riding can be concluded as follows in

Table 2. Identification of surf-riding/marginal surf-riding.

Description	Surf-Riding	Marginal Surf-Riding
Forward speed	${\mathit{V}}_{\mathit{sw}}$=${\mathit{V}}_{\mathit{w}}$	${\mathit{V}}_{\mathit{sw}} {\ge 0.9·\mathit{V}}_{\mathit{w}}$

where ${\mathit{V}}_{\mathit{sw}}$ is the projection of the initial hull celerity, and ${\mathit{V}}_{\mathit{w}}$ is the wave celerity.

:

When the initial speed of the ship is faster than the celerity of the encountered wave, the ship fails to climb the following wave crest and remains caught on the back face of the crest after passing through the last trough [24]. This condition can be called ‘wave blocking’ [25]. The definition has been shown in

Table 3. Identification of wave-blocking.

Description	Wave-Blocking
Forward speed	${\mathit{V}}_{\mathit{sw}} {\ge \mathit{V}}_{\mathit{w}}$
Change in speed	$\left({\mathit{V}}_{\mathit{sw}}{/\mathit{V}}_{\mathit{w}}\right)\mathit{\text{'}}<0$ Until ${\mathit{V}}_{\mathit{sw}}$= ${\mathit{V}}_{\mathit{w}}$

.

Surf-riding is the prerequisite for broaching. The broaching behavior was studied similarly to the tendency to surf-riding [24]. The equation could obtain the deflection of the target nozzle angle δ:

$$\delta ={\mathit{K}}_{\mathit{p}}{\mathsf{\xi }}_6+{\mathit{K}}_{\mathit{d}}{\dot{\mathsf{\xi }}}_6$$

(5)

where ${\mathit{K}}_{\mathit{p}}$ = 9.5 represents the proportional coefficient, and ${\mathit{K}}_{\mathit{d}}$ = 3.0 is the differential coefficient used in the PD control in this paper. To analyze the tendency to broaching, broaching and marginal broaching were defined as presented in Table 4.

Table 4. Identification of broaching/marginal broaching.

Description	Broaching	Marginal Broaching
Heading deviation	$\mathit{\psi }\ge 20°$	$\mathit{\psi }\ge 20°$
Rudder angle	$\delta ={\delta }_{\max }$	$\delta ={\delta }_{\max }$
Yaw rate	$\dot{{\mathsf{\xi }}_6 }>0$	$\dot{{\mathsf{\xi }}_6}>0$
Yaw acceleration	$\ddot{{\mathsf{\xi }}_6}>0$	-

where the ${\mathsf{\xi }}_6$ represents the yaw angle bias at each timestep and ${\dot{\mathsf{\xi }}}_6$ denotes the rate of course deviation and heading change. Here, the heading rate and the $\mathit{\psi }$ represented the heading of the ship and the $\mathit{\psi }$ is equivalent to ${\mathsf{\xi }}_6$. The one-hot encoding method of ships’ states based on the definition has been shown in Figure 3. One-hot encoding is a method used to represent categorical variables as binary vectors. Each category is mapped to a unique binary vector, where only one element is “1” (indicating the presence of that category) and all other elements are “0” (indicating the absence of other categories). This method is commonly used to convert categorical data into a numerical format that machine learning models can process effectively.

Table 4. Identification of broaching/marginal broaching.

Description	Broaching	Marginal Broaching
Heading deviation	$\mathit{\psi }\ge 20°$	$\mathit{\psi }\ge 20°$
Rudder angle	$\delta ={\delta }_{\max }$	$\delta ={\delta }_{\max }$
Yaw rate	$\dot{{\mathsf{\xi }}_6 }>0$	$\dot{{\mathsf{\xi }}_6}>0$
Yaw acceleration	$\ddot{{\mathsf{\xi }}_6}>0$	-

where the ${\mathsf{\xi }}_6$ represents the yaw angle bias at each timestep and ${\dot{\mathsf{\xi }}}_6$ denotes the rate of course deviation and heading change. Here, the heading rate and the $\mathit{\psi }$ represented the heading of the ship and the $\mathit{\psi }$ is equivalent to ${\mathsf{\xi }}_6$. The one-hot encoding method of ships’ states based on the definition has been shown in Figure 3. One-hot encoding is a method used to represent categorical variables as binary vectors. Each category is mapped to a unique binary vector, where only one element is “1” (indicating the presence of that category) and all other elements are “0” (indicating the absence of other categories). This method is commonly used to convert categorical data into a numerical format that machine learning models can process effectively.

