A Multi-Data Fusion-Based Bearing Load Prediction Model for Elastically Supported Shafting Systems

Abstract

To address the challenge of bearing load monitoring in elastically supported marine shafting systems, a multi-data fusion-based prediction model is constructed. In view of the small-sample nature of measured bearing load data, transfer learning is adopted to migrate the physical relationships embedded in finite element simulations to the measurement domain. A limited number of actual samples are used to correct the simulation data, forming a high-fidelity hybrid training set. The system—supported by air-spring isolators mounted on the raft—is divided into multiple sub-regions according to their spatial layout, establishing local mappings from air-spring pressure variations to bearing load increments to reduce model complexity. On this basis, a Stacking ensemble learning framework is further incorporated into the prediction model to integrate multi-source information such as air-spring pressure and raft strain, thereby enriching the model’s information acquisition and improving prediction accuracy. Experimental results show that the proposed transfer learning-based multi-sub-region bearing load prediction model performs significantly better than the full-parameter input model. Furthermore, the strain-enhanced Stacking-based multi-data fusion bearing load prediction model improves the characterization of shafting features and reduces the maximum prediction error. The proposed multi-data fusion modeling strategy offers a viable approach for condition monitoring and intelligent maintenance of marine shafting systems.

1. Introduction

The shafting system, serving as a critical component for power transmission, directly determines whether the entire mechanical system can operate continuously, reliably, and safely. Shafting plays an essential role in various power transmission applications, including industrial machinery [1], automotive drivetrains [2], and marine propulsion systems [3]. This is particularly evident in marine propulsion shafting, where alignment quality must ensure that the actual installed state matches the theoretical design and satisfies key design requirements, thereby keeping bearing loads and shaft segment stresses within allowable limits or achieving an optimal distribution. However, during actual ship operation, constraints, loads, boundary conditions, and external disturbances continuously vary, leading to dynamic changes in the shafting operating state. As a key aspect of shaft alignment, bearing load verification allows direct assessment of shaft alignment quality through the measurement of bearing loads [4,5,6,7]. However, under actual operating conditions, continuously varying constraints, loads, boundary conditions, and external disturbances cause dynamic changes in shafting behavior. As an essential aspect of alignment assessment, bearing load verification enables direct evaluation of alignment quality through load measurement. Bearing load measurement methods are generally classified into direct and indirect approaches. Direct methods employ force measurement devices such as jack-up techniques and load cells to obtain bearing loads explicitly, while indirect methods estimate loads by monitoring shaft-related parameters using approaches like the three-moment equation [8], transfer matrix method [9], or finite element analysis [10,11]. In practice, installing load sensors directly on bearings requires additional space, and jack-up methods are typically limited to installation and commissioning phases. Current research on bearing load identification and prediction focuses predominantly on rigidly supported shafting systems, with online monitoring of elastically supported shafting remaining relatively understudied [12,13,14]. Figure 1 presents a schematic of the elastically supported shafting structure. Compared with conventional rigidly supported configurations, bearing loads in an elastically supported shafting system are more sensitive to external disturbances, and the resulting elastic deformations introduce greater complexity into the alignment condition of the shafting mounted on the raft. Therefore, achieving accurate online prediction of bearing loads—which are difficult to measure directly—enables not only the assessment of whether they remain within acceptable limits, but also the analysis of the operational state of the shafting and the implementation of corresponding adjustments. This capability holds significant engineering importance for advancing intelligent operation and long-term health monitoring of propulsion shafting systems [15].

Machine learning methods have been widely applied to load prediction in aerospace, transportation, and bridge engineering [16,17,18]. For instance, in the field of bearing life prediction, Shutin et al. integrated physics-based and data-driven models to establish a mapping relationship among speed, temperature, current wear, bearing wear rate, and remaining useful life, enabling online prediction of wear and remaining service life for locomotive traction motor shaft bearings [19]. In the domain of structural load monitoring, Cooper et al. employed a double-layer feedforward neural network to establish a mapping between 15 strain data points and static loads at two locations on a wing rib structure [20]. Jiang et al. introduced a PCA-GA-BP neural network, integrating Principal Component Analysis (PCA) and Genetic Algorithm (GA) to optimize a Backpropagation (BP) neural network, which used flight parameters as input to predict flight loads [21]. These studies demonstrate that for structural systems characterized by intricate mechanisms and challenging modeling requirements, machine learning methods offer a way to circumvent cumbersome physics-based modeling by directly constructing nonlinear mappings between inputs and outputs. Traditional analytical methods based on linear assumptions (e.g., the influence coefficient method) or empirical look-up tables often fail to accurately characterize load transfer behavior in elastomeric support rafts under conditions involving geometric discontinuities and stiffness nonlinearity. Consequently, employing machine learning methods for bearing load prediction has emerged as a promising technical pathway. However, bearing load data for marine shafting is typically only collectible during the installation and commissioning phases, making it difficult to obtain continuous measurements during actual vessel operation. This results in a very limited volume of data available for training. Under such small-sample conditions, conventional machine learning models struggle to fully learn the underlying patterns within the data, significantly compromising their generalization capability and prediction accuracy [22,23].

