A Coupled SVM-NODE Model for Efficient Prediction of Ship Roll Motion

Abstract

Traditional analyses of ship roll damping and added moment of inertia rely on free roll decay and forced roll tests, but acquiring linear (small angles) and nonlinear (large angles) relationships demands extensive computational cases and parameter fitting, limiting efficiency. To address this, this study couples Support Vector Machine (SVM) and Neural Ordinary Differential Equation (NODE) networks: SVM solves for added moment of inertia, linear damping, and nonlinear damping, while NODE constructs a complete model for the roll motion equation. Using the DTMB5415 hull form, Computational Fluid Dynamics (CFD) simulations of forced roll build a “time-angle-moment” sample space, and the coupled model learns and predicts free roll decay under different initial angles. The results show that SVM effectively determines roll damping and added moment of inertia from constant-amplitude variable-frequency and constant-frequency variable-amplitude data, reducing required cases significantly. NODE’s simulation of free roll decay validates coefficient accuracy. Within a certain angle range, the SVM-NODE model meets rapid roll motion analysis needs, providing an innovative method for ship roll research and engineering.

1. Introduction

Ship roll motion is a complex dynamic process influenced by various forces, including inertial forces, damping forces, restoring forces, and external forces. Due to the unique curved lines of a ship’s hull, its roll damping exhibits both linear and nonlinear characteristics. It is generally accepted that when the roll angle is less than 5°, the roll damping and added moment of inertia approximately satisfy a linear relationship. When the roll angle is between 5° and the deck immersion angle, they exhibit a nonlinear relationship. In the ship engineering design stage, the International Maritime Organization (IMO) recommends using Ikeda’s simplified method for estimation. However, this method has limitations, such as overly conservative results, making it difficult to accurately reflect the actual characteristics of ship roll.

Currently, the primary methods for calculating parameters related to ship roll motion are towing tank model tests, empirical formulas, CFD numerical calculations, and machine learning. While towing tank model tests can realistically simulate actual ship roll conditions, they suffer from high costs and long cycles. Empirical formulas are simple to compute but lack accuracy. CFD numerical calculations can provide detailed flow field information but face challenges in balancing accuracy and computational efficiency. With the development of data engineering, research methods supported by machine learning algorithms, utilizing large amounts of data accumulated from model tests and CFD numerical calculations, are gradually being applied.

In the areas of towing tank model tests and CFD numerical calculations, researchers primarily explore two simulation methods: free decay and forced roll, aiming to obtain ship roll characteristics. Isar Ghamari et al. [1] proposed a potential-based hybrid solver, conducting roll decay numerical calculations through CFD simulation to extract linear and nonlinear damping terms. Simone Mancini et al. [2] validated the roll damping assessment of the DTMB5415 model using both towing tank model tests and CFD numerical methods, studying the free decay process under different initial heel angles to determine the optimal scheme for roll decay simulation. Jinrong Liang et al. [3] used the difference method on free roll decay curves to obtain roll angular velocity decay curves and improved the classical energy method using phase trajectory analysis, enhancing the prediction accuracy of the roll angle variation process during free decay. Waskito K T et al. [4] used three-dimensional frequency-domain Rankine Panel Method (RPM), New Strip Method (NSM), and Enhanced Unified Theory (EUT) to study ship roll motion, with each method employing roll decay tests to calculate roll damping. Researchers estimate roll damping and added moment of inertia by fitting data from roll angle decay curves over time, but limitations remain in handling nonlinear relationships and obtaining precise parameters.

In forced roll research, Xiaojian Liu et al. [5] designed forced roll experiments, studying the relationship between roll frequency, moment of inertia, and roll damping by fitting roll moments at various speeds. Sangming Xu et al. [6] used CFD to study the forced roll motion of a damaged ship, employing cubic spline interpolation to fit the added moment of inertia and damping coefficient for roll angles between 5° and 20°. Guorui Lai et al. [7] used a single-degree-of-freedom forced roll method to calculate the linear roll damping coefficient and quadratic roll damping coefficient of a ship. Sumislawski P et al. [8] estimated roll damping based on numerical simulations and model tests using Harmonic Excited Roll Motion (HERM) technology, employing different input excitation schemes to determine roll damping characteristics for different ship types. Kianejad SS et al. [9] used a technique based on HERM to calculate the roll motion and roll damping moment of a container ship model under different conditions; compared to decay tests, their results showed lower uncertainty. Zhang W et al. [10] used an equivalent roll damping method to study the parametric roll problem of the KCS container ship, transforming the nonlinear roll damping obtained from CFD into linear roll damping, and predicted the parametric roll of the KCS container ship in regular waves based on the Rankine Panel Method. Forced roll can obtain richer data on ship roll angles and forces. Since the excitation conditions can be actively controlled, it allows systematic study of the influence of factors like frequency and amplitude on ship roll, making it suitable for building accurate parametric models. However, whether for free decay or forced roll, existing methods rely on linear and nonlinear fitting, requiring a substantial data foundation, leading to numerous test and CFD calculation cases and low computational efficiency.

