## 3 Results and Discussion
### Deep Neural Network (DNN) Results  
In addition to ML models, the DNN algorithm was also utilized to evaluate the performance of the regression problem and compared with ML models. After generating the DNN structure and learning rate, the preprocessing, optimization techniques and Huber loss function were applied to investigate the performance of the regression problem. The results of all various steps are summarized in Figure 7. Figure 7 compares the performance  
Figure 6: The comparison of the performance results of ML based on application of FS with correlation coefficient and FS with CfsSubsetEval method. of different DNN optimization algorithms, namely SGD, Adam, Adamax, Nadam, and Rmsprop, across different data normalization techniques and dataset splits. As shown in Figure 7a, the combination of Z-score normalization and SGD optimizer algorithms resulted in the highest R\({}^{2}\) value of 0.9381 when the test rate was 20%. There was a slight decrease in R\({}^{2}\) performance values when the test rate was increased from 20% to 30%. As the test rate was 30%, the combination of robust normalization and Adam optimizer algorithms resulted in the highest R\({}^{2}\) value of 0.9271. In addition, when the test rate was 20%, the combination of Z-score normalization and SGD optimizer algorithms was found to have the lowest RMSE value of 18.3958, whereas the combination of robust normalization and Adam optimizer algorithms had the lowest RMSE value of 21.5702 for a 30% test rate in Figure 7b. The chosen combination of preprocessing methods, learning rate, architecture, and activation functions led to high prediction accuracy, as indicated by an R\({}^{2}\) value of approximately 0.93. This performance level is substantial given the relatively small dataset size and the complexity of the problem. The learning rate allowed the optimizer to make precise adjustments during training, while the network architecture, featuring a decreasing number of neurons, prevented overfitting by focusing on the most relevant patterns. Additionally, using the Huber loss function provided robustness against potential outliers, contributing to the model's overall effectiveness. In addition, when the test rate increased from 20% to 30%, the combination of Z-score normalization, Adam optimizer, ReLU activation and linear output function confirmed the effectiveness of the selected model design, showing that it can compete with the results in the condition with a 20% test rate. This suggests that the model maintained robust performance but was marginally slightly less accurate when exposed to a larger test dataset, demonstrating the trade-off between test size and predictive accuracy. By strategically selecting these hyperparameters and design elements, the DNN model can effectively predict the boride thickness, achieve high accuracy, and maintain robustness in different test scenarios. Figure 8 illustrated a comprehensive analysis of the model's performance over 1000 training epochs, showing both MAE and R\({}^{2}\) for the training and validation datasets. As seen in Figure 8, a rapid decrease in both training and validation for the MAE metric within the first 100 epochs was observed, indicating that the model was rapidly learning to minimize prediction errors. As training progressed, MAE levels decreased. This indicates that the model converges and additional epochs result in minimal improvements in prediction accuracy. The early stopping in MAE reflected the efficient learning supported by the chosen architecture and the relatively small dataset. During the early stages of training,  
Figure 7: The performance of various DNN optimization algorithms: (**a**) R\({}^{2}\) and (**b**) RMSE. the R\({}^{2}\) values for both the training and validation datasets showed a significant increase. This indicates that the model progressively captured more of the variance in the dataset, improving its ability to explain the relationship between the features and the target variable.