In this study, one-hot encoding was selected to represent the ship’s states (such as surf-riding, broaching and wave blocking) because it provides a simple and effective way to handle categorical data. By using one-hot encoding, we can easily distinguish between different states of the ship and allow the model to learn the relationships between these states and the associated motion features. This encoding method ensures that each state is treated as a separate entity, preventing the model from incorrectly interpreting relationships between states that are not inherently connected.

In addition, the features applied to define the occurrence of surf-riding, wave blocking and broaching will be named definition features. In this paper, to represent them simply, the marginal surf-riding and surf-riding will be marked in figures in the same way, as will marginal broaching and broaching.

These features were selected based on their ability to capture the critical dynamics associated with these complex phenomena, ensuring that the model can accurately distinguish between different ship motions and predict surf-riding or broaching events.

Rudder angle plays a crucial role in steering and maneuverability. In the context of surf-riding and broaching, it helps assess how the ship’s steering input influences its stability and ability to maintain a safe heading. A sharp change in rudder angle may be indicative of an impending broaching event or an attempt to maintain control during surf-riding. The ship speed-to-wave speed ratio is another critical feature, as it directly reflects the ship’s interaction with wave conditions. This ratio is essential for understanding whether the ship is moving in harmony with the waves (a condition favorable to surf-riding) or whether the wave forces are overwhelming the ship’s forward momentum, potentially leading to broaching. In addition, the roll, pitch and heave features provide insight into the rotational and vertical motion of the ship. These motion characteristics are essential in detecting the onset of broaching, where extreme values of roll or pitch can indicate a loss of stability, while heave captures the ship’s vertical movement in response to changing wave heights. The trajectory and heading features are particularly important for understanding the ship’s overall path and orientation in relation to the incoming waves. The ship’s heading, in combination with its trajectory, helps to predict whether it is likely to experience surf-riding, where the ship rides along the wave or is broaching, where it is misaligned with the wave and risks capsizing or losing control.

These features were integrated into the model because they reflect the fundamental characteristics of ship motion in different sea conditions, enabling the model to identify and predict these critical phenomena with greater precision.

2.3. Bi-LSTM Prediction Model

A Bidirectional-LSTM network is a type of recurrent neural network (RNN) designed to handle sequence data and capture long-range dependencies. Unlike traditional RNNs, which can suffer from issues such as vanishing gradients, LSTMs use unique gating definitions to maintain long-term information. Bi-LSTM extends the capabilities of standard LSTMs by processing the data in both forward and backward directions. This means that each unit in a Bi-LSTM network has access to both past (backward) and future (forward) feature data, providing a more comprehensive understanding of the sequence [26].

This paper combines the attitude motion features and state features as the training datasets with complex temporal patterns. Bi-LSTM models can incorporate future data points into the prediction process to effectively identify complex temporal patterns. So, the Bi-LSTM model will be more suitable for predicting the states of ships and motions than the LSTM model. The LSTM cell and the form of Bi-LSTM are shown in Figure 4 and Figure 5.

For both forward and backward LSTMs, the following equations define the LSTM cell operations. The forward LSTM can be denoted by the superscript $\to$ and the backward LSTM by $\leftarrow$. The feedforward formula is used as an example:

$$\begin{gathered} {\mathit{f}}_{\mathit{t}}^{\to }=\mathsf{\sigma }({\mathit{W}}_{\mathit{f}}^{\to }·[{\mathit{h}}_{\mathit{t}-1}^{\to },{\mathit{x}}_{\mathit{t}}]+{\mathit{b}}_{\mathit{f}}^{\to }) \\ {\mathit{i}}_{\mathit{t}}^{\to }=\mathsf{\sigma }({\mathit{W}}_{\mathit{i}}^{\to }·[{\mathit{h}}_{\mathit{t}-1}^{\to },{\mathit{x}}_{\mathit{t}}]+{\mathit{b}}_{\mathit{i}}^{\to }) \\ {\stackrel{\mathrm{~}}{\mathit{C}}}_{\mathit{t}}^{\to }=\tan \mathit{h}({\mathit{W}}_{\mathit{C}}^{\to }·[{\mathit{h}}_{\mathit{t}-1}^{\to },{\mathit{x}}_{\mathit{t}}]+{\mathit{b}}_{\mathit{C}}^{\to }) \\ {\mathit{C}}_{\mathit{t}}^{\to }={\mathit{f}}_{\mathit{t}}^{\to }\ast {\mathit{C}}_{\mathit{t}-1}^{\to }+{\mathit{i}}_{\mathit{t}}^{\to }\ast {\stackrel{\mathrm{~}}{\mathit{C}}}_{\mathit{t}}^{\to } \\ {\mathit{o}}_{\mathit{t}}^{\to }=\mathsf{\sigma }({\mathit{W}}_{\mathit{o}}^{\to }·[{\mathit{h}}_{\mathit{t}-1}^{\to },{\mathit{x}}_{\mathit{t}}]+{\mathit{b}}_{\mathit{o}}^{\to }) \\ {\mathit{h}}_{\mathit{t}}^{\to }={\mathit{o}}_{\mathit{t}}^{\to }\ast \tan \mathit{h}({\mathit{C}}_{\mathit{t}}^{\to }) \end{gathered}$$

