## 8 Simulation Results
The maximum average of performance improvement, in this case, is 5.2%. Additionally, the robust Fair loss function has semi-equal performance to the robust Cauchy loss function, and the maximum average of performance improvement is 4.4%. On the other hand, the traditional MSE loss function performs poorly compared to the others. As the table shows in the case of set C with contaminated data, the robust neural networks are insensitive to outliers. The robust loss function Fair has the best performance compared to the others. The average performance improvement, in this case, is 8.1%. In addition, the robust Cauchy loss function performs approximately the same as the robust Fair loss function, and the average performance improvement is 6.9% in this case. However, the traditional MSE loss function offers poor performance compared to its peers. Figure 4 shows, using Set A with high-quality noise-free data, that all models trained with robust and traditional loss functions have approximately equivalent responses to the ideal model. We noted that the estimated model gradually decreases with an increase in the input pattern. Figure 5 illustrates the validation loss of each model over the number of epochs during DNN training in three cases according to the loss function used--MSE, Cauchy, and Fair, respectively. It was found that the error gradually decreases after more training epochs in all cases. The best validation performance is 0.015838 at the 63 rd epoch, 0.0012387 at the 45 th epoch, and 0.018973 at the 43 rd epoch of the training network using the traditional loss functions MSE, Cauchy, and Fair, respectively. Figure 4: The predicted models in the robust loss functions Cauchy and Fair and the traditional loss function MSE in Set A. In Figure 6, it can be seen that in the case of using set B on the data corrupted with Gaussian noise, the predicted models using Cauchy and Fair robust loss functions have responses that are semi-equal and close to the ideal model, except that the model predicted using the conventional MSE loss function deviates from the ideal model. The performance of the training, in this case, is shown in Figure 7. We noticed that the loss decreases as the number of epochs increases. The best validation performance using the traditional MSE loss function is 0.06399 at the 46 th epoch. The best validation performance in the case of robust Fair loss function is 0.07479 at the 44 th epoch. Moreover, the best validation performance using the robust Cauchy loss function is 0.0040613 at the 54 th epoch. Figure 5: The best validation performance of DNN training in the case of Set A.  
Figure 6: The predicted models in robust loss functions Cauchy and Fair and traditional loss function MSE in set B. As illustrated in Figure 8, when using set C with data corrupted with outliers, the robust neural networks are immune to outliers and produce approximately ideal results. Otherwise, the model generated by traditional neural networks (MSE) has been strongly affected by the outliers. The performance of the training, in this case, is shown in Figure 9. We noted that the loss decreases with the increasing number of epochs. The best validation performance when using the traditional MSE loss function is 0.069745 at the 65 th epoch; the best validation performance when using the robust, fair loss function is 0.047096 at the 50 th epoch. Moreover, the best validation performance when using the robust Cauchy loss function is 0.0027879 at the 71 st epoch. Figure 3:0 Predicted models in the robust loss functions Cauchy and Fair and the traditional loss function MSE in the set C case. Figure 3:1 The best validation performance of DNN training in the case of set B. In Figure 10, the relationship between the performance represented by RMSE and the percentage of outliers is depicted. It can be observed that RMSE increases with an increasing percentage of outliers, which is significant when using the conventional MSE loss function.