An Optimized GWO-BPNN Model for Predicting Corrosion Fatigue Performance of Stay Cables in Coastal Environments

Abstract

Corrosion and fatigue damage of high-strength steel wires in cable-stayed bridges in coastal environments can seriously affect the reliability of bridges. Previous studies have focused on isolated factors such as corrosion rates or stress ratios, failing to capture the complex interactions between multiple variables. In response to the critical need for accurate fatigue life prediction of high-strength steel wires under corrosive conditions, this study proposes an innovative prediction model that combines Grey Wolf Optimization (GWO) with a Backpropagation Neural Network (BPNN). The optimized GWO-BPNN model significantly enhances prediction accuracy, stability, generalization, and convergence speed. By leveraging GWO for efficient hyperparameter optimization, the model effectively reduces overfitting and strengthens robustness under varying conditions. The test results demonstrate the model’s high performance, achieving an R² value of 0.95 and an RMSE of 140.45 on the test set, underscoring its predictive reliability and practical applicability. The GWO-BPNN model excels in capturing complex, non-linear dependencies within fatigue data, outperforming conventional prediction methods. Sensitivity analysis identifies stress range, average stress, and mass loss as primary determinants of fatigue life, highlighting the dominant influence of corrosion and stress factors on structural degradation. These results confirm the model’s interpretability and practical utility in pinpointing key factors that impact fatigue life. Overall, this study establishes the GWO-BPNN model as a highly accurate and adaptable tool, offering substantial support for advancing predictive maintenance strategies and enhancing material resilience in corrosive environments.

1. Introduction

With advancements in bridge design, cable-stayed bridges have become a widely used structure worldwide. Stay cables, made of high-strength steel wires, are critical load-bearing components that ensure the bridge’s overall performance and safety. Despite protective measures, these steel wires are still vulnerable to external corrosive factors [1]. The combined effects of corrosion and cyclic stress can lead to corrosion fatigue, significantly reducing the cables’ service life [2]. Corrosion fatigue is characterized by early crack initiation and rapid propagation at lower stress levels, particularly in harsh environments such as coastal or industrial areas [3,4]. Accurate methods for predicting the corrosion fatigue performance of high-strength steel wires are essential for ensuring the reliability of stay cables.

Recent studies have focused on the interaction between corrosion and fatigue in high-strength steel wires, aiming to develop accurate predictive models for corrosion fatigue life. It is well established that corrosive environments, such as marine atmospheres, high humidity, and industrial pollution, significantly accelerate fatigue crack propagation in steel wires, making this interaction a primary failure mechanism in stay cables. Experimental evidence indicates that corrosion fatigue life can be reduced by 40% to 60% under such conditions compared to non-corrosive environments [5].

To enhance prediction accuracy, models based on fatigue crack propagation have been widely applied. These models incorporate crack initiation, growth, and failure stages and are calibrated with experimental data for improved reliability [6]. Additionally, fracture mechanics and probabilistic models have been developed to adjust traditional S-N curves by integrating environmental and corrosion factors [7].

New methods, such as crack growth rate models and stress-corrosion estimation approaches, have also emerged. For example, models derived from the Paris equation account for the nonlinear effects of corrosive media on crack propagation [8]. Numerical simulations, particularly using finite element analysis (FEA), have proven effective in simulating crack growth under cyclic loads, further refining predictive accuracy [9]. However, current models have limitations, particularly in accounting for the coupling effects of multiple factors such as stress ratios, stress levels, and material properties. This often leads to inaccuracies when applied to real-world conditions. The lack of comprehensive integration underscores the need for more advanced predictive techniques, such as neural networks, to address these complexities in future research.

Machine learning (ML) techniques have become critical in material science for solving nonlinear problems, such as predicting the fatigue performance of metallic materials. These methods handle large datasets and complex interactions effectively, making them ideal for predicting fatigue life under various conditions. Common approaches, including decision trees, support vector machines, and neural networks, have been successfully applied to model fatigue behavior, achieving notable accuracy in predicting material performance [10]. For example, Yu developed an ML model to predict stress concentration factors from etch pits in steel wires, significantly improving accuracy over traditional models [11]. Similarly, Ma applied a Bayesian network framework to predict fatigue crack growth in corroded steel wires, enhancing reliability [12]. Despite these advancements, current models often suffer from limitations such as small datasets, inadequate feature analysis, and ineffective hyperparameter optimization, leading to suboptimal predictive accuracy and generalization [13]. Wang et al. [14] developed an XGBoost model to predict the fatigue performance of high-strength steel wires. However, XGBoost still has limitations in capturing complex nonlinear relationships, especially between factors such as stress and corrosion. In contrast, neural networks are better at modeling complex nonlinear relationships and are more flexible in capturing complex interactions between variables, making them more suitable for such tasks [15].