Transfer learning can significantly reduce a target model’s dependency on large amounts of target-domain data by transferring relevant knowledge from existing models, thereby enabling effective modeling even when only a small amount of target data is available [24]. For instance, Wei et al. developed a machine learning model based on a transfer learning framework to predict rotating bending fatigue strength. This approach first constructs a source domain model using a large, cost-effective tensile property dataset and then adapts it to the target domain with minimal fatigue samples [25]. In the field of bearing condition monitoring, transfer learning has also demonstrated notable engineering utility. To address the issue of low diagnostic accuracy for gearbox faults under imbalanced data and varying operating conditions, Wang et al. proposed a hybrid data-assisted multi-source domain transfer learning method, enabling the transfer of knowledge from multiple source domains to different target domains [26]. To tackle the challenge of acquiring measured fault data, Aduwenye et al. utilized massive simulation data generated by a rotor-bearing dynamics model to build a convolutional neural network source-domain model. By combining four transfer learning strategies with a small number of measured samples, they achieved efficient bearing fault diagnosis across domains and under varying operating conditions, with three of the strategies attaining perfect diagnostic accuracy [27]. For marine shafting systems characterized by complex hull structures and variable operating conditions, finite element simulation serves as an effective means of providing substantial data support at a relatively low cost. However, discrepancies often exist between simulation results and actual measurements. These differences stem partly from measurement errors in physical load tests and partly from the inherent inability of simulation models to fully replicate real-world structures. Therefore, by leveraging a transfer learning mechanism, a small amount of measured data can be used to correct finite element simulation results, constructing a hybrid dataset that balances physical rationality with data fidelity. This approach enables effective expansion of the training set under the constraint of limited measured samples.

Stacking ensemble learning is a machine learning methodology that enhances model performance by combining multiple base learners. Its core principle lies in leveraging the diversity and complementarity of different base learners in feature extraction and pattern recognition, integrating their strengths through an ensemble mechanism to improve model generalization capability, thereby achieving superior prediction accuracy and robustness compared to individual learners [28]. Xu et al. tackled complex power system fault classification and feature selection challenges by integrating multiple models through Stacking ensemble learning, significantly improving fault detection accuracy over single-model approaches. Separately, Tong et al. developed a multi-sensor feature fusion method for bearing fault diagnosis, where vibration signals undergo variational mode decomposition before combining 12 time-domain, 5 frequency-domain, and 5 multi-scale entropy features into a multi-domain feature set, using these fused features for deep feature learning and classification [29].

Addressing the challenge of bearing load monitoring in elastically supported marine shafting systems, this paper proposes an integrated solution model that combines transfer learning, multi-subdomain modeling, and Stacking ensemble learning. This model aims to overcome the limitations of fully data-driven machine learning prediction models—specifically, their insufficient accuracy and generalization capability under conditions of limited measured data samples. By utilizing a Stacking ensemble learning framework, the model effectively integrates two types of information: external excitation forces from air-spring pressure and the internal structural response from raft strain. This fusion constructs a Stacking ensemble learning-based bearing load prediction model suitable for small-sample scenarios in elastically supported shafting systems.

To achieve the research objectives outlined above, the subsequent sections of this paper are structured as follows: Section 2 details the overall methodology and technical specifics of the model construction. It separately introduces the establishment processes of the transfer learning-based multi-subdomain bearing load prediction model and the Stacking-based multi-data fusion model that incorporates air-spring pressure and raft strain. Section 3 presents experimental validation results, quantifying the prediction accuracy and robustness of the proposed model through comparisons with baseline models, such as the full-parameter input BP model. This section also analyzes the impact of different meta-learners on the ensemble model’s performance and discusses sources of model error, the limitations of idealized boundary conditions, and the advantages of multi-source information fusion. Finally, Section 4 summarizes the core findings of the paper, clarifies the engineering application value of the proposed modeling strategy.

2. Method

2.1. Overall Methodology for Model Construction

The workflow for establishing the Stacking ensemble learning-based bearing load prediction model for elastically supported shafting systems is illustrated in Figure 2. The model first introduces a transfer learning mechanism: a low-fidelity simulation dataset is obtained through a finite element model and used to train initial low-fidelity models. A small amount of measured data is then utilized to correct the outputs of these models, thereby constructing a high-fidelity hybrid dataset. This process reduces dependency on large-scale measured data. Building on this, a multi-subdomain partitioning strategy is employed to decompose the overall system into several local modeling units. To enhance the physical interpretability of the model, the concept of “bearing load increment” is introduced as the modeling variable, establishing a mapping between subdomain pressure changes and bearing load increments. For any given subdomain I and bearing load measurement point j, the “bearing load increment” $\mathsf{\Delta }{L}_{ij}$ is defined as the difference between the absolute values of the bearing load measured at point j after and before the air spring pressure change in that subdomain, namely

$$\mathsf{\Delta }{L}_{ij}=L_{ij}^{\mathrm{after}}-L_{ij}^{\mathrm{before}}$$

(1)

where $L_{ij}^{\mathrm{before}}$ and $L_{ij}^{\mathrm{after}}$ represent the bearing load recorded at measurement point j before and after adjusting the air spring pressure in subdomain I, respectively.