To enhance the processing capability for complex roll data and overcome the limitations of traditional polynomial fitting methods, many researchers have explored the application of machine learning. Zhihua Zeng et al. [11] proposed a method based on the Particle Swarm Optimization (PSO) algorithm to obtain roll damping and restoring moment coefficients from free roll decay test curves, iteratively adjusting coefficients to minimize the error between predicted and experimental values. Jinwei Sun et al. [12] used an asymptotic method with free roll decay time history data, representing nonlinear damping with a linear plus quadratic model and nonlinear restoring moment with an odd function to identify damping coefficients, achieving good consistency with experimental data. Yichen Jiang et al. [13] applied a step-by-step model for reduced-order prediction of roll motion: first using CFD to simulate hull roll motion to obtain moments, then using an energy equivalence method to analyze the correction function of roll damping with the Keulegan–Carpenter (KC) number, and finally correcting roll damping based on the Euler equation to improve prediction accuracy and computational efficiency. Kyle E et al. [14,15] combined low-fidelity physics and machine learning methods, training on unforced roll decay time-series CFD data to predict the roll motion of the ONRT appended model in waves. Jiameng Li et al. [16] used 1500 ship roll data points, training and predicting with BP neural networks and random forests, respectively; for small-angle roll conditions, BP neural networks were prone to overfitting. Chong Li et al. [17] used a Nonlinear AutoRegressive model with exogenous inputs (NARX) neural network prediction method to predict ship roll motion, training on 1001 sets of target ship data. Compared with standard Backpropagation (BP) neural networks and BP with Adaptive Particle Swarm Optimization (SAPSO-BP), the NARX method showed superiority in Mean Absolute Error (MAE), Mean Squared Error (MSE), and Root Mean Squared Error (RMSE). Dong Zhang et al. [18] established a roll damping analysis model based on standard model test data, using roll angle and angular velocity as independent variables, introducing the Prony-SS method for roll damping identification to study the effects of speed and fluid memory on damping; as speed increased, fluid memory effects caused the lift component of roll damping to dominate. S. Pongduang et al. [19] established an inverse problem solution for the first kind of Volterra-type integral equation based on motion time history data to characterize the nonlinear added mass moment of inertia and damping moment in large-amplitude roll motion, using regularization to enhance solution stability and zero-crossing detection technology to effectively identify the functional forms of nonlinear added inertia and damping moments.

Among various machine learning methods, Support Vector Machine (SVM) has unique advantages in ship motion prediction. Transforming the matrix form of the ship roll motion equation into a hyperplane solution, SVM can handle data from constant-amplitude variable-frequency and constant-frequency variable-amplitude roll, enhancing its ability to process small samples and nonlinear features. Han Liu et al. [20] used the 3D potential flow software ANSYS-AQWA to perform multi-condition numerical calculations of the roll motion response of the damaged DTMB5415 model, constructing a damaged ship roll motion database, and used Support Vector Regression (SVR) for modeling and prediction. Jiang, Y et al. [21] derived a coupled response model for ship turning and roll motion based on a linearized 4-Degree-of-Freedom (DOF) ship maneuvering equation. Using part of the 10°/10° zigzag test data for the DTMB 5415 model, they trained support vectors for ε-SVM and Least Squares SVM (LSSVM) to identify maneuvering indices in the response model. The identified indices were used to predict 10°/10° and 20°/20° zigzag tests. Comparison with test data verified the predictive capability and generalization performance of the response model and identification method. Liu, X et al. [22] researched deck motion prediction methods, introducing two models based on sinusoidal wave combinations and statistical power spectra. Considering the instantaneous, random, and nonlinear nature of deck motion and the strong mapping capability of the SVM algorithm, they used LSSVM for prediction, dividing it into information accumulation and information window stages to avoid matrix inversion. Genetic algorithms optimized parameters. Simulations and experimental tests compared various methods, showing that the online LSSVM method with genetic algorithms had better generalization ability. Xu Chang-Zhou et al. [23] proposed an online ship roll motion prediction method based on Automatic Moving Grid Search—Least Squares Support Vector Machine (AGS-LSSVM). They used the fourth-order Runge–Kutta method to solve the second-order nonlinear differential equation of ship roll motion to obtain simulated data, the Pierson-Moskowitz spectrum to simulate irregular waves, combined a sliding time window with LSSVM for model training, and used the AGS method to optimize hyperparameters online to predict roll motion 30 s into the future. The AGS-LSSVM algorithm achieved online hyperparameter updates, significantly improving ship roll motion prediction accuracy compared to fixed hyperparameter methods. The computation time of the AGS algorithm met the timeliness requirements for online prediction. Chen, C et al. [24] introduced a Nonlinear Least Squares Support Vector Machine (NLS-SVM) algorithm to estimate unknown damping coefficients in ship roll models in shallow water. Combining numerical simulations and free decay test data, they validated the algorithm’s effectiveness and accuracy in identifying roll models. Compared with traditional methods, NLS-SVM performed better in identifying nonlinear damping coefficients and showed good applicability under different water depths and speeds.

