## 2 Materials and Methods
### Artificial Neural Network (ANN) Model  
The present study proposes the use of a vanilla artificial neural network (ANN) to identify the optimal combination of process parameters that would prevent the delamination of composite materials during AWJ piercing. In this study, the network was trained using the backpropagation method using MATLAB's neural network toolbox (MATLAB R2024a). Backpropagation is a supervised learning technique used for training artificial neural networks (ANNs). It involves a two-phase cycle: forward propagation of input data to generate output and backward propagation of error to adjust weights and biases in the network. \begin{table}
\begin{tabular}{c c} \hline
**Parameters** & **Value** \\ \hline Cutting head & Carat 3.0 Gardella \\ Abrasive inlet angle & \(\Psi=40\) deg. \\ Orifice & Diamond, d\({}_{0}=0.35\) mm \\ Focusing tube & Ceratizit WJNS (df = 0.76 mm/lf = 101.6 mm) \\ \hline \end{tabular}
\end{table} 
Table 2: Cutting head characteristics. \begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline
**Run** & **P** & **SOD** & **ASS** & **t** & **Dext** & **Run** & **P** & **SOD** & **ASS** & **t** & **Dext** \\ \hline  & **MPa** & **mm** & **On/Off** & **mm** & **mm** & & **MPa** & **mm** & **On/Off** & **mm** & **mm** \\ \hline \(1\) & 250 & 5.5 & Off & 2 & 17.5 & 26 & 356.07 & 2.32 & Off & 2 & 25 \\ \(2\) & 250 & 5.5 & On & 2 & 12.5 & 27 & 250  
The network architecture consists of a hidden layer and an output layer. The sigmoid activation function was used in the hidden layer. This configuration captures the non-linear relationships and provides linear predictions suitable for regression tasks. The network architecture is composed of an input layer, which consists of neurons representing the process parameters; a hidden layer, which contains 10 neurons with a sigmoid activation function; and an output layer, which comprises neurons with linear activation functions to predict the extent of delamination. Initially, the data were imported from an Excel file where the training data were stored. The data were then divided into distinct subsets for training (70%), validation (15%), and testing (15%). A neural network was defined (Figure 3) and trained for a number of epochs until the change in error was very small. The performance of the trained network was then evaluated through metrics such as mean squared error and regression analysis. Finally, the results were analyzed using visualization techniques, including regression fit plots and error histograms. The main steps of modeling the process for AWJ piercing optimization are:  
1. Data input. The first step in the modeling process involves feeding the process parameters into the neural network. The dataset consists of predictors (input data) and responses (target data). A double array with 51 samples and 4 features representing the process parameters (water jet pressure P, standoff distance SOD, material thickness t, and abrasive start synchronization ASS) comprises the predictors. The response consists of a double array with 51 samples and 4 features representing the delamination extent (D\({}_{\text{ext}}\)). Each feature in the predictor array is represented as a neuron in the  
Figure 3: The neural network presentation. input layer of the network, which serves as the starting point for data processing. The data input (51 samples) was split, with 70% used for ANN model training, 15% for model validation, and 15% for model testing. 2. Hidden layer processing. After the input data enter the network, they are processed by the hidden layer, which contains 10 neurons. These neurons use a sigmoid activation function to introduce non-linearity into the model. Each hidden neuron calculates a weighted sum of the input values, adding a bias term to this sum. The sigmoid function is then applied to this weighted sum, which helps the network learn complex patterns and relationships between the input parameters and the desired output. 3. Output layer processing. The processed data from the hidden layer are then passed to the output layer.