By establishing the local mapping relationship from “subdomain pressure variation” to “bearing load increment,” each lightweight sub-model only needs to learn the net effect of pressure changes in a specific subdomain on the shafting load. This increment $\mathsf{\Delta }L$ directly quantifies the change in the output variable (load variation) caused by the change in the input variable (subdomain pressure variation), thereby establishing a purer and more well-defined local causal input–output relationship. The final bearing load is then predicted by aggregating the load increments from all subdomains relative to the initial air-spring pressure state of the shafting platform. This strategy reduces model complexity while improving modeling efficiency and interpretability. Furthermore, a Stacking ensemble learning framework is incorporated to fuse multi-source information, such as air-spring pressure and raft strain, constructing a multi-level, multi-modal feature-load mapping relationship. Ultimately, this enables more accurate and robust prediction of bearing loads under small-sample conditions.

2.2. Transfer Learning-Based Multi-Sub-Region Bearing Load Prediction Model

The transfer learning method adopted in this study aims to migrate the mapping relationship between air-spring pressure and bearing load, initially established through finite element simulation, to actual shafting test data. This enhances the learning efficiency and generalization performance of the prediction model under small-sample conditions. The model constructed via transfer learning consequently retains the physical priors extracted from simulation data while incorporating the true system characteristics reflected in the measured data.

As illustrated in Figure 3, this study takes an elastically supported propulsion shafting experimental platform as the research object. The initial subdomain division is primarily based on engineering heuristics: considering the asymmetric structure and the arrangement of main equipment, the 17 air springs are divided into six subdomains (S1–S6) according to their spatial positions relative to the bearings. For instance, subdomains S1–S5 are partitioned based on axial physical proximity, while air springs V16 and V17, located directly beneath the motor, exhibit significantly different load-bearing characteristics compared to the axially adjacent V13–V15 and are therefore grouped separately into subdomain S6. The core of this subdomain division lies in the subsequent multi-subdomain modeling strategy: during the training process, the independent submodel for each subdomain automatically learns and optimizes the weights of the input features from the air springs within that subdomain in a data-driven manner. This enables dynamic calibration of their actual contribution to bearing load prediction. This design not only achieves effective dimensionality reduction of input parameters but also ensures the model’s robustness against potential sensitivity biases arising from the initial grouping.

The development of the transfer learning-based multi-subdomain bearing load prediction model—abbreviated as the MSTL-BLP model—comprises two main stages, as illustrated in Figure 4. In the first stage, a finite element model of the elastically supported shafting system is established based on the physical test rig. Cyclic inflation/deflation conditions are designed for air springs in each sub-region to obtain low-fidelity simulation data $D_l$. Using $D_l$, a BP neural network is constructed to map the sub-region air pressure $Q_l$ to the bearing load increment $F_l$, yielding a low-fidelity prediction model $M_{l1}-M_{l6}$.

Subsequently, $F_{Ml1}-F_{Ml6}$ is concatenated with the air pressure $Q_h$ from the corresponding region in $D_h$ to serve as the input for the high-fidelity prediction model, with the actual bearing load variation $F_h$ from $D_h$ as the output. Following network parameter initialization, a backpropagation neural network is used to build the calibrated model $M_{c1}-M_{c6}$. The subdomain air pressure–bearing load increment prediction model $M_{h1}-M_{h6}$ is then formed by serially integrating models $M_{l1}-M_{l6}$ and $M_{c1}-M_{c6}$. The parameters of $M_{h1}-M_{h6}$ thus incorporate knowledge transferred from the simulation dataset as well as features learned from the actual measured data.

In the second stage, the training data $D_{tr}$ from multi-sub-region mixed inflation conditions, specifically the air pressure $Q_{tr}$, is input into the corresponding sub-region prediction model $M_{h1}-M_{h6}$ to obtain the predicted bearing load increment $F_{y1}-F_{y6}$ for each region. The load increment predictions $F_{y1}-F_{y6}$ from all six sub-regions are then used as input to the bearing load prediction model, with the actual bearing load values $F_{tr}$ from $D_{tr}$ as the output. Following the initialization of the BP neural network parameters, a genetic algorithm is applied to optimize the model, resulting in the sub-region load increment-to-final bearing load prediction model $M_{s1}$. The topology of the constructed neural network is shown in Figure 5. By serially integrating the $M_{h1}-M_{h6}$ and $M_{s1}$ models, the final transfer learning-based multi-sub-region bearing load prediction methodology is established.