After solving the parameters related to the ship roll motion equation, ship motion prediction is required. Since the ship roll motion equation is an Ordinary Differential Equation (ODE), researchers often use Reduced-Order Models (ROM) to construct NODE (Neural Ordinary Differential Equation). Gonzalez Rojas et al. [25] applied a latent ODE generation method to simulate the evolution of temporal coefficients provided by Proper Orthogonal Decomposition (POD), adding an extra neural ODE block after Variational Autoencoder (VAE) sampling. After training, latent trajectories were inferred by redefining time boundaries in the ODE solver. S. Pawar et al. [26] proposed a deep learning-based non-intrusive reduced-order modeling framework for data-driven reduced-order modeling of fluid flow dynamic systems. The computational cost of full-order simulations (i.e., DNS or even LES) remains prohibitively high due to the large number of degrees of freedom required to resolve all flow features, especially when traditional methods require repeated model evaluations over large parameter ranges. ROM methods can reduce computational burden and serve as surrogate models for efficient computational analysis of fluid systems. Portwood et al. [27] used Neural ODEs for data-driven modeling of fluid turbulence. They extracted turbulent kinetic energy k and its dissipation rate ε from Direct Numerical Simulation (DNS) data as ground truth and applied the continuous-time Neural ODE framework to model these time series. Compared to existing models, the Neural ODE model predicted the evolution of the dissipation rate more accurately for different p₀ values, with errors maintained at 1–2%, nearly two orders of magnitude lower than existing models. Fukami et al. [28] proposed the use of probabilistic neural networks for fluid flow surrogate modeling and data recovery, constructing a framework based on Gaussian distributions. They validated the framework’s capabilities in model reduction, prediction, and spatial field recovery using four typical cases: the shallow water equation, 2D cylinder flow, NACA0012 airfoil wake, and NOAA sea surface temperature dataset, also quantifying uncertainty to aid model interpretation. Maulik et al. [29] studied the performance of Long Short-Term Memory networks (LSTMs) and Neural ODEs (NODEs) in learning latent space representations of the dynamics governed by the advection-dominated viscous Burgers equation. The reduced-order space was obtained via POD, formulated in a non-intrusive manner. Validated based on two test problems, results showed LSTMs and NODEs could more effectively reproduce the effects of missing scales than intrusive Galerkin projection. They also introduced a method using scalable Bayesian optimization for selecting machine learning architectures. Maulik et al. [30] explored the application of probabilistic neural networks in fluid flow surrogate modeling and data recovery. This framework assumes target variables are sampled from a Gaussian distribution conditioned on inputs, predicting distribution hyperparameters to compute the objective function. Using four classic cases (shallow water equations, 2D cylinder flow, NACA0012 airfoil wake, NOAA SST dataset), they demonstrated that the network can not only build machine learning-based fluid flow surrogate models but also quantify uncertainty, aiding model interpretation. Ricky T. Q. Chen et al. [31] addressed the need for continuous ODE solutions by parameterizing the hidden layer derivative with a neural network, constructing a NODE continuous normalizing flow model trained via maximum likelihood estimation, explicitly trading numerical accuracy for speed.

Based on this, this study aims to couple the SVM learning algorithm and the NODE algorithm for analysis. Building on ship data from constant-amplitude variable-frequency and constant-frequency variable-amplitude roll, a coupled model for roll coefficient calculation and roll motion prediction is constructed. The structure of this paper is as follows: Section 2 introduces the coupled SVM-NODE model; Section 3 introduces the data foundation obtained from forced roll calculations; Section 4 covers data training and hyperparameter analysis; Section 5 presents prediction results and conclusion analysis.

2. Construction of the Coupled SVM-NODE Model for Ship Roll Motion Prediction

The solution to the ship roll motion problem in traditional methods is achieved through the ship roll motion equation. The coupled SVM-NODE model will also be constructed based on the roll motion equation. This section first introduces the ship roll motion equation, then explains the SVM and NODE models, and finally describes how to build the coupled SVM-NODE model to predict ship roll motion.

2.1. Ship Roll Motion Equation

The real ship motions can be expressed in six degrees of freedom. Due to the variation in the hull surface curvature, roll motion is usually coupled with sway and yaw motions. However, coupling of roll with other degrees of freedom is generally neglected because a good balance is achieved between simplicity and accuracy when using one degree-of-freedom [32]. In this paper, in order to discuss the problem of nonlinear roll damping, the equation of ship roll motion is expressed in the single-degree-of-freedom form as Equation (1) [5,33].

$$\left(I+I_{xx}\right)\ddot{\varphi }+N\dot{\varphi }+B\left|\dot{\varphi }\right|\dot{\varphi }+mgh\varphi =M$$