However, neural networks often face issues such as slow convergence and local minima during training, particularly in manual hyperparameter optimization. To address these challenges, this study proposes using GWO to optimize BP neural networks, enhancing convergence speed and improving overall performance by efficiently searching for optimal hyperparameters.

In this study, we first employed Pearson’s correlation coefficient to identify the primary factors influencing the fatigue behavior of steel wires, narrowing them down to five key features: stress ratio, weight loss, stress range, average stress, and loading frequency. Using these features, we constructed five predictive models: the traditional SN fitting model, ridge regression, XGBOOST, a BP neural network (BPNN), and a BPNN optimized with the Grey Wolf Optimization (GWO) algorithm.

The results of the predictive modeling show that the GWO-optimized BPNN model outperforms traditional regression models and the unoptimized neural network, delivering improved accuracy and stability, as indicated by higher R² values and reduced error rates. The integration of the GWO algorithm effectively enhances hyperparameter tuning, convergence speed, and model robustness, resulting in a more reliable predictive performance. Among the evaluated factors, corrosion-induced weight loss, average stress, and stress range were identified as the most critical influences on fatigue life.

The primary innovations of this study include the novel application of GWO to optimize BPNN for fatigue life prediction under corrosive conditions, achieving higher accuracy and robustness. This approach provides practical insights into key fatigue factors and establishes an efficient optimization framework that minimizes overfitting. Overall, the model offers a reliable tool for assessing the lifespan of corroded steel wires, with valuable implications for structural health monitoring and maintenance planning.

2. Methods

As illustrated in Figure 1, the methodology begins with data collection from literature and experiments, followed by feature selection and dimensionality reduction to focus on key factors such as stress range, stress ratio, average stress, frequency, corrosion rate, and fatigue life. The dataset is then split into training (80%) and test (20%) sets for model development and evaluation. Multiple models, including Ridge Regression, XGBoost, BPNN, and the proposed GWO-BPNN, are compared using evaluation metrics like RMSE, MAPE, and R², with GWO-BPNN achieving the best performance in predicting fatigue life under corrosive conditions.

2.1. Database of Steel Wire

In practical applications, the fatigue life of high-strength steel wires—defined as the number of cycles to failure during fatigue testing—is influenced by various factors, including corrosion, load frequency, stress ratio, material strength (ultimate tensile strength), and fatigue stress range. Localized corrosion, including pitting, crevice corrosion, and pore corrosion, can cause stress concentrations, which significantly reduce the fatigue life of wires [16]. Although detecting and quantifying these forms of localized corrosion remains challenging, weight loss measurements provide a relatively accessible method for assessing corrosion severity. Hence, developing a comprehensive database that incorporates key variables such as weight loss, stress range, and load frequency is essential for creating more accurate fatigue life prediction models.

For this investigation, high-strength galvanized steel wires with diameters of 5 mm (1670 MPa) and 7 mm (1770 MPa) were selected to undergo accelerated corrosion and fatigue testing.

Table 1. Component elements of high tensile steel wire.

Chemical Elements	Fe/%	C/%	Mn/%	Si/%	Cr/%	V/%	S/%	P/%
Mass percentage/(%)	97.96	0.82	0.73	0.25	0.20	0.03	0.001	0.011

illustrates the detailed chemical composition of these wires. To replicate aggressive environmental conditions, we employed the CASS test in accordance with established procedures [17]. The pH of the test solution was maintained between 3.1 and 3.3, and it consisted of glacial acetic acid, sodium chloride, distilled water, and copper chloride. The specific concentrations of the solution components are outlined in

Table 2. Solution composition for accelerated corrosion test.

	${\mathbf{H}}_2\mathbf{O}$	$\mathbf{NaCl}$	$\mathbf{CuCl}\cdot {2\mathbf{H}}_2\mathbf{O}$	${\mathbf{CH}}_3\mathbf{COOH}$
Every 1000 mL	943.7 mL	50 g	0.26 g	6 mL
Percentage	94.37%	5%	0.026%	0.6%

.

During the corrosion tests, the severity of degradation was assessed by measuring the weight loss of the steel wire samples, which served as an indirect indicator of the extent of corrosion. Precise measurements were taken using an electronic balance (LP-C3003, Leaping, manufactured in Jiangsu, China, with a precision of 0.01 g), allowing for accurate calculation of the corrosion weight loss rate. This rate was then used as the primary metric to quantify the corrosion damage.

Following the corrosion tests, fatigue tests were conducted on the steel wire samples using an electronic universal testing machine (JITAI-100KN, JITAI, manufactured in Beijing, China), with a maximum load capacity of 100 kN. The fatigue test parameters, as detailed in

Table 3. Fatigue test details.