(6)

where ${\mathit{f}}_{\mathit{t}}^{\to }$, ${\mathit{i}}_{\mathit{t}}^{\to }$, ${\mathit{o}}_{\mathit{t}}^{\to }$, respectively, represent gate activation vectors of the forget gate, input gate, and output at time step $\mathit{t}$, ${\mathit{W}}^{\to }$ and ${\mathit{b}}^{\to }$, respectively, represent the weight matrix and bias for each gate, ${\mathit{h}}_{\mathit{t}-1}^{\to }$ denotes the hidden state from the previous time step in the forward LSTM, ${\mathit{x}}_{\mathit{t}}$ is the input at the current time step, ${\mathit{C}}_{\mathit{t}}^{\to }$ is the cell state vector at time step $\mathit{t}$, and ${\stackrel{\mathrm{~}}{\mathit{C}}}_{\mathit{t}}^{\to }$ is the candidate cell state vector at time step $\mathit{t}$.

The final output of the Bi-LSTM at each time step $\mathit{t}$ is a concatenation of forward and backward hidden states:

$${\mathit{h}}_{\mathit{t}}= [{\mathit{h}}_{\mathit{t}}^{\to };{\mathit{h}}_{\mathit{t}}^{\leftarrow }]$$

(7)

where $[;]$ denotes the concatenation operation.

2.4. The Selected Features

Feature selection is a crucial step in model development as it directly impacts the quality and performance of predictive models. Currently, there are various feature selection methods proposed for optimizing feature sets in machine learning applications. These methods aim to identify the most relevant features while reducing dimensionality and minimizing redundancy. Common approaches include filter methods, which assess the relevance of features based on statistical measures such as correlation or mutual information [27]; wrapper methods [28], which evaluate feature subsets by testing their performance within a predictive model; and embedded methods, which perform feature selection during the model training process itself. Additionally, methods such as Unbalanced Incomplete Multiview Unsupervised Feature Selection with Low-Redundancy Constraint in Low-Dimensional Space are designed to address high-dimensional data and redundancy, ensuring that the selected features retain significant and non-redundant information [29]. Each of these methods has its advantages and is typically chosen based on the specific requirements of the task, such as the complexity of the data and the desired model performance.

In the context of this study, the feature selection process was tailored specifically to capture the dynamics of surf-riding and broaching phenomena, which is critical to understanding the vessel’s performance in specific sea conditions. The features were selected by considering both the motion characteristics of the ship and the criteria related to surf-riding and broaching behaviors.

The selected ship motion features in this study include key ship motion characteristics that are crucial for modeling surf-riding and broaching behaviors. These features are as follows: roll, which represents the ship’s rotational movement around its longitudinal axis and is important for assessing stability; pitch, the rotational movement around the transverse axis, contributing to the vessel’s overall attitude; heave, the vertical displacement of the ship, indicating the impact of wave height on the vessel; rudder angle, which governs the ship’s heading and maneuverability; the ship speed-to-wave speed ratio, critical for evaluating the ship’s interaction with waves and its tendency to surf or broach; trajectory, the path traced by the ship, which reflects its movement and stability in dynamic conditions; and heading, which determines the ship’s orientation relative to the waves and wind. These features were selected to ensure a comprehensive and effective prediction of ship behavior in various wave conditions, with a particular emphasis on surf-riding and broaching phenomena.

3. The Proposed Bi-LSTM Model and Performance Evaluation

This section will present the architecture of the Bi-LSTM model and the methodology used for performance evaluation. The Bi-LSTM model is designed to leverage the bidirectional processing of time sequences, capturing context from both past and future states.