(1)

where $I$ is the moment of inertia about the ship’s longitudinal axis through the center of gravity, $\mathrm{k}\mathrm{g}·{\mathrm{m}}^2$; $I_{xx}$ is the added moment of inertia about the ship’s longitudinal axis, $\mathrm{k}\mathrm{g}·{\mathrm{m}}^2$; $N$ is the linear roll damping coefficient, $\mathrm{N}·\mathrm{m}·\mathrm{s}$; $B$ is the nonlinear roll damping coefficient, $\mathrm{N}·\mathrm{m}·{\mathrm{s}}^2$; $m$ is the ship mass, $\mathrm{k}\mathrm{g}$; $h$ is the metacentric height, $\mathrm{m}$; $g$ is the gravitational acceleration; $M$ is the hull roll moment, $\mathrm{N}·\mathrm{m}$; $\varphi$ is the roll angle, $\mathrm{r}\mathrm{a}\mathrm{d}$; $\dot{\varphi }$ is the roll angular velocity, $\mathrm{r}\mathrm{a}\mathrm{d}/\mathrm{s}$; $\ddot{\varphi }$ is the roll angular acceleration, $\mathrm{r}\mathrm{a}\mathrm{d}/{\mathrm{s}}^2$.

When using the forced roll method, it can be divided into amplitude variation and frequency variation. Conditions where the roll amplitude changes while the frequency remains constant are termed constant-frequency variable-amplitude. Conditions where the roll frequency changes while the amplitude remains constant are termed constant-amplitude variable-frequency.

The motion equation for constant-frequency variable-amplitude is:

$$\left\{\begin{array}{c}\varphi ={\varphi }_0·A·t·sin\left({\omega }_{0}t\right)\\ \dot{\varphi }={\varphi }_0·A·\left(sin\left({\omega }_{0}t\right)+{\omega }_{0}tcos\left({\omega }_{0}t\right)\right)\\ \ddot{\varphi }={\varphi }_0·A·{\omega }_0·\left(2·cos\left({\omega }_{0}t\right)-{\omega }_{0}tsin\left({\omega }_{0}t\right)\right)\end{array}\right.$$

(2)

The motion equation for constant-amplitude variable-frequency is:

$$\left\{\begin{array}{c}\varphi ={\varphi }_0·sin\left({A·\omega }_0·t^2\right)\\ \dot{\varphi }={2\varphi }_0·A·\omega ·t·cos\left(A·{\omega }_0·t^2\right)\\ \ddot{\varphi }={2\varphi }_0·A·{\omega }_0·\left(cos\left({A·\omega }_0·t^2\right)-2t·sin\left({A·\omega }_0·t^2\right)\right)\end{array}\right.$$

(3)

In Equations (2) and (3): ${\varphi }_0$ is the roll angle amplitude, $\mathrm{r}\mathrm{a}\mathrm{d}$; $A$ is the amplification factor; ${\omega }_0$ is the roll circular frequency, $\mathrm{r}\mathrm{a}\mathrm{d}/\mathrm{s}$.

Under forced roll data, $\varphi$, $\dot{\varphi }$, $\ddot{\varphi }$ are all time-series functions. Equation (1) can be written in the following matrix form:

$$\left[\begin{array}{ccc}{I}_{xx}& B& N\end{array}\right]\left[\begin{array}{c}\ddot{\varphi }\\ \left|\dot{\varphi }\right|\dot{\varphi }\\ \dot{\varphi }\end{array}\right]+\left(I·\ddot{\varphi }+mgh·\varphi -1·M\right)=0$$

(4)

2.2. SVM Model

The roll motion equation (Equation (1)) is rearranged into a matrix form for SVM regression:

$$X·\Theta =Y$$

(5)

where

• input matrix $X=\left[\ddot{\varphi } \dot{\varphi } \left|\dot{\varphi }\right|\dot{\varphi } \varphi \right]$ (time-series data of angular acceleration, angular velocity, nonlinear damping term, and roll angle),
• $\mathsf{\Theta }={\left[-\left(I+I_{xx}\right), -N, -B, -mgh\right]}^T$, parameter vector to be solved,
• target vector $Y=-M$ (external moment).

SVM treats the problem as finding an optimal hyperplane that minimizes the prediction error of the external moment M while maximizing the margin. For regression tasks (since we aim to fit continuous parameters), the SVM model is defined as:

$$f\left(x\right)={\omega }^T·\varphi \left(x\right)+b$$

(6)

where kernel function $\varphi \left(x\right)$ mapping the input vector $x$ (from $X$) to a high-dimensional feature space, weight vector $\omega$, bias term $b$.