Stress Range (MPa)	Average Stress (MPa)	Stress Ratio	Frequency (Hz)
360	540	0.5	8

, included a stress ratio of 0.5, a stress range of 360 MPa, and a loading frequency of 8 Hz. A total of 20 datasets were collected, combining both corrosion and fatigue test results, to examine the relationship between corrosion-induced damage and the fatigue life of the high-strength steel wires.

In order to enhance the dataset for machine learning purposes, we supplemented the 20 experimental datasets with an additional 466 corrosion–fatigue datasets sourced from prominent academic sources [18,19,20,21,22,23,24,25]. This resulted in a total of 486 data points, encompassing fatigue life (in cycles) along with several independent variables, including wire diameter, corrosion degree (measured by weight loss), material strength (standard tensile strength), average fatigue stress, stress range, stress ratio, and loading frequency. The distribution of each feature is shown in Figure 2, while the overall dataset is illustrated in Figure 3.

The following expressions represent the stress range (S), stress ratio (R), weight loss (ω), and average fatigue stress (σ_m):

$$S={\sigma }_{\max }-{\sigma }_{\min }$$

(1)

$$R={\displaystyle \frac{{\sigma }_{\min }}{{\sigma }_{\max }}}$$

(2)

$$\omega ={\displaystyle \frac{\mathsf{\Delta }m}{m}}$$

(3)

$${\sigma }_m={\displaystyle \frac{{\sigma }_{\max }+{\sigma }_{\min }}{2}}$$

(4)

In this study, we evaluated the correlations between features by calculating the Pearson correlation coefficient matrix, as shown in Figure 4. The results indicate a high correlation between certain features, which may introduce data redundancy and negatively impact the model’s generalization ability. Specifically, the correlation coefficient between wire strength and corrosion degree is −0.831681, which approaches the threshold for a highly negative correlation. This suggests a considerable overlap in the information provided by these two features, potentially leading to multicollinearity issues. As a result, it would be prudent to retain one of these features while eliminating the other to reduce the influence of redundant features on the model. Additionally, the correlation between wire strength and diameter with respect to fatigue life is relatively low. Features with low correlation are less likely to contribute significantly to the model’s predictive performance. Therefore, it is reasonable to consider excluding features with minimal contributions to further optimize the model.

Based on this analysis, and to reduce feature redundancy while enhancing the model’s generalization performance, this study will exclude the factors of wire diameter and strength. Instead, it will focus on the four key factors: corrosion rate, average stress, stress range, stress ratio, and loading frequency. This approach is expected to effectively reduce model complexity, improve computational efficiency, and lower the risk of overfitting.

2.2. Numerical Fitting Model

In the field of metal fatigue research, the Baskin fitting equation is one of the most widely used methods, as shown in Equation (5). This equation allows researchers to analyze the effect of corrosion range on fatigue life [19,26]. Similarly, based on the existing dataset, a simplified S-N equation for steel wires can be developed. Equation (6) presents the S-N fitting equation for the fatigue life of high-tensile steel wires, taking corrosion factors into account.

$$\lg (N)=C-m\lg (S)$$

(5)

$$\lg (N)=2.49-0.76\omega \lg (S)$$

(6)

2.3. Data-Driven Model

2.3.1. Ridge Regression

Ridge regression extends linear regression by incorporating an L2 regularization term, which serves to penalize large coefficients and optimize the model. This regularization helps mitigate overfitting, enhances the model’s generalization capability, and increases robustness. The ridge regression formulation is represented by Equation (7):

$$Y={\beta }_0+{\beta }_1{X}_1+{\beta }_2{X}_2+\dots +{\beta }_n{X}_n+\epsilon -\lambda \times \left|\right|w|{|}^2$$

(7)

2.3.2. Random Forest

Random Forest (RF) is an ensemble learning method designed for both classification and regression tasks. It combines multiple decision trees to improve prediction accuracy and reduce overfitting through two main techniques: bootstrap aggregating (bagging) and random feature selection.

RF creates N bootstrapped datasets by sampling with replacement from the original dataset. Each decision tree T_i is trained independently, and the final prediction is an aggregation of individual tree outputs:

$$\widehat{y}={\displaystyle \frac{1}{N}}{\sum }_{i=1}^N{T}_i\left(x\right)$$

(8)

2.3.3. Support Vector Machine

Support Vector Machine (SVM) is a supervised learning algorithm widely used for classification and regression problems. Its core objective is to identify a hyperplane in a high-dimensional space that maximally separates data points from different classes. For linearly separable data, SVM constructs the hyperplane with the largest possible margin between classes, defined by the support vectors—data points closest to the hyperplane.

For non-linear data, SVM employs the kernel trick, which maps the input space to a higher-dimensional space where the data becomes linearly separable. Commonly used kernels include linear, polynomial, and radial basis function (RBF). The choice of kernel function and its parameters significantly impacts the performance of the model.