3.1. The Evaluation of the Dropout Rate in Each Layer

In this study, model performance will be assessed using three evaluation metrics: Mean Absolute Error (MAE), Root Mean Squared Error (RMSE) and the Coefficient of Determination (R²). MAE provides a straightforward measure of the average absolute difference between the predicted and actual values, offering a direct interpretation of prediction accuracy. RMSE, by squaring errors before averaging and then taking the square root, emphasizes larger errors more than MAE, thereby being more sensitive to outliers. On the other hand, R² quantifies the proportion of the variance in the dependent variable that can be explained by the independent variable, serving as a measure of how well the model fits the data overall. These metrics comprehensively evaluate model accuracy, error magnitude and explanatory power. Assuming that $\widehat{{\mathit{y}}_{\mathit{i}}}$ is the prediction value and ${\mathit{y}}_{\mathit{i}}$ is the real value, the equations can be denoted as follows:

$$\mathrm{MAE}={\displaystyle \frac{1}{\mathit{n}}}{\sum }_{\mathit{i}=1}^{\mathit{n}}\left|\widehat{{\mathit{y}}_{\mathit{i}}}-{\mathit{y}}_{\mathit{i}}\right|$$

(8)

$$\mathrm{RMSE}=\sqrt{{\displaystyle \frac{1}{\mathit{n}}}{\sum }_{\mathit{i}=1}^{\mathit{n}}{(\widehat{{\mathit{y}}_{\mathit{i}}}-{\mathit{y}}_{\mathit{i}})}^2}$$

(9)

$${\mathit{R}}^2=\sqrt{1-{\displaystyle \frac{{\sum }_{\mathit{i}=1}^{\mathit{n}}{(\widehat{{\mathit{y}}_{\mathit{i}}}-{\mathit{y}}_{\mathit{i}})}^2}{{\sum }_{\mathit{i}=1}^{\mathit{n}}{(\stackrel{\mathrm{-}}{\mathit{y}}-{\mathit{y}}_{\mathit{i}})}^2}}}$$

(10)

3.2. The Effect of the Dropout Rate in Each Layer

Applying the dropout layer after each Bi-LSTM layer can diminish the possibility of overfitting by randomly dropping a fraction of the units during training, encouraging the network to learn more robust features. This regularization technique also improves the generalization of new data by reducing the model’s reliance on specific neurons. Consequently, it can lead to a better overall model performance and stability. Three constructions of the model, based on different dropout rates, have been prepared to test the performance. The specific constructions of the three models are shown in

Table 5. The specific construction of the models.

Model	Dropout-Layer-1	Dropout-Layer-2	Dropout-Layer-3
Model #3.1	0.5	0.3	0.1
Model #3.2	0.3	0.2	0.1
Model #3.3	0.2	0.1	-

. A navigation case with surf-riding, wave blocking and broaching conditions will be employed to train and test the model. The total time of the case is 100. The model will be trained using 80% of the data and the rest will validate the model’s prediction performance.

Figure 6 shows the prediction values of three models with different dropout rates and Figure 7 shows the three metrics value of the performance. The data in the blue frame have been used to train the models, and the data in the green frame are the prediction results. The occurrence of the ship’s states (surf-riding, wave blocking, and broaching) has been marked with different colors. In addition, the blue dotted box represents the data segment used to train the model, and the green dotted box represents the prediction result.

Figure 6 shows the prediction results of ships’ attitude motions and states. The content within the “Train the Model” box in the diagram refers to the dataset used for training the model, while the remaining portion is used for testing the model. After the model is trained, it starts making predictions using the data in the “Learn New” section, with the prediction results displayed in the “Prediction” section. It can be seen that the differences between the predictions and real data in the three models are similar. The prediction difference in the three models in the ship heading is more apparent than that of the other features. The results demonstrate a tendency for the model’s performance to decrease with the dropout rate increase in each dropout layer. The small gap in prediction performance between adjacent models means that as the number of datasets increases, the dropout rate can be appropriately increased to avoid overfitting problems during model training.

Figure 7 shows the MAE, RMSE, and R² results in different models. As the dropout rate increases, the MAE and RMSE tend to increase while the R² value decreases. This trend indicates that higher dropout rates lead to poorer model performance, reinforcing the need to carefully balance dropout to prevent overfitting without compromising accuracy. The comparison among the three models also highlights that while the MAE and RMSE values are relatively close, the decrease in R² value with an increased dropout rate is more pronounced, signifying a reduction in the model’s capability to understand the variance in the ship’s attitude motions and states. This also emphasizes the importance of optimizing dropout rates for robust and reliable predictions in ocean engineering applications.

The analysis of training and validation errors reveals a clear correlation between model complexity and overfitting in

Table 6. The comparison of the training errors and validation R².