In this study, a linear kernel function is used (as confirmed in Section 4.1), simplifying the model to:

$$f\left(x\right)={\omega }^T·x+b$$

(7)

The goal is to solve for $\omega$ and $b$ such that the predicted moment $\widehat{M}=f\left(x\right)$ matches the actual moment $M$ from CFD data, with the loss function minimized:

$$\begin{array}{c}min\\ \omega , b\end{array}\left\{{\displaystyle \frac{1}{2}}{‖\omega ‖}^2+C{\sum }_{i=1}^n{L}_{ϵ}\left(y_i-f\left(x_i\right)\right)\right\}$$

(8)

where

• $C$ = penalty parameter,
• $L_{ϵ}=ϵ-\mathrm{i}\mathrm{n}\mathrm{s}\mathrm{e}\mathrm{n}\mathrm{s}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{e} \mathrm{l}\mathrm{o}\mathrm{s}\mathrm{s} \mathrm{f}\mathrm{u}\mathrm{n}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}$, which ignores errors within $ϵ$ to avoid overfitting.

The SVM optimization problem is transformed into a quadratic programming problem using Lagrange multipliers. By introducing Lagrange multipliers $a_i$ and $a_i^{*}$, the dual form of the problem is solved to obtain the optimal $\omega$ and $b$.

Once $\omega$ and $b$ are determined, the parameter vector $\mathsf{\Theta }$ in Equation (5). is derived by comparing coefficients with the roll motion equation. Specifically:

• The coefficient of $\ddot{\varphi }$ in the SVM output corresponds to $-\left(I+I_{xx}\right)$
• The coefficient of $\dot{\varphi }$ corresponds to $-N$
• The coefficient of $\left|\dot{\varphi }\right|\dot{\varphi }$ corresponds to $-B$
• The coefficient of $\varphi$ corresponds to $-mgh$

This allows direct calculation of $I_{xx}$, $N$, and $B$ (since $I$ and $mgh$ are known from ship parameters.

2.3. NODE Model

In the coupled SVM-NODE model, the core function of the Neural Ordinary Differential Equation (NODE) is to construct a complete dynamic model and predict the free roll decay process based on the roll motion equation parameters solved by SVM. The mapping relationship between them is established on the order reduction in the roll motion equation and the continuous-time modeling characteristics of NODE, as follows:

The core parameters solved by SVM include:

• Total moment of inertia $\left(I+I_{xx}\right)$,
• Linear damping coefficient $N$,
• Nonlinear damping coefficient $B$.

In the free roll state, the external moment $M=0$, and Equation (1) is simplified to:

$$\ddot{\varphi }=-{\displaystyle \frac{1}{I+I_{xx}}}\left(B\left|\dot{\varphi }\right|\dot{\varphi }+N\dot{\varphi }+mgh\varphi \right)$$

(9)

To adapt to the ODE solving form of NODE, the second-order Equation (9) needs to be reduced to a system of first-order equations. Define state variables:

• $\varphi$ (roll angle),
• $v=\dot{\varphi }$ (roll angular velocity, i.e., the first derivative of $\varphi$).

Then, the roll angular acceleration $\dot{\varphi }$ can be expressed as the first derivative of $v$, i.e., $\dot{v}$. Equation (9) is transformed into:

$$\left\{\begin{array}{c}\dot{\varphi }=v\\ \dot{v}=-{\displaystyle \frac{1}{I+I_{xx}}}\left(Nv+B\left|v\right|v+mgh\varphi \right)\end{array}\right.$$

(10)

This system of first-order equations fully describes the dynamic characteristics of roll motion, where:

• The first equation defines the derivative relationship between state variables.
• The second equation quantifies the relationship between the rate of change in angular velocity and the current state $\left(\varphi ,v\right)$ through parameters solved by SVM $\left(I+I_{xx}, N, B\right)$ and known parameters $mgh$.

The core of NODE is to parameterize the derivative function of ODE with a neural network, realizing the modeling of continuous-time dynamic processes. In this model, NODE directly maps the above first-order system Equation (10), specifically as follows:

• Mapping between Input Layer and State Variables: The input of NODE is the state vector $\mathrm{s}={\left[\varphi ,v\right]}^{\mathrm{T}}$ at the current moment, corresponding to the roll angle and angular velocity.
• Mapping between Hidden Layer and Dynamic Parameters: NODE learns the derivative $\dot{s}$ of the state vector through the neural network $f\left(s, t; \theta \right)$, where $\theta$ is the network parameter. The functional form of this neural network is designed to strictly correspond to system (11):

$$f\left(s, t; \theta \right)=\left[\begin{array}{c}\dot{\varphi }\\ \varphi \end{array}\right]=\left[\begin{array}{c}v\\ -{\displaystyle \frac{1}{I+I_{xx}}}\left(Nv+B\left|v\right|v+mgh\varphi \right)\end{array}\right]$$

(11)

At this time, the parameters solved by SVM $\left(I+I_{xx}, N, B\right)$ are embedded into the calculation logic of the neural network as fixed parameters, directly determining the form of the derivative function.

• Mapping between Output Layer and Time Integration: NODE performs time-domain integration on $f\left(s,t;\theta \right)$ through numerical integration methods (such as the Implicit Adams method used in Section 4.2). Starting from the initial state (e.g., $\varphi ={\varphi }_0$, $v=0$) at $t=0$, it obtains the state $s\left(t\right)$ at any time, i.e., the time history curve of roll angle $\varphi \left(t\right)$.