The backpropagation (BP) neural network is a multilayer feedforward neural network trained using the error backpropagation algorithm. Its core principle lies in employing gradient descent to iteratively adjust network weights and biases through forward propagation for output computation and backward propagation for error signal transmission, thereby approximating complex nonlinear mapping relationships [30,31]. The input signal propagates layer by layer from the input layer to the output layer via weighted summation and nonlinear transformation. For the $j$-th neuron, its input $z_j$ is calculated as follows:

$$z_j={\displaystyle \sum _i{w}_{ji}}{a}_i+b_j$$

(2)

The net input $z_j$ of the neuron is subsequently transformed nonlinearly by the activation function $f(z)$, yielding the final output $a_j$ of the neuron:

$$a_j=f(z_j)=f({\displaystyle \sum _i{w}_{ji}}{a}_i+b_j)$$

(3)

In this study, the hidden layers adopt the ReLU function as the activation function. After the forward propagation yields the network output, it is compared with the ground truth labels, and the error is computed using MSE loss function:

$$E_{MSE}={\displaystyle \frac{1}{2N}}{\displaystyle \sum _{i=1}^N\Vert y_i-z_i{\Vert }^2}$$

(4)

where $N$ represents the number of samples.

Subsequently, the error signal is propagated backward through the network. The gradients of the loss function $E$ with respect to each layer’s parameters ($w_{ji}$ and $b_j$) are computed via the chain rule. Starting from the output layer, the gradient of the loss with respect to the output is calculated first, then backpropagated layer by layer. The parameter update rule is as follows:

$$w_{ji}^{new}=w_{ji}^{old}-\eta {\displaystyle \frac{\partial E}{w_{ji}}}$$

(5)

$$b_{ji}^{new}=b_{ji}^{old}-\eta {\displaystyle \frac{\partial E}{b_{ji}}}$$

(6)

where $\eta$ denotes the learning rate, a hyperparameter controlling the step size of parameter updates.

When employing the genetic algorithm (GA) to optimize the BP neural network, the GA is first utilized to automatically search for optimal hyperparameters—such as learning rate, hidden layer depth, and neuron count—thereby mitigating the time-consuming and locally optimal tendencies of traditional manual trial-and-error methods [32]. Subsequently, based on the optimized hyperparameter configuration, the GA is invoked again to perform global optimization of the network’s initial weights and biases. After GA optimization, the error backpropagation mechanism is applied to fine-tune the network parameters. This approach effectively combines the global exploration capability of the genetic algorithm with the local precise search characteristics of the BP algorithm, enhancing the overall model performance and robustness.

2.3. Stacking Ensemble Learning-Based Model with Multi-Data Fusion for Bearing Load Prediction

In elastically supported shafting platforms, variations in air-spring pressure act as the primary external disturbance on the system, directly influencing the shafting boundary conditions and the overall deformation of the raft. However, the bearing load, as the final response, does not correspond directly to the pressure input. Their mapping relationship is mediated by the complex deformation of the raft, resulting in a highly coupled pressure-deformation-load interaction. To meet requirements for weight reduction and installation, the raft often incorporates geometric discontinuities such as openings and stiffeners, leading to a non-uniform spatial distribution of its structural stiffness. This stiffness heterogeneity causes local raft deformation to become a nonlinear “decoupling” link between air pressure and bearing load: identical pressure changes, when transmitted through this heterogeneous structure, can induce highly nonlinear deformation responses, making it difficult to reliably infer the final load from pressure data alone. Consequently, relying solely on external pressure excitation, models struggle to accurately capture the load response patterns and are prone to unstable accuracy under small-sample conditions. To address this issue, it is necessary to introduce observational signals that directly reflect the true internal state of the structure. The strain on the raft near the bearings is a direct internal structural response under load, possessing a shorter and more well-defined mechanical transfer path to the bearing load. By simultaneously incorporating strain information, the model can bypass the difficult-to-model nonlinearity of the “pressure–complex deformation” link and directly establish a more robust mapping relationship between “strain and bearing load,” thereby enhancing prediction accuracy and robustness.

To monitor the local structural response in characteristic regions of the raft, strain sensors are installed on the upper surface of the raft near the bearings. Their measurement data can quantify the degree of local deformation at these positions. It is important to note that the aforementioned internal structural characteristics cannot be obtained solely by analyzing external system disturbances (such as pressure changes) but must be effectively characterized through local strain responses.

$${\epsilon }_{\mathrm{local}}={\displaystyle \frac{F}{k_{\mathrm{local}}\cdot L}}={\displaystyle \frac{{\sigma }_{\mathrm{local}}\cdot A}{k_{\mathrm{local}}\cdot L}}$$

(7)

• ${\epsilon }_{\mathrm{local}}$: Local strain (dimensionless)
• $k_{\mathrm{local}}$: Local stiffness (N/m)
• $F$: Force applied to the local region (N)
• $L$: Characteristic length of the local region in the force direction (m)
• ${\sigma }_{\mathrm{local}}$: Local stress (Pa)
• $A$: Effective load-bearing area of the local region (m²)

To further address the challenge of limited measured data in marine shafting systems, this paper proposes a multi-data fusion model for bearing load prediction based on Stacking ensemble learning. The model simultaneously acquires air-spring pressure values and strain data from characteristic regions, effectively integrating external disturbance variations and internal structural responses within a Stacking framework. This enables correction of predictions from the MSTL-BLPM model, achieving more accurate bearing load estimation for elastically supported shafting platforms under small-sample conditions.