SVM also incorporates the concept of a soft margin to handle overlapping classes by introducing a regularization parameter C. This parameter balances the trade-off between maximizing the margin and minimizing classification errors.

2.3.4. XGBoost

XGBoost offers a notable computational efficiency advantage compared to ridge regression and Adaboost. By leveraging greedy algorithms and parallel processing, it significantly accelerates both training and prediction. XGBoost excels at handling high-dimensional sparse data and effectively manages missing values. Additionally, the inclusion of L1 and L2 regularization helps mitigate overfitting while enhancing generalization. XGBoost excels in handling large and complex datasets, which makes it well suited for various real-world applications, such as regression, classification, and ranking tasks. Its capabilities allow it to efficiently manage and model intricate data structures [27].

Let $D={\left\{\left(x_i,y_i\right)\right\}}_i^n$ represent a dataset consisting of i samples, each containing n features. The prediction model is defined as follows:

$${\widehat{y}}_i={\displaystyle {\sum }_{k=1}^K{f}_k(x_i)},f_k\in \phi$$

(9)

$$\phi =\left\{f(x)=w_s(x)\right\}(s:R^m\to T,w_s\in R^T)$$

(10)

XGBoost addresses overfitting by optimizing the loss function, which is expressed as follows:

$$Obj={\displaystyle {\sum }_{i=1}^{m}l(y_i,{\widehat{y}}_i^{(t-1)}+f_i(x_i))+\mathsf{\Omega }}(f_k)$$

(11)

$$\mathsf{\Omega }(f_k)=\gamma T+1/2\lambda {\omega }^2$$

(12)

2.3.5. BP Neural Networks

The BP neural network model, developed based on studies of neural communication and learning mechanisms in the brain [28,29], is structured with an input layer, multiple hidden layers, and an output layer, as shown in Figure 5. Its core mathematical foundation lies in the processes of forward propagation and backpropagation. The neuron’s mathematical model can be formulated as:

$$o_{\mathrm{j}}=f\left(ne{t}_j)=f(W_j^{T}X\right)$$

(13)

The fundamental equation of Forward Propagation (FP) is used to compute the output of each neuron in the neural network. This calculation can be expressed as Equation (14):

$$z_j^{(l)}={\sum }_{i=1}^{n(l-1)}{\omega }_{ij}^{(l)}{a}_i^{(l-1)}+b_j^{(l)}$$

(14)

Backpropagation is a critical component of the neural network algorithm, and it is mathematically described by Equation (15):

$${\delta }_j^{(l)}=(a_j^{(l)}-y_j)\sigma \prime (z_j^{(l)})$$

(15)

The typical structure of a neural network is illustrated below:

Figure 5. Typical structure of a BP neural network.

Figure 5. Typical structure of a BP neural network.

2.3.6. Optimized BP Neural Network

In traditional BP neural networks, although the backpropagation algorithm optimizes model weights, hyperparameters such as the number of hidden layers, the number of neurons per layer, and the learning rate often require manual tuning. Additionally, the model tends to overfit in complex tasks. To address these issues, this study introduces Dropout and the Grey Wolf Optimizer (GWO) to optimize the BP neural network, aiming to accelerate the hyperparameter search and improve the model’s generalization and prediction accuracy.

During neural network training, especially in complex models with multiple layers, the model tends to overfit the training data. Dropout is a common regularization technique that randomly drops a portion of neurons during each training iteration to reduce co-adaptation between neurons, thus improving the model’s generalization. Specifically, in each training cycle, each neuron has a certain probability of being ignored (i.e., “dropped out”), which effectively prevents the model from over-relying on the training data. Figure 6a illustrates the structure of a BP neural network with Dropout applied to its hidden layers. In this figure, the Dropout layers are inserted after each hidden layer, where certain neurons (represented by hollow circles) are randomly deactivated during each training iteration. This deactivation process forces the model to adapt without relying on specific neurons, reducing the risk of overfitting. In this study, Dropout is applied in the hidden layers of the BP neural network. By adjusting the dropout probability, we control the proportion of neurons being dropped in each layer, which reduces the complexity of the network and mitigates overfitting. As shown in Figure 6a, the network retains its original structure, but during each training cycle, different neurons are dropped out based on the predefined probability. This not only improves the model’s performance on the test set but also enhances its robustness to unseen data by preventing the network from becoming overly dependent on specific neurons.