2.4. SVM-NODE Coupled Model

Taking the roll motion equation (Equation (1)) as the core, a hyperplane is constructed to form the solution conditions for the SVM model. Using CFD-calculated forced roll data, the added moment of inertia $I_{xx}$, linear roll damping $N$, and nonlinear roll damping $B$ are solved, providing the motion equation with complete coefficients, forming a solvable ODE $f\left(t,\varphi ,M\right)$. The roll motion equation is reduced in order, and the reduced-order system is used to construct the solution conditions for the NODE model. The initial state of free roll decay at any angle is used as the initial condition for the NODE model, and time history integration is performed within the range $\left[\begin{array}{cc}{t}_0& t_1\end{array}\right]$. The framework diagram of the SVM-NODE coupled model is shown in Figure 1.

3. Forced Roll Calculation Based on Standard Hull Form

The forced roll calculations in this chapter are performed using the CFD method. The roll data will be used to train the coupled SVM-NODE model.

3.1. Calculation Model

The DTMB5415 ship model is used for the calculations, and model parameters are shown in

Table 1. Parameters of DTMB5415 model [2].

Parameter	Unit	Value
Scale Ratio		51
Length Between Perpendiculars $L_{PP}$	m	2.788
Waterline Beam $B_{WL}$	m	0.374
Molded Beam $B$	m	0.403
Draft $T$	m	0.121
Displacement $\Delta$	kg	63.50
Block Coefficient $C_B$		0.505
Longitudinal Center of Gravity $L_{CG}$	m	1.375
Vertical Center of Gravity $V_{CG}$	m	0.148
Roll Moment of Inertia $I_{xx}/B$		0.370
Metacentric Height $h$	m	0.038

.

CFD simulations were performed using Star-CCM+(18.06.006.R8) software. An overset grid approach was employed to handle the rotational motion of the hull during roll, consisting of a stationary background grid and a rotating sub-grid around the ship’s longitudinal axis, with donor-acceptor cell interpolation ensuring accurate data exchange between grids [34]. The SIMPLE semi-implicit method and a segregated flow solver were utilized for the computations. For air-water interface capture, the Volume of Fluid (VOF) method with a High-Resolution Interface Capturing (HRIC) scheme was adopted to resolve sharp free surface gradients. The k-ω SST turbulence model was used to solve the Reynolds stresses.

For the computational domain boundaries (Figure 2), a velocity inlet boundary is set in front of the ship, using a uniform incoming flow velocity (set to 0); a pressure outlet boundary is set behind the ship, maintaining the ambient pressure; the sides and top of the computational domain are also set as pressure outlet boundaries and the bottom is set as a no-slip wall boundary. The hull surface adopts the no-slip wall boundary condition. In addition, to avoid the impact of wave reflection on the calculation results, appropriately wide damping wave absorption zones are set on both sides of the computational domain.

3.2. Grid Uncertainty Analysis

Grid Uncertainty analysis was performed using the multiple solutions (m) method. At least three solutions (m = 3) were used, uniformly refined with cell increments $\Delta x_k$, defining a constant refinement ratio $r_k$:

$$r_k=\Delta x_m/\Delta x_{k_{m-1}}$$

(12)

The convergence ratio ($R_k$) is defined by considering the solution change (${\epsilon }_{ijk} ={ S}_{ki} -{ S}_{kj}$) between the fine ($S_{k1}$), medium ($S_{k2}$), and coarse ($S_{k3}$) solutions for input parameter $k$:

$$R_k={\epsilon }_{{21}_k}/{\epsilon }_{{32}_k}$$

(13)

According to ITTC guidelines (ITTC Procedures and Guidelines, 2024) [35,36], three different cases for $R_k$ may occur:

(1)

Monotonic convergence: $0 < R_k < 1$;

(2)

Oscillatory convergence: $R_k < 0$, $\left|R_k\right|<1$;

(3)

Divergence: $\left|R_k\right|>1$.

Based on the grid independence study, an overset grid with refinement schemes was used, as shown in Figure 3. The computational domain size was: ${2L}_{PP}$ ahead of the ship, ${2L}_{PP}$ behind the ship, ${1.2L}_{PP}$ on each side, and a vertical height of $L_{PP}$. The calculation used 1-DOF. To avoid wave reflection effects, damping wave absorption zones ${0.5L}_{PP}$ wide were set on both sides. Numerical results for three grid resolutions—coarse (Mesh A, 1.36 million cells), medium (Mesh B, 3.32 million cells), and fine (Mesh C, 5.86 million cells)—are shown in Figure 4. The calculated $R_{km}$ curves are shown in Figure 5.

In Figure 4, the roll angle over time is generally consistent across the three grid schemes, with deviations mainly at peaks and troughs. The medium and fine grids are close, while the coarse grid shows relatively larger differences. Convergence analysis of the three grids (Figure 5) shows $R_{km}$ mostly within [−1, 1]. Larger absolute values of $R_{km}$ correspond to peaks and troughs, showing strong regularity over time, satisfying convergence requirements. Therefore, the Mesh B scheme is used for subsequent calculations.