Model	Model #3.1	Model #3.2	Model #3.3
Training errors	0.427	0.553	0.630
Validation errors	0.281	0.428	0.500

. As the model complexity increases from Model #3.1 to Model #3.3, both training and validation errors rise. However, the gap between training and validation errors also widens, particularly for Model #3.2 and Model #3.3, suggesting that these models are overfitting to the training data. Model #3.1, with a smaller gap between training and validation errors, indicates a better generalization and less overfitting, likely due to the effective use of dropout regularization. In contrast, the larger gap in Models #3.2 and #3.3 suggests that these models are not generalizing to unseen data as well, highlighting the importance of balancing model complexity and regularization techniques to prevent overfitting.

The analyses of Figure 6 and Figure 7 demonstrate that the dropout rate setting for predicting special attitude motions should be based on the size of the dataset. Based on a comprehensive consideration, the dropout rate of Model #3.2 is chosen as the dropout rate of this paper.

3.3. The Effect of Different Layer Numbers

The number of layers in the LSTM or Bi-LSTM neural network significantly impacts its performance, especially in complex tasks like predicting ships’ attitude motions based on different types of features. Increasing the number of layers generally enables the model to capture more complex and imperceptible patterns and relationships within the data. However, this also introduces the risk of overfitting, where the model becomes too tailored to the training data, leading to its poor performance on new or unseen data. Thus, careful consideration is needed to determine the optimal number of layers to balance model complexity and generalization. In this section, three models with different numbers of layers will be constructed to compare and analyze the performance based on the results and evaluation metrics. Model #3.4 has two Bi-LSTM layers, Model #3.5 has three Bi-LSTM layers, and Model #3.6 has four layers.

Figure 8 illustrates the prediction results of the three models for ships’ attitude motions and states. It is evident that the model’s performance does not improve linearly with an increase in the number of layers. Models with more layers demonstrate better performance in predicting the occurrence of complex ship states, such as surf-riding, wave-blocking, and broaching. However, models with fewer layers perform better in predicting specific motion values, such as precise attitude motions. This suggests that deeper models excel at capturing broader patterns and complex events but may struggle with accurately predicting finer details. Conversely, shallower models are more effective at estimating specific values but may fail to capture the nuances of more intricate phenomena.

Figure 9 presents the MAE, RMSE, and R² values for the different models. Among the models, Model #3.5, which has three layers, achieves the best performance, with the lowest MAE and RMSE values and the highest R² value. This demonstrates that the three-layer model strikes an optimal balance between capturing complex events and providing precise predictions of ships’ attitude motions and states.

In comparison, Model #3.4, with two layers, exhibits higher MAE and RMSE values and a lower R² value than Model #3.5. While it performs well in predicting specific motion values, it lacks sufficient depth to adequately capture broader patterns and events, such as surf-riding and broaching. On the other hand, Model #3.6, with four layers, also shows a decline in performance relative to Model #3.5, despite its increased depth. This performance drop can be attributed to the increased complexity of the deeper model, which introduces challenges such as overfitting and difficulty in optimization during training. As a result, Model #3.6 exhibits higher MAE and RMSE values and a lower R² value, reinforcing the observation that simply adding more layers does not guarantee better performance.

To further understand these trends, an analysis of training versus validation errors was conducted to investigate overfitting. Models with fewer layers, such as Model #3.4, exhibit smaller gaps between training and validation errors, indicating better generalization but a limited capacity to capture complex behaviors. In contrast, deeper models, such as Model #3.6, show significant divergence between training and validation errors, suggesting overfitting to the training data. The three-layer Model #3.5 demonstrates the most balanced behavior, with training and validation errors closely aligned, indicating an optimal configuration that mitigates overfitting while capturing both specific motion values and complex state occurrences.

From

Table 7. The comparison of the training errors and validation R².

Model	Model #3.1	Model #3.2	Model #3.3
Training errors	0.538	0.547	0.540
Validation errors	0.411	0.428	0.407

, it can be observed that the training errors are consistently lower than the validation errors across all models, which is expected as models tend to perform better on the data they were trained on. However, the gap between training and validation errors remains relatively small for each model, indicating that none of the models are heavily overfitting to the training data. The differences in error values between the models are minimal, suggesting that the models are similarly able to generalize to new data.

In conclusion, the three-layer Model #3.5 was chosen for this study as it provides the most reliable balance between model complexity and prediction accuracy, excelling in both motion value predictions and the identification of ship state occurrences.

3.4. The Specific Construction of the Proposed Model

To address the temporal dependencies in the prediction of complex ship states, a Bi-LSTM architecture was implemented in this study. The model leverages Bi-LSTM’s bidirectional learning capability to effectively capture changes in features occurring both before and after critical state transitions. This design is particularly advantageous for identifying temporal patterns, such as the rudder angle approaching its maximum before broaching or the negative gradient of the ship velocity-to-wave velocity ratio prior to wave blocking.