Specifically, this study employs the transfer learning-based multi-subdomain bearing load prediction model as Base Learner 1, and constructs a strain model using a GA-BP neural network as Base Learner 2 to introduce the mapping relationship between raft strain and bearing load. During the training process, 5-fold cross-validation is applied to both base learners to enhance the stability and generalization capability of model training. Finally, a Residual meta-learner effectively integrates the predictions from the two base learners, forming a complete ensemble prediction model, whose structure is illustrated in Figure 6.

The Residual meta-learner operates on the core principle of deep residual networks, where its key mechanism allows input signals to bypass nonlinear transformation layers via skip connections within each residual block. This architecture enables the network to focus on learning residual mappings between inputs and outputs rather than complete transformations, effectively mitigating the gradient vanishing problem in deep networks. Consequently, the meta-learner can stably train deep architectures to capture complex nonlinear relationships among base learner predictions [33]. Compared to other common meta-learner architectures, the Linear variant is limited to learning simple linear weighted combinations and cannot capture complex nonlinear interactions among base learners. While the Attention mechanism can dynamically adjust the weights of base learners, its structural complexity and large parameter count make it prone to overfitting on small datasets. In contrast, the Residual meta-learner maintains strong expressive capability with a more compact structure, demonstrating superior training stability and generalization performance in data-scarce scenarios.

3. Results and Discussion

3.1. Simulation Results of Finite Element Model

Based on an elastically supported shafting test platform of a marine vessel, this study establishes a three-dimensional solid finite element model, as shown in Figure 7. The shafting system consists of six shaft segments and two bearings. Both the entire shafting system and the main engine are rigidly mounted on a lightweight-designed raft frame, which is supported by 17 asymmetrically arranged air spring isolators at the bottom. The maximum dimension of the platform is approximately $9\mathrm{m}\times 3.5\mathrm{m}$. The bearings are fixed to the raft frame through bearing housings, while the tail end of the shafting extending beyond the raft is simplified as a free end. To accurately simulate the support characteristics of the bearings, axial and radial spring elements are introduced at the contact interfaces between the bearings and the raft frame. These springs equivalently represent the bearing support effects on the shafting system through their stiffness properties. Specifically, the thrust bearing is simplified as one spring element, while the radial bearing, due to its larger size, is represented by two spring elements for equivalent modeling. The primary objective of the finite element model in this study is to provide accurate static load distribution data of the shafting system corresponding to various stable air pressure states for subsequent transfer learning and neural network training. Although the inflation and deflation of the air springs are inherently dynamic processes, the operational conditions investigated and data collected in this paper pertain to the state of the system after pressure adjustments are completed and stability is achieved. The analysis focuses on this series of discrete static equilibrium positions. Within the framework of static analysis, the final distribution of static loads among the support points is determined solely by the stiffness characteristics of the system; damping does not influence the static equilibrium solution. Therefore, employing spring elements that only equivalently represent stiffness while omitting damping in the finite element modeling is a reasonable simplification tailored to the goal of static load prediction in this research. Additionally, the influence of other equipment on the dynamic behavior of the shafting system is simulated using distributed mass points attached to the transmission shaft and the raft frame, along with high-stiffness spring elements. The 17 air spring isolators located beneath the raft frame are equivalently modeled as spring elements with one end connected to the bottom of the raft frame and the other end fixed. Their key parameters are listed in

Table 1. Equivalent spring stiffness.

Equivalent Component	Spring Stiffness
Air Spring Isolators	X-direction: 700–800 N/mm; Y-direction: 700–800 N/mm; Z-direction: 500–720 N/mm
Thrust Bearing	X-direction: 2 × 106 N/mm; Y-direction: 2 × 106 N/mm; Z-direction: 5.6 × 106 N/mm
Radial Bearing	Y-direction: 4.8 × 104 N/mm; Z-direction: 4.5 × 105 N/mm

. It should be noted that the stiffness of the air spring isolators in Table 1 is presented as a range. This is because the stiffness of the air springs varies nonlinearly with the internal pressure. This range corresponds to their normal operational pressure interval. The specific values are determined based on the pressure-stiffness curve obtained from load-bearing tests and are applied in the finite element model to accurately reflect their mechanical behavior.

Selected operating conditions were analyzed to compare uncorrected low-fidelity simulation data with measured data, with the results summarized in

Table 2. Comparison between low-fidelity simulation data and measured data.