3.3. Time Step Uncertainty Analysis

In implicit solvers the time step is decided by the flow features. For periodic phenomena (roll decay, vortex shedding, etc.) use at least 100 time steps per period [37]. Measured roll period is 1.37 s of the DTMB5415 mode [12]. So a recommended value of 0.014 s is promoted. A time step convergence study with three increments (0.002 s, 0.0014 s, 0.001 s) using a refinement ratio of $\sqrt{2}$ is launched.

In Figure 6, the time-history calculation results for different time steps are very close. With reference to the grid independence analysis, a convergence analysis of the time steps $R_{kt}$ was conducted, as shown in Figure 7. After 25 s, roll degree of hull enters a relatively stable phase. The time step is more sensitive within a small range, but overall, it does not have a significant impact on the calculation results, $\left|R_{kt}\right|<1$, the timesteps satisfy the convergence conditions. A similar pattern can be observed in the grid independence analysis shown in Figure 5. To accelerate the computation, a time step of 0.002 s was selected for subsequent calculations [2].

3.4. Condition Definition

The ship model is subjected to inertial forces, damping forces, restoring forces, and external forces during roll motion. Damping forces include linear and nonlinear components. At small roll angles, the nonlinear damping term is not significant due to low roll angular velocity $\dot{\varphi }$, but its proportion increases as the angle increases. Since free roll decay tests experience rapid roll angle decay, they are not conducive to solving the nonlinear damping term. This paper uses forced roll tests as the data foundation to solve for the added moment of inertia $I_{xx}$, linear roll damping $N$, and nonlinear roll damping $B$. The simulation conditions are defined in

Table 2. Forced Roll Simulation Conditions.

Condition Name	Initial Roll Angle ${\mathit{\varphi }}_0$	Angle Amplification Factor ${\mathit{K}}_{\mathit{\varphi }}$	Initial Circular Frequency ${\mathit{\omega }}_0$	Frequency Amplification Factor ${\mathit{K}}_{\mathit{\omega }}$
	$\mathbf{r}\mathbf{a}\mathbf{d}$		$\mathbf{r}\mathbf{a}\mathbf{d}/\mathbf{s}$
Variable Amplitude Roll	$\pi /180$	1.1	4.425	-
Variable Frequency Roll	$\pi /9$	-	2.95	1.1

.

The forced roll motion angle is expressed as:

$$\left\{\begin{array}{cc}\varphi ={\varphi }_0·K_{\varphi }·t·sin\left({\omega }_0·t\right)& \left(variable amplitude roll\right)\\ \varphi ={\varphi }_0·sin\left({\omega }_0·K_{\omega }·t^2\right)& \left(variable frequency roll\right)\end{array}\right.$$

(14)

The motion of the hull and the cylindrical motion domain is specified by Equation (14). Hull motion angles and roll moments are monitored.

3.5. Variable Frequency Forced Roll

In the variable frequency forced roll condition, slamming and spray caused by nonlinearities increase significantly as frequency rises, as shown in Figure 8. Figure 9 shows the damping zones applied on both sides suppressing wave reflection effects.

The time history of angle and moment for variable frequency forced roll is shown in Figure 10. A phase lag is evident between moment and angle. Analysis of peak/trough extremes (Figure 11) shows that as time progresses and roll frequency increases, the time difference exhibits distinct multi-stage patterns: low at low frequencies, decreases noticeably with increasing frequency in mid-range, and significantly low at high frequencies. This characteristic phase shift reflects the strong nonlinearity during variable frequency forced roll and is effective for solving roll motion equation coefficients.

3.6. Variable Amplitude Forced Roll

The variable amplitude roll calculation used the hull natural frequency analyzed in reference [2] as the computational frequency. As the roll amplitude increases gradually, the bow wave making is less intense compared to variable frequency roll, as seen in Figure 12. The overall wave-making amplitude is small in the free surface flow field (Figure 13).

Figure 14 shows the time history of angle and moment during variable amplitude roll. Compared to Figure 10, angle and moment are more concomitant, with peak/trough occurrences closely aligned. Time difference analysis (Figure 15) shows an overall low level of time difference, decreasing slightly as amplitude increases. At larger amplitudes where nonlinear effects are more significant, the time difference variation begins to increase.

4. Validation of the Coupled SVM-NODE Model for Ship Roll Motion Prediction

This chapter utilizes the forced roll calculation data from Section 3. The SVM model learns the time history data from variable amplitude and variable frequency roll to obtain the added moment of inertia, linear damping, and nonlinear damping coefficients for roll motion. The NODE model is then used to construct a complete roll motion equation solution model to predict ship roll motion. The effectiveness of the coupled SVM-NODE model is validated by comparing it with CFD data.

