## 3 Methodology
### Artificial Neural Network  
An artificial neural network (ANN) is a machine learning technique suitable for supervised learning. It can be used to solve regression problems because of its ability to map complex relationship in a nonlinear dataset. In this study, a feedforward neural network (FFNN) was used. The network comprised an input layer, hidden layer(s), and an output layer. The network was trained with the FEA results of the X52, X65 and X80 pipelines with a single corrosion defect subjected to internal pressure and longitudinal compressive stress. In the ANN, the true UTS of the pipes, corrosion defect parameters, and longitudinal compressive stress were used as inputs, and their corresponding failure pressure was the target output. The inputs and outputs were normalised to prevent inputs with large values from dominating other inputs. The X52, X65, and X80 FEA results were randomly divided into a training set (70%), a validation set (15%), and a test set (15%). The validation and test sets were used to prevent overfitting. The Levenberg-Marquardt (LM) backpropagation algorithm was used as the learning algorithm to train the ANN model. The LM backpropagation algorithm is more efficient than other learning algorithms due to its second-order convergence rate. It converges faster in fewer epochs [27]. The architecture of the neural network was reconfigured through trial and error in order to prevent overtraining and improve prediction accuracy. Hyperparameters such as the number of hidden neurons and the number of hidden layers were tuned so that the mean squared error, MSE <1 \(\times\) 10\({}^{-5}\). The ANN development framework is shown in Figure 5.  
The training process of the FFNN is further described in Figure 6. The weight and biases of the link between neurons were randomly initialised instead of the default zero value. The normalised inputs were moved through weighted links between the input neurons and hidden neurons. They were then calculated based on the activation function of the neurons, as well as their weights and biases. The performance of the neural network was measured using a cost function. The error between the predicted output and expected output was calculated based on the cost function. The weights and biases of the neurons were subsequently updated to minimise errors. The performance of the ANN was evaluated through the coefficient of determination (\(R^{2}\)) in Equation (1), mean squared error (MSE) in Equation (2), and mean absolute error (MAE) in Equation (3). \[R^{2}=\left(\frac{\sum_{i=1}^{N}(y_{i}-\overline{y}_{i})\left(\hat{y}_{i}- \overline{y}_{i}\right)}{\sqrt{\sum_{i=1}^{N}(y_{i}-\overline{y}_{i})^{2}\sum _{i=1}^{N}\left(\hat{y}_{i}-\overline{y}_{i}\right)^{2}}}\right)^{2} \tag{1}\]  
\[\text{MSE}=\frac{1}{N}\sum_{i=1}^{N}(\hat{y}_{i}-y_{i})^{2} \tag{2}\]  
\[\text{MAE}=\frac{1}{N}\sum_{i=1}^{N}|y_{i}-\hat{y}_{i}| \tag{3}\]  
where \(\hat{y}_{i}\) and \(y_{i}\) are the actual and predicted output values for the \(i\)th output, respectively; \(\overline{\hat{y}}_{i}\) and \(\overline{y}_{i}\) are the average of actual and predicted output, respectively; and \(N\) is the number \begin{table}
\begin{tabular}{c c c c} \hline
**Specimen** & **Failure Pressure from Burst Tests (MPa)** & **Failure Pressure Predicted in FEM (MPa)** & **Absolute Percentage Difference (\%)** \\ \hline LD & 19.8 & 20.1 & 1.50 \\ LF & 15.0 & 15.5 & 3.67 \\ Test 5 & 28.6 & 29.2 & 2.10 \\ Test 6 & 28.7 & 29.6 & 3.14 \\ \hline \end{tabular}
\end{table} 
Table 4: Validation of FEM results against burst test for pipes only subjected to internal pressure and combined loadings of internal pressure and longitudinal compressive stress. of samples. \(R^{2}\) or the squared correlation coefficient is the evaluation of goodness-of-fit for the predicted value against actual value, where an \(R^{2}\) value of 1.00 corresponds to a perfect fit. The MSE is the sum of squared difference between the predictions and actual values.