## 4 Neural Network Model  
In this subsection, a virtual sensor emerges as a data-driven alternative to traditional physical sensors, enabling the continuous monitoring and analysis of weld bead geometry without requiring direct physical contact with the weld zone. This virtual sensor harnesses the power of machine learning algorithms, particularly neural networks, to establish a correlation between the process parameters and weld bead characteristics. The sensor's core ability is to extract meaningful insights from process monitoring data, encompassing parameters like wire feed speed, voltage, current, travel speed, energy, and pyrometer temperature, as can be seen in Figure 9a, following the same procedure as for the mold data. There are forty-six control point in the nine tests performed, as described in Table 2. These data form the input layer of the neural network model, which serves as the sensor's analytical engine. The welding process is closely influenced by several key input parameters for the neural networks model. Firstly, voltage plays a crucial role in controlling arc length and heat input. Energy per unit length, another critical factor resulting from the current, voltage, and traverse speed, directly influences the heat input to the weld, thus affecting the depth of the penetration and the bead shape. In addition, the wire feed speed dictates the deposition rate of the wire. Conversely, the transverse speed, which governs the movement of the torch perpendicular to the welding direction, can alter the shape and width of the bead, depending on the speed employed. Current levels regulate the heat generated in the arc. Finally, the temperature, a product of the welding process, has a profound influence on the material properties and geometry of the bead. Figure 8: The mold manufactured by arc-DED used on Invar material. The neural network model, trained on a comprehensive dataset of weld bead measurements which were obtained using physical sensors and profilometers, learns to map the intricate relationships between process parameters and weld bead geometry. This trained model constitutes the virtual sensor's intelligence, thus enabling it to predict weld bead geometry for new process conditions without any physical interaction. The neural network model at the heart of the virtual sensor employs a series of interconnected nodes, mimicking the structure of the human brain. These nodes process information, performing complex calculations and passing the processed information to subsequent nodes in a hierarchical manner. The input layer of the model receives the process monitoring data, represented as vectors of numerical values. These vectors are transformed into features that capture the essence of the process conditions. The transformed features are then fed into the hidden layers of the neural network, where they undergo a series of mathematical operations. The hidden layers, typically comprising multiple layers, introduce non-linearities into the model, enabling it to capture the complex relationships between the features and the  
Figure 9: Zero bead estimation: (**a**) input parameters and (**b**) actual values versus predicted values in the test observations (\(0.2\times 9\times 46\)) using the neuronal network model. -weld bead geometry. The output of the hidden layers is then passed through an output layer, which generates the predicted weld bead geometry, represented as a parabola that accurately represents the bead's shape and dimensions. The neural network model predicts the height and width of the weld beads based on various input parameters, such as average, voltage, speed, gas, energy, wire, and temperature. The code follows the following methodology, using the libraries (sklearn.model_selection and tensorflow). These libraries are used for data manipulation, data splitting, and model training, respectively. The code employs experimental test data with different welding conditions.