The hyperparameters of traditional BP neural networks are typically adjusted using methods like grid search or random search, both of which can be time-consuming and often susceptible to local optima. To overcome these limitations, the Grey Wolf Optimizer (GWO) is introduced. GWO mimics the hunting behavior of grey wolves, where Alpha wolves lead, while Beta and Delta wolves assist in guiding the pack, including Omega wolves, towards the global optimum [30]. In this study, GWO is utilized to optimize the BP neural network’s hyperparameters, such as the number of hidden layers, the number of neurons in each layer, the learning rate, and the dropout probability. As depicted in Figure 7, the process begins with the population initialization, where several “wolf” individuals are generated based on predefined hyperparameter ranges. During each iteration, the fitness of each individual is evaluated by calculating the average cross-validation loss. The positions of the Alpha, Beta, and Delta wolves are updated accordingly, leading the rest of the individuals (Omega wolves) toward better solutions. Through this collaborative movement and adjustment, GWO ensures that the search gradually converges toward the global optimum. After 50 iterations, the optimal combination of hyperparameters is determined, as illustrated in Figure 7. This efficient search process, driven by GWO’s hierarchical and collaborative structure, allows the BP neural network to be optimized more effectively compared to traditional methods, providing a faster and more reliable way to achieve high-performance results.

In the improved BP neural network, the temporary retreat method and the gray wolf optimizer work together. The network structure is constructed dynamically by using the regularization effect of the temporary retreat method to prevent overfitting, while the grey wolf optimizer further improves the performance of the model by searching for optimized combinations of hyperparameters, such as the optimal number of hidden layers, learning rate, and so on. The complete algorithm flow of this study is shown in Algorithm 1.

Figure 8 clearly illustrates the search range of four core hyperparameters and their corresponding optimal values obtained through optimization: the number of hidden layers, the learning rate, the number of neurons per layer, and the dropout probability.

First, the number of hidden layers is set within the range of three to seven layers. Selecting an appropriate number of hidden layers is crucial for the model’s expressiveness. Too many layers may result in overfitting and inefficient training. Second, the search range for the learning rate spans from 1 × 10⁻⁵ to 0.1. The learning rate governs the step size of model parameter updates, which has a significant influence on the convergence speed and training efficiency. A learning rate that is too high may cause the model to fail in convergence, whereas a rate that is too low can excessively prolong the training time. For the number of neurons per layer, the search range was set between 10 and 150 neurons. The number of neurons per layer directly impacts the neural network’s capacity and expressive power. Lastly, the dropout probability was explored within a range of 0 to 1, which serves to regulate overfitting by randomly dropping neurons during training.

In conclusion, the GWO hyperparameter optimization results depicted in Figure 8 present a well-balanced parameter combination for BP neural networks. By effectively searching across a broad parameter space, GWO autonomously identifies the optimal hyperparameters that balance model complexity with generalization ability. This ensures that the neural network’s training process is both more efficient and capable of significantly improving prediction accuracy.

Algorithm 1. Neural Networks Optimized by the Gray Wolf Algorithm.

1. Data Loading and Processing --Load the data; --Split the data into training and test sets; --Standardize the feature data. 2. Define the Neural Network Model --Dynamically construct the neural network based on the hyperparameters. 3. Define the Five-Fold Cross-Validation Objective Function --Use KFold to split the training set into 5 folds; --Map the hyperparameters from the Grey Wolf Optimizer solution; --Perform five-fold cross-validation training and validation; --Calculate and return the average loss from the cross-validation. 4. Use Grey Wolf Optimizer for Hyperparameter Optimization --Define the search space; --Execute 50 iterations of hyperparameter optimization. 5. Get the Optimal Solution --Output the optimal hyperparameter combination; --Rebuild the neural network using the optimal hyperparameters. 6. Train the Model with Optimal Hyperparameters --Initialize the neural network; --Train the model using the full training set for 1000 epochs. 7. Output the Results --Output the results after model optimization.

2.4. Evaluation of Indicators

In order to quantitatively evaluate the models’ predictive performance, three metrics are employed, as shown in

Table 4. Performance indicators.

Indicator	Expression Formula
$R^2$	$R^2={\displaystyle \frac{SSR}{SST}}={\displaystyle \frac{{\displaystyle {\sum }_{i=1}^n{\left({\widehat{y}}_i-\overline{y}\right)}^2}}{{\displaystyle {\sum }_{i=1}^n{\left(y_i-\overline{y}\right)}^2}}}=1-{\displaystyle \frac{SSE}{SST}}=1-{\displaystyle \frac{{\displaystyle {\sum }_{i=1}^n{\left(y_i-{\widehat{y}}_i\right)}^2}}{{\displaystyle {\sum }_{i=1}^n{\left(y_i-\overline{y}\right)}^2}}}$
$R_{MSE}$	$R_{MSE}=\sqrt{{\displaystyle \frac{1}{n}}{{\displaystyle {\sum }_{i=1}^n(y_i-\widehat{y})}}^2}$
$M_{APE}$	$M_{APE}={\displaystyle \frac{1}{N}}{\displaystyle {\sum }_{j=1}^N\frac{\left|y_i-\widehat{y}\right|}{y_i}}\times 100\%$

Here, y denotes the actual values, $\overline{y}$ represents the average value, and $\widehat{y}$ represents the predicted values.