The Bi-LSTM network comprises three Bi-LSTM layers with respective units of 150, 125, and 50, interleaved with dropout layers to prevent overfitting (

Table 8. The specific construction of the proposed Bi-LSTM model.

Model Layer	Specific Parameter
Bi-LSTM_layer#1	units: 150 predict step (advanced time step): 50
Dropout_layer#1	rate: 0.3
Bi-LSTM_layer#2	units: 125
Dropout_layer#2	rate: 0.2
Bi-LSTM_layer#3	units: 50
Dropout_layer#3	rate: 0.1
Dense	Units: $\mathrm{N}$(Motion features + State features)
Activation	LeakyReLU
MinMaxScaler	feature_ range: (0, 1)
Train-Test Split	Train ratio in total dataset: 80%; Test ratio in total dataset: 20%;
Time Step	The same as (predict step)
Epoch	50
Batch_size	100

). The dense output layer employs the LeakyReLU activation function to handle gradients efficiently, and the data features are scaled to the range (0, 1) using MinMaxScaler for uniformity. To evaluate performance, the dataset was split into 80% training and 20% testing, and the model was trained for 50 epochs with a batch size of 100.

The training and evaluation of the Bi-LSTM model were conducted using the open-source TensorFlow framework on a system equipped with a single NVIDIA V100 GPU (32 GB). This computational setup ensured efficient and optimized training, with each epoch taking approximately 3 to 4 minutes on average. The model demonstrated superior prediction accuracy compared to other architectures, achieving the lowest MAE and RMSE, along with the highest R² values, on the testing dataset. These results confirm the suitability of the Bi-LSTM architecture for accurately predicting ships’ attitude motions and dynamic states under various conditions.

4. Discussion

In this section, the proposed model was employed to predict the attitude motions and states in different databases. Then, the prediction performance of the model will be compared with different methods.

4.1. The Performance of the Model on Different Datasets

The occurrence of surf-riding, wave-blocking, and broaching should be determined by the definition mentioned in Section 2.2. The specific information on the different navigation conditions is shown in

Table 9. The specific conditions of navigation.

Dataset	${\mathbf{V}}_{\mathbf{sw}}/{\mathbf{V}}_{\mathbf{w}}$	$\mathbf{Fr}$	$\mathsf{\lambda }/\mathbf{H}$	$\mathsf{\beta }$
Dataset #4.1	0.73	0.35	5.45	30°
Dataset #4.2	0.91	0.35	2.33
Dataset #4.3	0.94	0.45	3.27

and

Table 10. The occurrence of the ship states in different cases.

Dataset	${{\mathbf{O}}_{\mathbf{MS}}/\mathbf{O}}_{\mathbf{S}}$	${\mathbf{O}}_{\mathbf{W}}$	${\mathbf{O}}_{\mathbf{MB}}/{\mathbf{O}}_{\mathbf{B}}$
Dataset #4.1	×	×	×
Dataset #4.2	√	×	×
Dataset #4.3	√	√	√

. In the table, O_MS represents the occurrence of a marginal surf-riding event, O_MB represents the occurrence of a marginal broaching event and O_W represents the occurrence of a wave-blocking event. Figure 10 shows the visualization of the states of the ships in different conditions. Each plot contains the nozzle deflection angle δ, the heading angle $\mathit{\psi }$, the ratio of ship velocity and wave celerity and the trajectory of the ship. The occurrence and duration of surf-riding, wave-blocking and broaching is labeled. These cases cover most conditions and are helpful for the subsequent training of LSTM neural network prediction models.

The motion features and state features are combined as the input features to train the model in this section. The prediction effect of the Bi-LSTM model under different navigation conditions could be observed through full-feature learning. Figure 11 shows the prediction results in different datasets, and the occurrence of surf-riding, wave blocking and broaching has been marked in the figures based on the state features. Figure 12 shows the three evaluation metrics of the prediction results. Before the first red dotted line is the data used to train the model, between the two red dotted lines is the input validation data set, and after the second red line is the prediction.

Figure 11 shows that the model achieves good performance under different navigation conditions, with minor deviations between the prediction results and the real data across all conditions. The most apparent deviation occurs in Dataset #4.3. While the performance of Dataset #4.3 is slightly lower compared to Datasets #4.1 and #4.2, it excels in capturing the occurrences and ends of the ship’s states. The variation in predictive performance from Dataset #4.1 to Dataset #4.3 further highlights that as the complexity of the ship’s navigation conditions increases, the model’s predictive accuracy also changes. This highlights the importance of introducing state features, which enable the model to better capture and adapt to these variations in the complexity of the navigation states.

