## 3 Results and Discussion
### Discussion  
Based on 45 training data and 16 validation data, the current ANN model with eight neurons in a sub-layer successfully predicted the normalized \(K_{\text{I}}\) values and \(a_{\text{C}}\) for a pressurized pipe with an infinitely long internal surface crack in the longitudinal direction. The differences from FEA results were lower than 2.2% and higher than 0.59% for normalized \(K_{\text{I}}\) values and \(a_{\text{C}}\), respectively. It is understood that the performance of the ANN model depends on various parameters, with the impact of the number of training data and validation data being considered one of the most significant [15]. If additional training data and validation data are utilized in the construction of the ANN model, an enhancement in the model's performance can be expected.  
The influences of interpolation methods (i.e., linear and nonlinear interpolations) on the estimated \(K_{\text{I}}\) in pressurized pipes with infinitely long internal surface cracks in the longitudinal direction have been investigated by [PERSON] et al. [27]. They found that cubic spline interpolation (i.e., a mathematical technique that uses piecewise cubic polynomials to create a continuous curve representing a set of discrete data points) can offer a better prediction of normalized \(K_{\text{I}}\) from API 579-1/ASME FFS-1 when compared to linear interpolation and other nonlinear interpolations. However, the normalized \(K_{\text{I}}\) values estimated by cubic spline interpolation was still significantly different from those obtained from FEA, i.e., the maximum difference was up to 7.3% within the nonlinearity zone. Given its lower accuracy compared to the estimation using the present ANN model, it would be more appropriate to estimate the \(K_{\text{I}}\) in pressurized pipes with infinitely long internal surface cracks in the longitudinal direction using the ANN model.  
As the main practical applications of this work, the ANN model can be applied for predicting \(K_{\text{I}}\) in a pressurized pipe with an infinitely long internal surface crack in the longitudinal direction. Because the severity of a crack was quantified by \(K_{\text{I}}\), the ANN model can be applied to any pipes made of linear-elastic materials. In addition to the cases presently investigated, determining the normalized \(K_{\text{I}}\) values and \(a_{\text{C}}\) for pipes exhibiting alternative crack shapes (e.g., circumferential cracks, inclined cracks, free cracks), diverse pipe  
Figure 9: Relationships between \(K_{\text{I}}\) and \(a\) obtained from FEA, API 579-1/ASME FFS-1, and ANN.  
geometries (e.g., elbow pipes, T-shape pipes), and/or varied mechanical loading scenarios (e.g., bending, tension, torsion) presents significantly greater complexity. Generally, the normalized \(K_{\rm I}\) values and \(a_{\rm C}\) for these problems can be numerically calculated using FEA. However, FEA demands advanced numerical skills and prolonged computational time. Consequently, the application of ANN could yield benefits in estimating the normalized \(K_{\rm I}\) values and \(a_{\rm C}\) for these complex problems. As a result, further investigation is imperative for guiding future research in this field.