. R² measures the proportion of variance in the dependent variable explained by the model, with values ranging from 0 to 1. A value closer to 1 suggests stronger explanatory power. The MAPE (Mean Absolute Percentage Error) evaluates the average percentage error in predictions, where a value closer to 0 indicates lower prediction error and improved performance. The RMSE (Root Mean Square Error) assesses the average deviation between predicted and observed values. A lower RMSE indicates more accurate predictions and better model performance.

3. Results and Discussion

3.1. Training Results for Each Model

This section presents a comparison of the average performance of different models on the training and test sets.

Table 5. The average performance of each model on the training set.

	Mathematical Model	Ridge Regression	Random Forest	SVM	XGBoost	BPNN	Optimized BPNN
$R^2$	0.48	0.52	0.89	0.32	0.88	0.87	0.96
$R_{MSE}$	477.81	421.22	214.66	732.54	276.80	208.92	120.68
$M_{APE}$	361.54%	307.31%	/	/	75.66%	42.69%	22.87%
Training time (S)	/	/	/	/	/	5418	751

The training time includes the time required for optimal hyperparameter tuning.

shows the performance of each model on the training set, while

Table 6. The average performance of each model on the test set.

	Mathematical Model	Ridge Regression	Random Forest	SVM	XGBoost	BPNN	Optimized BPNN
$R^2$	0.42	0.47	0.82	0.24	0.86	0.81	0.95
$R_{MSE}$	493.38	439.70	317.29	351.41	289.64	229.81	140.45
$M_{APE}$	385.15%	326.46%	/	/	81.27%	45.92%	26.07%

presents the results on the test set. The evaluation metrics used include the coefficient of determination (R²), root mean squared error (R_MSE), and mean absolute percentage error (M_APE).

As observed in Table 5, the optimized BPNN demonstrates superior performance on the training set, achieving the highest R² value of 0.96, significantly outperforming the other models. Additionally, the optimized BPNN shows the lowest error rates with an R_MSE of 120.68 and an M_APE of 22.87%. The XGBoost model also exhibits high performance with an R² of 0.88 and moderate error rates (R_MSE of 276.80 and M_APE of 75.66%), though it still falls short compared to the optimized BPNN. In addition to its superior performance metrics, the GWO-BPNN significantly reduces training time. Specifically, the traditional grid search BPNN requires 5418 s for training, including hyperparameter tuning, while the GWO-BPNN completes training in only 751 s.

On the test set, as shown in Table 6, the optimized BPNN maintains its strong performance, achieving an R² value of 0.94, along with an R_MSE of 150.45 and an M_APE of 26.07%. XGBoost follows closely with an R² of 0.86, performing well but still inferior to the optimized BPNN. The standard BPNN shows slightly lower performance with an R_MSE of 229.81 and M_APE of 45.92%.

Figure 9 further illustrates the distribution of R² values for each model on the test set using boxplots. The optimized BPNN demonstrates both a higher median and average R² compared to other models, and it exhibits a narrower range of variance, suggesting greater robustness and generalization capability.

Table 7. Comparison with other scholars’ models.

	R2	RMSE	MAPE
Recommended optimized BPNN	0.95	140.45	26.07%
Model recommended by Guo	0.93	153.52	/
Model recommended by Wang	0.94	143.73	28.62%

provides a comparison between the optimized BPNN model developed in this study and models proposed by other scholars, such as Guo and Wang. As illustrated, the optimized BPNN achieves the highest R² value of 0.95, indicating superior predictive accuracy when compared to the models from Guo (R² = 0.93) and Wang (R² = 0.94). Additionally, the optimized BPNN also demonstrates better performance in terms of R_MSE, with a value of 140.45, which is lower than the R_MSE values reported by Guo (153.52) and Wang (143.73). In terms of M_APE, the optimized BPNN model shows a significant improvement, with an M_APE of 26.07%. This is lower than Wang’s model (28.62%), further confirming the robustness of the optimized BPNN in minimizing prediction errors. Although Guo’s model did not report an M_APE, the results still highlight the enhanced predictive performance of the proposed optimized BPNN, particularly in comparison to other models.

In summary, the optimized BPNN consistently outperforms the other models on both the training and test sets, highlighting its effectiveness for this particular task.

3.2. Predictive Performance of Optimized BP Neural Networks

Figure 10 illustrates the relationship between predicted and actual values for both the training and test datasets. The plot includes a perfect fit line (dashed gray line) representing the ideal case where all predictions perfectly match the actual values, as well as a ±30% range (green dashed lines) to assess the acceptable margin of prediction error.

