## 4 Conclusions  
In this study, an artificial neural network (ANN) model was developed and employed to predict the normalized stress intensity factor (\(K_{\rm I}\)) for a pressurized pipe with an infinitely long internal surface crack in the longitudinal direction. The predictions of normalized \(K_{\rm I}\) from both API 579-1/ASME FFS-1 and the ANN model were compared with those obtained from the finite element analysis (FEA), which provides an accurate solution for normalized \(K_{\rm I}\). The key findings related to the ANN model formulation are as follows:  
* Among the ANN models with 6 to 12 neurons in a sub-layer, the model with 8 neurons exhibited the lowest mean squared error (MSE) and the smallest maximum difference in validation data. Therefore, the suitable number of neurons for the present ANN model was determined to be eight in the sub-layer.
* The ANN model that selected nonlinear validation data demonstrated a lower MSE and a smaller maximum difference in testing data compared to uniformly selecting validation data among the training data. This observation suggested that choosing nonlinear validation data improved the performance of the ANN model.
* Regarding the performance of the ANN model:
* The ANN model successfully predicted normalized \(K_{\rm I}\) values, with differences from FEA results lower than 2.2%. Thus, the applicability of the ANN model for predicting \(K_{\rm I}\) in pressurized pipes with infinitely long internal surface cracks in the longitudinal direction was confirmed. On the other hand, the bilinear interpolation (BiLerp) of API 579-1/ASME FFS-1 failed to accurately predict normalized \(K_{\rm I}\) values, resulting in differences up to 4.3% within the linear zone and up to 24% within the nonlinearity zone.
* The ANN model also effectively predicted the critical crack size (\(a_{\rm C}\)), showing a difference of 0.59% higher than the \(a_{\rm C}\) obtained from FEA. Conversely, the BiLerp of API 579-1/ASME FFS-1 inaccurately predicted \(a_{\rm C}\), being 4.13% lower than the \(a_{\rm C}\) obtained from FEA. This reaffirmed the applicability of the present ANN model for predicting both normalized \(K_{\rm I}\) and \(a_{\rm C}\) in pressurized pipes with infinitely long internal surface cracks in the longitudinal direction.  
P.S.: writing--review and editing, software, methodology, conceptualization, visualization, validation, and investigation. N.N.: software, methodology, conceptualization, and investigation. C.K.: writing--original draft, methodology, conceptualization, resources, funding acquisition, supervision, and project administration. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the National Research Council of Thailand (grant number: N41A640152/2564), and the Thammasat Postdoctoral Fellowship (grant number: TUPD5/2566).  
Not applicable.  
The raw and processed data generated during this study will be made available upon reasonable request.  
The authors would like to acknowledge the support from [PERSON].  
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.