## 4 Application  
### Development of Deep Learning-Based Prediction Model  
The ANN model developed in this study was a shallow neural network with one hidden layer. Attempts have been made to develop better prediction models through deep learning-based models using deep neural networks with two or more hidden layers to improve predictive performance. In shallow neural networks, when the number of nodes in the hidden layer exceeded 10, the R\({}^{2}\) values of the validation models were consistently below 0.80. Consequently, when developing a deep-learning model with two hidden layers, the number of nodes in each hidden layer ranged from 2 to 10.  
Figure 8 illustrates the R\({}^{2}\) values for the training, validation, and all datasets of the deep learning-based prediction models with two hidden layers. Similar to the method used to denote shallow neural network models, a deep neural network model with two hidden layers is expressed as [number of nodes in the input layer = 10] \(\times\) [number of nodes in  
Figure 7: Correlation between predicted and measured yield strength: (**a**) multiple regression— Equation (1); (**b**) linear regression; (**c**) robust linear regression; (**d**) linear SVM; (**e**) Matern 5/2 GPR; and (**f**) ANN—10 \(\times\) 5 \(\times\) 1 models.  
the first hidden layer = 2-10] \(\times\) [number of nodes in the second hidden layer = 2-10] \(\times\) [number of nodes in the output layer = 1]. Figure 8 confirms that the deep-learning models with the highest R\({}^{2}\) values for the training models were 10 \(\times\) 3 \(\times\) 3 \(\times\) 1 and 10 \(\times\) 5 \(\times\) 5 \(\times\) 1, with three or five nodes in the hidden layers. Considering the validation results, the R\({}^{2}\) value of the 10 \(\times\) 5 \(\times\) 5 \(\times\) 1 model was higher than that of the 10 \(\times\) 3 \(\times\) 3 \(\times\) 1 model, indicating that the 10 \(\times\) 5 \(\times\) 5 \(\times\) 1 model was the best among the deep-learning models with two hidden layers.  
Both the shallow neural network with one hidden layer and the deep neural network with two hidden layers achieved high prediction accuracy when the number of nodes was set to five. Consequently, the number of nodes in the hidden layers was fixed at five, and the R\({}^{2}\) values were obtained by increasing the number of hidden layers from one to five, as shown in Figure 9.  
When the number of hidden layers was set to 1, the R\({}^{2}\) value of the training model was 0.94. When the number of hidden layers increased to five, the R\({}^{2}\) value remained constant at 0.94. However, with four hidden layers, the R\({}^{2}\) value reached its peak at 0.96. Conversely, during model validation, the 10 \(\times\) 5 \(\times\) 1 model with one hidden layer exhibited the highest R\({}^{2}\) value, i.e., 0.86. As the number of hidden layers increases, the R\({}^{2}\) value progressively decreases.  
Therefore, despite simultaneously considering changes in the number of nodes and adding two or more hidden layers, the prediction accuracy of the validation model tends to decrease because of overfitting during the training process [48; 49]. In this study, the yield strength prediction model for weld metals based on austenitic stainless steel confirmed that the artificial neural network model with a shallow neural network structure of 10 \(\times\) 5 \(\times\) 1 was the best.  
Figure 8: R\({}^{2}\) values for deep-learning prediction models.