No.	Aft Radial Bearing Load (kg)			Forward Radial Bearing Load (kg)		Thrust Bearing Load (kg)
	Measured Data	FEA Results	Error	Measured Data	FEA Results	Error	Measured Data	FEA Results	Error
1	2495	2471	−1.0%	1842	1836	0.3%	3646	3675	−0.8%
2	2683	2541	−5.3%	1758	1639	7.2%	3685	3604	2.2%
3	2775	2600	−6.3%	1660	1542	7.7%	3703	3574	3.6%
4	2865	2663	−7.1%	1567	1430	9.6%	3727	3543	5.2%

. Under the influence of external disturbance forces induced by air pressure variations, the errors in bearing loads differ significantly across different test cases, with the maximum recorded error reaching 9.6% and the minimum error as low as 0.3%. The non-uniformity in errors can be attributed to several factors. First, the idealized simplifications and boundary condition settings commonly adopted in finite element modeling of complex structures introduce a certain degree of modeling inaccuracies. Second, although the simulated and measured air pressure variations are set to be identical, the resulting external disturbance forces acting on the shafting system may still differ, which further contributes to bearing load discrepancies. Finally, measurement errors in practical testing also play a role in the observed deviations.

3.2. Data Acquisition Method and Dataset Establishment

The strain sensor arrangement in the elastically supported shafting test platform is shown in Figure 8, with all sensors installed on the upper surface of the raft frame around the bearings. In traditional shafting alignment methods, bearing loads are typically measured using jack-up methods. To facilitate experimental research, this platform is equipped with load cells installed beneath the bearings and pressure sensors integrated into the air spring isolators, enabling direct acquisition of bearing loads and pressure values. First, cyclic inflation/deflation tests were conducted within each sub-region according to the established sub-region division, while maintaining constant pressure in other sub-regions. The initial pressure values [$Q_{h1}1$, $Q_{h2}1$, … $Q_{h17}1$] and three bearing load values [$F_{h1}1$, $F_{h2}1$, $F_{h3}1$] were recorded for each test condition. Subsequent test conditions were similarly recorded as [$Q_{h1}2$, $Q_{h2}2$, … $Q_{h17}2$] and [$F_{h1}2$, $F_{h2}2$, $F_{h3}2$]. A total of 60 test conditions across six sub-regions were compiled into dataset $D_h$. For simulation conditions, similar test patterns were adopted with enhanced inflation/deflation precision and reduced acquisition difficulty, generating 340 conditions that were compiled into dataset $D_l$. In the second phase, with the platform initially leveled, the pressures in sub-regions 1, 2, and 3 were gradually decreased and increased sequentially. After re-leveling the platform, the same pressure variation procedure was applied to sub-regions 5 and 6. Throughout these tests, pressure values [$Q_{tr1}$, $Q_{tr2}$, … $Q_{\mathrm{tr}17}$], strain measurements [${\epsilon }_{trC1}$, ${\epsilon }_{trC2}$, … ${\epsilon }_{trC9}$], and bearing loads [$F_{tr1}$, $F_{tr2}$, $F_{tr3}$] are collected, forming datasets $D_{tr}$ and $D_{te}$ with a total of 60 data groups. The detailed information of the established dataset is shown in

Table 3. Dataset information.

Name	Data Volume	Purpose
Dl	340	Simulation Dataset
Dh	60	Measured Correction Dataset
Dtr	40	Measured Training Dataset
Dte	20	Test Dataset

.

3.3. MSTL-BLP Model Prediction Results

To evaluate the predictive performance of the proposed MSTL-BLP Model for complex shafting platforms under small-sample conditions, this study compares it with two benchmark models: the Full-Parameter Input BP Model (abbreviated as FP-BP Model) and the Finite-Element-data-based Full-Parameter Input BP Model (abbreviated as FEFP-BP Model). As a baseline, the FP-BP Model employs the same training dataset and transfer learning strategy as the MSTL-BLP Model. Their core distinction lies in the processing of input features: the FP-BP Model directly uses the pressure values from all 17 air springs as input, whereas the MSTL-BLP Model adopts a subdomain partitioning strategy. In contrast, the FEFP-BP Model differs from the FP-BP Model in that it does not utilize the transfer learning strategy and is trained solely on finite element simulation data. Given that all three models involve multiple BP neural network stages, the Genetic Algorithm (GA) is employed to optimize the models with a larger number of input parameters, thereby obtaining optimal neural network parameters.

Table 4. Parameters of the bearing load prediction model in the final stage for both models.

Model Name	Activation Function	Learning Rate	First Hidden Layer Nodes	Second Hidden Layer Nodes	Training Epochs
FP-BP Model	ReLU	0.096	50	44	100
FEFP-BP Model	ReLU	0.041	47	25	100
MSTL-BLP Model	ReLU	0.053	35	20	100

lists the parameters of the final-stage bearing load prediction models for the three approaches after GA optimization. Figure 9 presents a comparison of the training loss for the three models after the GA has optimized key parameters, such as the learning rate and the number of nodes in the hidden layers, to their optimal values.

Figure 10a–c present the predictive performance of the three models for the radial bearing stern load, radial bearing bow load, and thrust bearing load, respectively.