As shown in Figure 10, the majority of data points from both the training and test sets fall within the ±30% range, indicating that the optimized BP neural network model is capable of providing accurate predictions for most samples. The model demonstrates a strong fit for higher actual values, with many points lying close to the perfect fit line, especially in the training set (blue points). Some minor deviations are observed in the test set (orange points), but overall, the model maintains high predictive accuracy.

Figure 11 illustrates the comparison between actual and predicted values on the test set. The results indicate that, for most samples, the predicted values closely approximate the actual values, demonstrating high accuracy and stability in the model’s predictions. In the lower value range (approximately 0 to 500), there is a strong concordance between actual and predicted values, with minimal prediction error, suggesting an excellent model fit within this interval. However, in a limited number of cases, particularly for samples with high actual values (e.g., exceeding 1000), a noticeable discrepancy emerges between predicted and actual values. This deviation may be attributed to the scarcity of samples in this range or the intrinsic variability of the data at higher magnitudes.

In summary, the model exhibits commendable predictive accuracy across the test set, effectively capturing the overall trend and variability of the data, with only minor errors observed in a few high-value samples. These results underscore the robustness and reliability of the model, affirming its suitability for predicting the performance of high-strength steel wires.

3.3. Analysis of Optimized BP Neural Network

Figure 12 presents the training and test loss curves (in terms of mean squared error (MSE)) over epochs for both the optimized BPNN and the standard BPNN. As shown in the figure, both models exhibit a rapid decrease in loss during the initial training phase, indicating effective learning from the data. The optimized BPNN achieves a lower final training and test loss compared to the standard BPNN, reflecting the improved performance and enhanced generalization ability due to optimization.

The training loss of the optimized BPNN converges more smoothly and reaches a lower value than that of the standard BPNN, suggesting better model stability. Similarly, the test loss for the optimized BPNN remains consistently lower across epochs, indicating reduced overfitting and greater robustness in handling unseen data. Overall, Figure 12 highlights the superiority of the optimized BPNN in minimizing prediction error, thus validating the effectiveness of the optimization techniques applied.

Figure 13 presents the frequency distribution and cumulative distribution function (CDF) of prediction errors in fatigue life. The histogram reveals that the majority of prediction errors are clustered around zero, indicating that the model generally achieves a high level of accuracy in predicting fatigue life. The CDF further illustrates the cumulative probability distribution of these errors, reinforcing that the model maintains low prediction errors for most samples. The error distribution shows a high frequency within the range of −200 × 10³ to 200 × 10³, with a concentration near zero, demonstrating the model’s robustness and stability in its predictions. However, there are a few outliers with larger errors, particularly around −800 × 10³, which may be attributed to unique characteristics in certain data points or inherent data variability. These outliers suggest potential areas for further investigation or model refinement to mitigate extreme deviations. Overall, this figure indicates that the model performs well in predicting fatigue life, with errors predominantly close to zero. The cumulative distribution confirms that the majority of samples exhibit minimal prediction errors, underscoring the model’s reliability and predictive accuracy.

Figure 14 provides an in-depth analysis of each feature’s contribution and importance in predicting fatigue life, as captured by SHAP values. The horizontal axis displays SHAP values, representing both the direction and magnitude of each feature’s impact on model output: positive SHAP values indicate an increase in the prediction, while negative values suggest a decrease. Each point corresponds to a SHAP value for a specific prediction, with color coding reflecting the feature’s actual value (high in red, low in blue). In this figure, w exhibits the largest range of SHAP values, making it the most influential feature in predicting fatigue life. The wide variability in SHAP values for w suggests a complex, non-linear relationship with the output, where lower values of w (blue points) generally increase the predicted fatigue life, while higher values (red points) reduce it. This pattern implies that w may capture material or structural characteristics crucial to fatigue resistance, with lower values linked to conditions that enhance durability. Avg Stress also has a notable impact, though its influence is more balanced compared to w. Lower values of Avg Stress (blue points) are associated with longer predicted fatigue life, while higher values (red points) slightly reduce it. This suggests that Avg Stress interacts with other stress-related factors in the dataset, affecting fatigue life in a complex manner. The feature S shows a moderate spread of SHAP values, indicating a consistent yet less dominant effect compared to w and Avg Stress. Lower values of S (red points) are generally linked to higher SHAP values, suggesting a positive effect on fatigue life. This stable influence implies that S contributes to extended fatigue life predictions without significant non-linear interactions. In contrast, Frequency exhibits a relatively narrow range of SHAP values centered around zero, indicating minimal influence on the model’s output. Although lower values of Frequency (blue points) slightly improve predictions, its overall impact remains minor, suggesting it may play an indirect or secondary role through interactions with more impactful features. Finally, R shows the smallest range of SHAP values, reflecting its minimal effect on predictions. The clustering of R’s SHAP values around zero suggests that changes in R do not significantly alter the predicted fatigue life, as its influence is overshadowed by stronger predictors like w and Avg Stress. In summary, Figure 13 highlights w as the most influential feature, followed by Avg Stress and S, while Frequency and R have comparatively minimal impact. The broad SHAP ranges for w and Avg Stress underscore their key roles in capturing the primary factors driving fatigue life predictions, enhancing the model’s interpretability by clarifying the structural and material factors that influence fatigue life outcomes.

