## 3 Results and Discussion
### Comparison between the Normalized \(K_{\text{I}}\) from API 579-1/ASME FFS-1 and ANN Model  
The cracked pipes, requiring the interpolation of normalized \(K_{\text{I}}\) from API 579-1/ASME FFS-1, were divided into two groups: the linear interpolation (Lerp) group and the bilinear interpolation (BiLerp) group. In the Lerp group, pipes with \(t\)/\(\text{R}_{\text{i}}\) values not matching those provided in API 579-1/ASME FFS-1 were selected. Since the \(a\)/\(t\) of the selection corresponded to those in API 579-1/ASME FFS-1, Lerp was utilized to estimate normalized \(K_{\text{I}}\) values between two \(t\)/\(\text{R}_{\text{i}}\) points. On the other hand, in the BiLerp group, the cracked pipes with \(a\)/\(t\) and \(t\)/\(\text{R}_{\text{i}}\) values not exactly corresponding to those in API 579-1/ASME FFS-1 were chosen. As a result, BiLerp was employed to estimate normalized \(K_{\text{I}}\) values between two \(a\)/\(t\) points and two \(t\)/\(\text{R}_{\text{i}}\) points. The normalized \(K_{\text{I}}\) values of the Lerp and BiLerp groups were also estimated using an ANN model. Subsequently, the predictions of normalized \(K_{\text{I}}\) from both API 579-1/ASME FFS-1 and the ANN model were compared with those obtained from FEA, which represents the accurate solution for \(K_{\text{I}}\).  
The normalized \(K_{\text{I}}\) values obtained from the Lerp of API 579-1/ASME FFS-1 and the ANN model were compared to those obtained from FEA, as listed in Table 3. The normalized \(K_{\text{I}}\) values obtained from the Lerp of API 579-1/ASME FFS-1 and the ANN model exhibit strong agreement with the FEA results. The differences in normalized \(K_{\text{I}}\) within the linearity and nonlinearity zones are both less than 1.6%. This confirms the suitability of both the Lerp of API 579-1/ASME FFS-1 and the ANN model for predicting \(K_{\text{I}}\) in pressurized pipes with infinitely long internal surface cracks in the longitudinal direction.  
For the BiLerp group, the normalized \(K_{\text{I}}\) values obtained from the BiLerp of API 579-1/ASME FFS-1 and the ANN model were compared to those obtained from FEA, as listed in Table 4. The BiLerp of API 579-1/ASME FFS-1 fails to accurately predict the normalized \(K_{\text{I}}\) values, leading to differences of up to 4.3% within the linearity zone and up to 24% within the nonlinearity zone. However, the ANN model continues to successfully predict the normalized \(K_{\text{I}}\) values, with differences within the linearity and nonlinearity zones both remaining below 2.2%. This reaffirms the applicability of the ANN model for predicting \(K_{\text{I}}\) in pressurized pipes with infinitely long internal surface cracks in the longitudinal direction.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**ANN**} & \multicolumn{3}{c}{**MSE**} & \multicolumn{3}{c}{**Maximum Difference (\%)**} \\ \cline{2-7}  & **Training** & **Validation** & **Testing** & **Training** & **Validation** & **Testing** \\  & **Data** & **Data** & **Data** & **Data** & **Data** & **Data** \\ \hline A & 0.00002 & 0.00510 & 0.00513 & 0.53 & 2.04 & 3.97 \\ B & 0.00022 & 0.01040 & 0.00309 & 0.40 & 2.44 & 2.16 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Influence of validation data on the performance of ANN model.