## 7 Data Driven-Based Prediction of the Ultimate Axial Load  
In this paper, based on the BS 7910 FAD method and the stress failure criterion, the ultimate axial load of a X65 pipeline with cracks under 2160 different working conditions and different crack parameters, pipeline geometric parameters, and material parameters has been calculated. The investigated values for each influencing factor are listed in Table 2. In order to accurately assess the effects of the different influencing factors on the prediction results of the ultimate axial load of a X65 pipeline with cracks, this section conducts a parallel calculation under different working conditions based on different combinations of material yield ratio, misalignment, wall thickness, and crack size values. In total, 2160 sets of data were produced.  
In this paper, the LIBSVM toolkit was used to train the above database and establish the support vector regression (SVM) model for ultimate axial load prediction. According to  
Figure 14: Effect of material properties on the ultimate axial load of pipeline girth weld cracks. (**a**) \(a/B=0.1\); \(2c/\pi D=0.01\); (**b**) \(a/B=0.3\); \(2c/\pi D=0.07\).  
the operation process of the LIBSVM toolkit, the ranges of the penalty function c and kernel function g are selected based on experience, and the above parameters are optimized by the PSO algorithm. The PSO algorithm is a population intelligence-based optimization algorithm with good convergence and wide applicability, especially for more complex engineering problems. The objective function is the mean square error (MSE) of the prediction results and finite element results. When \(\text{MSE}\leq 10^{-4}\), the optimization of the parameters is completed. The optimized values are given as \(c_{best}\) and \(g_{best}\), and the SVR model is trained by these parameters, in order to accurately predict the evaluation results.  
Error analysis is an important method for evaluating the accuracy of prediction models. The predictive accuracy of the developed SVR model can be determined based on statistical criteria, such as MSE and R. To assess the accuracy of the evaluation results, MSE and R are taken as the error analysis indices, which can be obtained using the following equations:  
\[\text{MSE}=\frac{1}{m}\sum_{i=1}^{m}(y_{i}-\hat{y}_{i})^{2} \tag{12}\]  
where \(m\) is the total number of samples, \(y_{i}\) refers to the actual results, and \(\hat{y}_{i}\) denotes the prediction results.  
\[r_{xy}=\frac{n\sum x_{i}y_{i}-\sum x_{i}\sum y_{i}}{\left(n\sum x_{i}{}^{2}- \left(\sum x_{i}\right)^{2}\right)^{0.5}\left(n\sum y_{i}{}^{2}-\left(\sum y_ {i}\right)^{2}\right)^{0.5}} \tag{13}\]  
where \(x\) is the actual result and \(y\) the prediction result.  
As can be observed in the prediction error diagram of the PSO-SVR model (Figure 15), the error between most prediction results and finite element calculation results was less than 10%. As can be seen in Table 3, the correlation coefficients of the two models were higher than 99%, and the predicted results were found to be in good agreement with the numerical simulation ones, indicating that the proposed model has high prediction accuracy and reliability. Consequently, the prediction method proposed in this paper can be utilized for the safety evaluation of X65 pipelines with crack defects.  
Figure 15: Evaluation prediction error diagram. (**a**) Option 1; (**b**) Option 2.