Figure 15 illustrates the importance of each feature in predicting fatigue life, with w identified as the most influential factor, followed closely by Avg Stress. These two features capture critical information about fatigue life and exert substantial influence on the model’s output, indicating their strong correlation with fatigue resistance. The feature S also plays a significant role, though its impact is secondary to w and Avg Stress, suggesting it contributes meaningfully to predictions but to a lesser extent. In contrast, Frequency and R exhibit minimal influence on the model’s predictions, implying they have limited relevance in determining fatigue life within this dataset. This distribution of feature importance highlights the primary roles of w, Avg Stress, and S as key drivers in the model, while Frequency and R appear to play marginal roles. These findings enhance the model’s interpretability by clarifying which factors are most critical in influencing fatigue life predictions.

Figure 16 shows a 3D partial dependence plot depicting the combined effect of Avg Stress and S on predicted fatigue life. The vertical axis represents partial dependence, indicating how changes in Avg Stress and S jointly impact the model’s output. The plot reveals a complex, non-linear relationship, where lower values of Avg Stress and S are generally associated with higher partial dependence values, suggesting that lower stress levels contribute positively to fatigue life. As both Avg Stress and S increase, partial dependence values tend to decrease, indicating a negative influence on predicted fatigue life, likely due to the correlation of higher stress levels with accelerated material degradation. Notably, the steep decline around S = 400 likely points to dataset limitations. Sparse or uneven data distribution in this region may lead the model to misinterpret trends, resulting in abrupt changes in partial dependence. Thus, this drop may not reflect the true relationship but rather the model’s extrapolation in areas with limited data support.

4. Conclusions

Based on the findings of this study, we draw several important conclusions regarding the predictive modeling of fatigue life in high-strength steel wires, particularly under corrosive conditions. This research developed an optimized Backpropagation Neural Network (BPNN) model enhanced by Grey Wolf Optimization (GWO), leading to notable improvements in prediction accuracy and generalization capability.

(1)

The GWO-BPNN model demonstrates substantial advantages in accuracy, stability, robustness, and adaptability across varied test conditions. Compared to traditional models and the standard BPNN, GWO-BPNN achieves excellent predictive performance, with an R² of 0.95 and an RMSE as low as 140.45 on the test set. This high level of accuracy and consistency underscores the model’s potential for real-world applications in the health monitoring and predictive maintenance of high-strength steel wire structures in corrosive environments, highlighting its wide applicability in engineering practice.

(2)

The integration of GWO into BPNN has proven particularly effective for hyperparameter optimization, enabling efficient exploration and convergence within the parameter space. This optimization enhances the model’s predictive power by reducing overfitting and achieving optimal configurations that capture the intricate dependencies in fatigue data. Consequently, the GWO-based optimization not only improves the model’s accuracy but also enhances its robustness, ensuring reliable performance under diverse stress conditions and material states over time.

(3)

Furthermore, sensitivity analysis clarifies the relationships among key predictors, identifying weight loss, stress range, and average stress as primary determinants of fatigue life. The analysis highlights particularly strong correlations between weight loss, average stress, and fatigue degradation, underscoring the dominant influence of corrosion and stress factors on fatigue life. These findings align well with observed patterns of material fatigue behavior, emphasizing the model’s practical value in identifying key contributors to structural durability.

In conclusion, this study establishes the GWO-BPNN model as a highly accurate and robust tool for predicting the fatigue life of corroded high-strength steel wires. The model’s optimized configuration, supported by insights from sensitivity analysis, underscores its potential to advance predictive maintenance strategies and enhance material resilience in corrosive environments. Moreover, while the current study focuses on predicting the fatigue life of individual steel wires, it opens possibilities for extending the approach to estimate the fatigue life of cable-stayed systems, including ropes and strands. Modern cable-stayed systems utilize complex rope weaves, whose strength characteristics depend on a multitude of factors such as cable composition, load distribution, and environmental conditions. These factors were not explicitly considered in the current model, but future research could explore integrating such complexities to enhance the predictive capabilities for entire cable-stayed systems. This direction offers significant potential for improving the durability and reliability of critical infrastructure.