Figure 12 shows the MAE, RMSE, and R² values of the prediction under different conditions. It is evident from the figure that Dataset #4.3 exhibits the highest MAE and RMSE values and the lowest R² value, indicating the largest deviations between the predicted and actual values. Despite this, Dataset #4.3 still effectively captures critical events like surf-riding, wave blocking and broaching, as previously mentioned. In contrast, Datasets #4.1 and #4.2 display a better overall prediction performance, with lower MAE and RMSE values and higher R² values. This suggests that the model performs more accurately under less complex navigation conditions. However, the introduction of state features significantly enhances the proposed model’s ability to adapt to and predict the increased complexity in Dataset #4.3. The observed changes in predictive performance across the datasets underscore the model’s sensitivity to varying navigation conditions. As the complexity of the ship’s navigation state increases from Dataset #4.1 to Dataset #4.3, the prediction accuracy of the model diminishes slightly but remains robust due to the introduction of the state features. The robust performance of the proposed model across different datasets based on the analyses of Figure 11 and Figure 12 demonstrates its capability to handle and predict effectively under diverse and challenging maritime conditions.

4.2. The Performance of the Model Compared with Other Methods

In former studies, the prediction of several states’ occurrence has mainly relied on predicting the motions of ships in sailing. Then, assessing whether surf-riding, wave-blocking or broaching events occur based on several motion features (such as $\delta$, $\mathit{\psi }$, and ${\mathit{V}}_{\mathrm{sw}}$/${\mathit{V}}_{\mathit{w}}$). In this section, two models are employed: one is the proposed Bi-LSTM model, and the other is the conventional LSTM model. Three prediction strategies based on the two models will be compared, and their specific methods are shown in

Table 11. The specific information of different methods.

Methods	Model	Motion Features	State Features
Method #4.1	LSTM	√	×
Method #4.2	Bi-LSTM	√	×
Method #4.3	Bi-LSTM	√	√

. Both models used Dataset #4.1 to Dataset #4.3 as the training datasets, with the new dataset shown in

Table 12. The specific conditions of navigation.

${\mathbf{V}}_{\mathbf{sw}}/{\mathbf{V}}_{\mathbf{w}}$	$\mathbf{Fr}$	$\mathsf{\lambda }/\mathbf{H}$	$\mathsf{\beta }$
1.14	0.55	2.33	30°

serving as the validation set. The state-prediction model needs more preprocessing of datasets than the motion-prediction model before producing a prediction.

Figure 13 shows the prediction results of three different methods based on two models. The conventional LSTM model of ship motion predicted the data of ship attitude motions and then identified the state based on the definition. In Method #4.3, the features ($\delta$, $\mathit{\psi }$, and ${\mathit{V}}_{\mathrm{sw}}$/${\mathit{V}}_{\mathit{w}}$) were replaced by the state features. Figure 14 shows the three evaluation metrics of the prediction results.

Figure 13 presents a comparison between the predicted and actual data, demonstrating the effectiveness of the proposed Bi-LSTM model in predicting ship attitude motions. The model, trained on the three datasets described in Section 4.1, performed well on unseen data, particularly in predicting attitude motions. However, all three models faced challenges in accurately identifying the occurrence of wave blocking.

Among the methods evaluated, Method #4.3 exhibited the best performance in terms of both prediction accuracy and the timing of state occurrences, aligning more closely with the real data than the other models. The bidirectional learning capability of the Bi-LSTM model enables it to better capture temporal changes in the ship’s state, optimizing the model’s predictive abilities. By replacing the original three features with state features that directly determine the ship’s condition, the model is able to output state predictions directly rather than indirectly, leading to an improved accuracy in predicting surf-riding and broaching events.

In contrast, when state features are not included, the performance of both the LSTM and Bi-LSTM models are similar. However, the Bi-LSTM model, due to its bidirectional nature, requires more computational resources and longer training times compared to LSTM models. While LSTM models are computationally simpler, they are prone to challenges such as the difficulty in capturing long-term dependencies and the susceptibility to vanishing gradients. These limitations are more pronounced in complex and dynamic maritime scenarios.

Methods #4.1 and #4.2 show significant deficiencies when compared to Method #4.3 in terms of both prediction accuracy and the timing of state occurrences. These methods fail to capitalize on the enhanced feature set and the bidirectional capabilities of the Bi-LSTM model, resulting in less accurate and reliable predictions.

Figure 14 shows the MAE, RMSE, and R² values for each method, clearly illustrating that Method #4.3 consistently outperforms Methods #4.1 and #4.2 across all evaluation metrics. Specifically, Method #4.3 demonstrates the lowest MAE and RMSE values, indicating more precise predictions with smaller errors. Additionally, Method #4.3 achieves the highest R² value, reflecting its superior ability to explain the variance in the ship’s attitude motions and states.