Table 5. Comparison of prediction errors between the three models under test conditions.

Model	Error Type	Aft Radial Bearing	Forward Radial Bearing	Thrust Bearing
FEFP-BP Model	MAE (kg)	99.1	165.2	42.3
	RMSE (kg)	131.7	226.1	61.8
	MRE	3.8%	10.3%	1.2%
	ME (kg)	394	434.6	150
FP-BP Model	MAE (kg)	58.4	76.6	10.7
	RMSE (kg)	65.9	84.5	13.5
	MRE	2.1%	4.9%	0.3%
	ME (kg)	204	239	38
MSTL-BLP Model	MAE (kg)	17.60	25.8	4.5
	RMSE (kg)	22.3	33.6	5.8
	MRE	0.6%	1.6%	0.1%
	ME (kg)	69	141	15

compares the prediction errors of these models under the test conditions.

3.4. SMDF-BLP Model

Analysis of the MSTL-BLP Model results indicates that, while it shows significant improvement over the FP-BP Model, the maximum error (ME) for both the stern and bow radial bearing loads remains relatively high. The SMDF-BLP Model addresses this by employing a stacking ensemble learning framework. In this architecture, the MSTL-BLP Model serves as Base Learner 1, and a strain-based model serves as Base Learner 2. A meta-learner is then used to fuse the predictions from these two base learners. The model parameters for Base Learner 1 in the SMDF-BLP Model are identical to those of the aforementioned MSTL-BLP Model, while the parameters for Base Learner 2 are listed in

Table 6. Architectures of the Base Learner Models.

Model Name	Activation Function	Learning Rate	First Hidden Layer Nodes	Second Hidden Layer Nodes	Training Epochs
SMDF-BLP Model	ReLU	0.053	35	20	100
Strain Model	ReLU	0.048	44	32	100

. Figure 11 compares the training losses of the two base learners after their respective hyperparameters, such as learning rate and number of hidden layer nodes, were optimized to their best values by the genetic algorithm. The predictive performance of the SMDF-BLP Model is compared with that of the MSTL-BLP Model in Figure 12, with a quantitative error comparison provided in

Table 7. Comparison of prediction errors between the two models under test conditions.

Model	Error Type	Aft Radial Bearing	Forward Radial Bearing
MSTL-BLP Model	MAE (kg)	17.60	25.8
	RMSE (kg)	22.3	33.6
	MRE	0.6%	1.6%
	ME (kg)	69	141
SMDF-BLP Model	MAE (kg)	6.05	4.5
	RMSE (kg)	7.88	5.9
	MRE	0.2%	0.3%
	ME (kg)	14	22

.

Figure 13 presents the prediction results of the SMDF-BLP Model with different meta-learner architectures for the forward radial bearing loads, demonstrating the influence of meta-learner structure on prediction performance.

3.5. Discussion

For the FP-BP Model and the FEFP-BP Model, the FEFP-BP Model exhibited significant prediction failures in several test conditions within the test sample, as detailed in

Table 8. Failure Cases of the FEFP-BP Model Under Test Conditions.

Test Sample No.	Pressure Condition Characteristics
3, 6, 12, 16, 20	In the stern region, air springs V1 and V3 are inflated, while V2 and V4 are in a low-pressure state. All other air springs remain at their normal operating pressure.
11	In the bow region, air springs V16 and V17 are inflated, while V13–V15 are in a low-pressure state. All other air springs remain at their normal operating pressure.

. These failures consistently occurred under operating conditions where the target air spring was inflated while neighboring springs remained at low pressure. Such scenarios essentially constitute highly nonlinear coupled boundary conditions: localized abrupt changes in pressure distribution induce asymmetric geometric deformation of the raft, thereby significantly altering the load transfer path. This phenomenon reveals a fundamental limitation in models trained solely on idealized simulation data when it comes to capturing the nonlinear responses of actual structures.

In contrast, the FP-BP Model, trained on high-fidelity finite element simulation data corrected by the transfer learning algorithm, demonstrated an overall notable improvement in predictive performance. Furthermore, for the specific test conditions where the FEFP-BP Model failed, the predictions of the FP-BP Model were consistently closer to the true values. Specifically, the Root Mean Square Error (RMSE) at the radial and thrust bearings decreased from 131.7, 226.1, and 61.8 to 65.9, 84.5, and 13.5, respectively. Correspondingly, the Maximum Error (ME) was reduced from 394, 434.6, and 150 to 204, 239, and 38.

However, the FP-BP Model, characterized by a complex neural network architecture with multiple input parameters, still appears constrained by the inherent limitation of insufficient training under small-sample conditions due to its structural complexity. This is particularly evident in the remaining significant errors at the radial bearings. This indicates that relying solely on external pressure parameters is still inadequate for fully characterizing the underlying mechanics governing the load response.