4.1. SVM Training Results

The SVM model was trained on the variable amplitude and variable frequency roll time history data. Since it is forced roll, backtesting focuses primarily on the moment. The kernel function used was Linear, penalty parameter C = 5 × 10⁻³, tolerance error $epsilon\left(\epsilon \right)=0.001$. Backtesting results for variable frequency are shown in Figure 16, and for variable amplitude in Figure 17.

Figure 16 and Figure 17 show that SVM fits both forced roll conditions well. With the same SVM parameters, variable frequency roll backtesting is relatively better. Variable amplitude roll backtesting shows oscillations near extrema, becoming more intense over time. The roll motion equation coefficients calculated for the two conditions are shown in

Table 3. SVM Calculated Coefficient Results (Model Scale).

Condition	Added Moment of Inertia ${\mathit{I}}_{\mathit{x}\mathit{x}} \mathbf{N}\mathbf{m}\mathbf{s}$	Linear Roll Damping $\mathit{N} {\mathbf{N}\mathbf{m}\mathbf{s}}^2$	Nonlinear Roll Damping $\mathit{B} {\mathbf{k}\mathbf{g}\mathbf{m}}^2$
Variable Frequency Roll	2.02196664 × 10−3	8.26139151× 10−5	4.49492591× 10−2
Variable Amplitude Roll	1.701961× 10−2	7.5673× 10−4	2.627401× 10−2

(model scale).

4.2. NODE Prediction Results Analysis

The coefficients from Table 3 for both conditions were input into the NODE model. Prediction time was set to 0–10 s. The integration method chosen was Implicit Adams, with relative tolerance $rtol=1\times {10}^{-3}$ and absolute tolerance $atol=1\times {10}^{-6}$. CFD simulations of free decay under different initial roll angles were also performed. Results using coefficients from both conditions and the CFD method are compared in Figure 18, Figure 19 and Figure 20.

Figure 18, Figure 19 and Figure 20 show that compared to CFD results, the roll decay amplitudes predicted by NODE solving the roll motion equation are larger. This is attributed to the CFD using a 1DOF model, reducing coupling decay from other degrees of freedom. At small angles, the NODE method, using coefficients solved from either variable frequency or variable amplitude SVM data, fits the CFD time history curves well. At larger roll angles, the initial period and amplitude are similar, but the results diverge over time.

Further analysis of half-period and single amplitude during decay is shown in Figure 21 and Figure 22. In the decay period comparison, the error between NODE and CFD is stable around 2% for small initial angles. At larger angles, CFD-calculated periods show outlier fluctuations which the NODE roll motion model fails to fully capture. This is partly because SVM training sample amplitudes did not reach 22.5°, limiting model adaptability to large angles. Additionally, the DTMB5415 model’s deck approaches immersion at 22.5°, causing green water phenomena and increased computational complexity. In roll amplitude comparison, NODE and CFD results are closer for smaller initial angles, but the error increases over time. Comparatively, results using coefficients solved from variable amplitude forced roll are closer to CFD results than those from variable frequency forced roll.

5. Conclusions

This paper constructed a coupled SVM-NODE model based on forced roll data input. The SVM model optimizes the solution for roll motion parameters including added moment of inertia, nonlinear damping coefficient, and linear damping coefficient. The NODE model solves the reduced-order roll motion equation for time history calculation, achieving weak coupling within the SVM-NODE model. Taking DTMB5415 as the research object, both CFD and the coupled SVM-NODE method were used to analyze single-degree-of-freedom roll decay under different initial roll angles. The conclusions are as follows:

(1)

Model Accuracy: The coupled SVM-NODE model demonstrates good predictive performance across different roll angles. For small-angle roll (small initial roll angles), the model’s results highly align with CFD simulation results, indicating its ability to accurately capture ship motion characteristics during small-angle roll. Under large-angle roll conditions, although deviations exist between model results and CFD simulations, the overall trends are consistent, and the errors are within an acceptable range. This confirms the model’s reliable analytical capability in complex large-angle roll scenarios, providing an effective reference for ship roll performance assessment.

(2)

Data Processing and Parameter Solving: The Support Vector Machine (SVM) method shows significant advantages in processing forced roll data. By analyzing data from constant-amplitude variable-frequency and constant-frequency variable-amplitude roll conditions, SVM efficiently solves for roll damping and added moment of inertia. Compared to traditional methods, it substantially reduces the number of required computational cases. This significantly lowers the cost and time consumption of experiments and CFD calculations while improving the efficiency and accuracy of parameter solving, offering a new pathway for rapid acquisition of ship roll motion parameters.

(3)

Model Applicability: The simulation of the free roll decay process by the Neural Ordinary Differential Equation (NODE) network effectively verifies the accuracy of the roll damping coefficients and added moment of inertia obtained by SVM. Within a certain angle range, this coupled model meets the demand for rapid analysis of ship roll motion. It holds high practical value for both performance prediction during ship design stages and roll motion assessment during actual navigation, providing an innovative and effective method for research and engineering applications in the field of ship roll.