The superior performance of Method #4.3 can be attributed to several key factors. First, the Bi-LSTM model’s bidirectional learning capability enables it to capture both past and future states effectively, which is particularly crucial for complex maritime events where significant changes occur before and after these events. The ability to process information in both directions enhances the model’s capacity to predict these changes more accurately than traditional LSTM models. Secondly, by replacing the original features with state features that directly determine the ship’s condition, the model can provide direct predictions with reduced complexity. This direct approach minimizes errors associated with indirect prediction methods, leading to an improved overall performance.

In comparison, Methods #4.1 and #4.2 exhibit considerable shortcomings. These methods, lacking the bidirectional capabilities and the enhanced feature set of Method #4.3, fail to achieve the same level of prediction accuracy. Method #4.1 and Method #4.2 show higher MAE and RMSE values and lower R² values, indicating less precise predictions and a diminished ability to explain the variance in the data.

Moreover, while the LSTM and Bi-LSTM models perform similarly in the absence of state features, the Bi-LSTM model requires more computational resources and longer training times due to its complexity. The LSTM model, although simpler, faces limitations such as the difficulty in capturing long-term dependencies and the susceptibility to vanishing gradients, which can lead to performance degradation, particularly in highly dynamic maritime scenarios.

In conclusion, the results clearly indicate that Method #4.3, which incorporates the Bi-LSTM architecture and direct state feature outputs, offers superior prediction accuracy, robustness, and overall performance. This method is particularly effective in predicting ship attitude motions and states across a range of conditions, and its incorporation of bidirectional learning and state features makes it a valuable tool in the field of ocean engineering.

5. Conclusions

To enhance the safety and stability of ship navigation, improve the accuracy of attitude motion predictions, and better understand the complexities of maritime conditions, this paper introduces a hybrid method combining the Bi-LSTM model with surf-riding and broaching definitions. The model utilizes definitions of surf-riding, wave-blocking, and broaching to classify ship states, which are then encoded into one-hot representations for learning by the Bi-LSTM neural network. Additionally, the proposed method is compared with conventional approaches based on LSTM and Bi-LSTM models. The effectiveness of the method is validated using ship navigation data from simulations of autopilot systems under various wave conditions. The key conclusions of this study are as follows:

(1)

The proposed hybrid method outperforms conventional prediction strategies based on LSTM models, achieving more accurate predictions of ship attitude motions and states.

(2)

The bidirectional learning capability of the Bi-LSTM model enhances its ability to capture significant temporal dependencies, enabling more precise and timely predictions of complex maritime events under diverse conditions. The model also demonstrates strong generalization, effectively handling data predictions across a wide range of sea states and navigation scenarios.

(3)

The inclusion of state features based on the occurrence definitions of surf-riding, wave-blocking and broaching optimizes the model’s ability to predict ship attitude motions and states. This shift from indirect to direct output for state prediction improves the model accuracy by eliminating intermediate prediction steps, leading to more precise and timely forecasts. As a result, the model can better anticipate and respond to critical maritime events, significantly contributing to safer and more stable ship navigation.

Limitations:

(1)

A limitation of the study is that the high dimensionality of the final state vectors increases the complexity of neural network training. This issue could be addressed by employing dimensionality reduction techniques to simplify the state vectors, thereby improving the efficiency of training and reducing computational overhead.

(2)

The model’s robustness across different ship types and sizes has not been fully explored, as it was primarily tested on trimaran vessels. Additionally, variations in initial conditions, such as ship displacement and hull shape, may impact the model’s predictions, and this will be further investigated in future studies.

(3)

Furthermore, environmental factors, such as wind and currents, which can significantly influence ship behavior, were not considered in the current study.

For practical implementation, the proposed model requires extensive real-world data collection from the target vessel, including ship motion, environmental factors, and control inputs. Initially, simulation-based training can be used to cover a wide range of scenarios, including rare events, followed by online learning for real-time adaptation to changing conditions. The model can be integrated with existing autopilot systems to provide real-time predictions, with redundancy and fail-safes incorporated for safety. A human-in-the-loop approach ensures that operators can monitor and override control commands if necessary. Rigorous testing and validation in both simulated and real-world environments are essential for ensuring model reliability. While the current model does not incorporate specific ship type parameters, it is designed to be adaptable across different vessels. Future research will explore variations in ship types, initial conditions, and environmental factors to enhance the model’s robustness, applicability and accuracy for real-world maritime control systems.