Compared to the aforementioned models, the MSTL-BLP Model, which integrates multi-subdomain partitioning with transfer learning, demonstrates significant improvements in prediction accuracy across all three bearing measurement points. The RMSE values are substantially reduced, and the relative errors are generally below 2%. However, this model still exhibits specific prediction shortcomings at the radial bearings, manifesting as relatively high Maximum Errors (ME) under certain operating conditions. The ME values for the stern and bow radial bearing measurement points are 69 and 141, respectively. Further analysis reveals that these prediction shortcomings predominantly occur when the inflating air spring itself is at a low pressure, or when neighboring springs are collectively in a low-pressure state. This indicates that the model’s adaptability to sudden changes in boundary conditions remains insufficient. Low pressure may cause excessive downward deflection of the raft, potentially triggering internal mechanical limits within the air springs and consequently leading to an abrupt change in the structural constraint state. Problems induced by such nonlinearities and discontinuous boundaries in the actual system are challenging to fully simulate using traditional finite element methods and are also difficult for neural networks relying solely on pressure inputs to adequately learn. Figure 14 illustrates the variation in the corresponding bow radial bearing load during the inflation process of air spring V11. The operating condition marked with a circle in the figure corresponds to test sample No. 5.

To overcome the aforementioned limitations, this study further introduces the SMDF-BLP Model, which incorporates strain information through Stacking ensemble learning. This model further reduces the RMSE for bow and stern radial bearing load predictions to 7.88 and 5.9, respectively, while also decreasing the Maximum Error (ME) to 14 and 22. This clearly demonstrates that strain signals, which directly reflect local structural deformation, can effectively compensate for modeling errors arising from boundary condition uncertainties and shaft alignment variations. Consequently, more robust load prediction is achieved under the same small-sample conditions.

Furthermore, Figure 12 compares the predictive performance of the SMDF-BLP Model for bow radial bearing loads using three commonly employed meta-learner architectures. The comparison data indicates that the Residual meta-learner outperforms the other methods in small-sample Stacking ensemble learning, achieving MAE and RMSE values of 3.6 and 5.1, respectively. These figures are significantly lower than those of the Linear meta-learner (19.5 and 28.2) and the Attention meta-learner (34.4 and 46.6). This result further confirms that, for complex, coupled small-sample prediction problems, a meta-learning mechanism capable of stable training while preserving nonlinear representation capacity plays a crucial role. In contrast, simple linear fusion or attention mechanisms prone to overfitting demonstrate clear inadequacies for such tasks.

The idealized boundary conditions adopted in the modeling process of this study exhibit inherent discrepancies from the structural deformations experienced by vessels during actual navigation. This constitutes one of the primary reasons why the model still yields prediction errors under certain complex operating conditions. Specifically, within the shafting system studied here, the bearing supports are simplified as linear spring elements, and the air springs are treated as rigidly mounted to the ground. This simplification fails to adequately account for the dynamic hull deformations—induced by wave loads, hull girder bending, etc.—during navigation and the consequent shafting alignment variation they cause. Although the introduction of strain data has partially compensated for the influence of local structural deformation, the model has not explicitly incorporated the theoretical mechanisms related to shaft displacement. In practice, hull deformation can cause the shafting system to gradually deviate from its initial alignment state, thereby inducing additional bending moments at the bearings. These moments superimpose with the vertical support forces directly generated by variations in air-spring pressure, collectively forming the actual monitored composite bearing load [34]. However, the multi-subdomain modeling strategy employed in this paper is primarily established based on static pressure–load increment mapping relationships. It does not yet systematically consider the dynamic modulation effect that alignment state changes—caused by shaft displacement—exert on the load transfer path. Consequently, under extreme operating conditions involving significant hull deformation, the model’s predictive accuracy for such composite loads still has room for further improvement.

4. Conclusions

This study integrates transfer learning, multi-subdomain modeling, and Stacking ensemble learning to construct a multi-data fusion model for bearing load prediction under small-sample conditions. Transfer learning reduces the model’s dependence on measured data, multi-subdomain partitioning alleviates the model complexity arising from high-dimensional inputs, and Stacking ensemble learning enables the effective fusion of multi-source information, thereby enriching the model’s capacity to characterize the structural features of the shafting platform. Using an elastically supported shafting platform as the experimental subject, the performance of four models was analyzed, leading to the following conclusions:

(1)

The transfer learning-based multi-subdomain bearing load prediction model significantly reduces model complexity through subdomain partitioning compared to the full-parameter FP-BP model. Its high prediction accuracy demonstrates the high fidelity of the hybrid training set generated by correcting low-fidelity simulation data via transfer learning, proving its effectiveness and practicality in handling multi-parameter coupling problems in complex shafting platforms under small-sample conditions, though partial prediction failures remain.

(2)

The SMDF-BLP Model, enhanced by incorporating strain information and a Stacking ensemble learning framework, effectively compensates for the limitation of using only air pressure parameters in characterizing complex structural conditions through multi-source information fusion. By utilizing strain data from characteristic regions to correct abrupt changes in bearing loads, it achieves improved predictive performance.