## 4 Comparisons
### Results of the Comparison  
It should be noted that the neural network trained by steepest descent (SD) (19) had \(l=6\) neurons in its hidden layer, a tuning factor of \(\alpha=0.0004\), and a number of epochs of \(e=40\). It should be noted that the neural network trained by steepest descent with mini-batches (SDMB) in [9; 10; 11; 12] and using (19), (22), (23) had \(l=6\) neurons in its hidden layer, mini-batches with a size of \(y=32\), a tuning factor of \(\alpha=0.0004\), and a number of epochs of \(e=40\). It should be noted that the neural network trained by the Hessian (H) in [17; 18; 19; 20] and using (15), (18), (14), (9), (10) had \(l=6\) neurons in its hidden layer, a tuning factor of \(\alpha=0.0004\), and a number of epochs of \(e=40\). It should be noted that the neural network trained by the Hessian with mini-batches (HMB) in this investigation and using (15), (18), (14), (9), (10), (22), (23) had \(l=6\) neurons in its hidden layer, mini-batches with a size of \(y=32\), a tuning factor of \(\alpha=0.0004\), and a number of epochs of \(e=40\). Figure 2 shows the cost maps during the training of the neural network with steepest descent (SD), steepest descent with mini-batches (SDMB), Hessian (H), and Hessian with mini-batches (HMB). As we can see, the Hessian with mini-batches provides better tuning when it comes to training the neural network and tends to converge more directly (with the help of the information provided from the second derivative) than with the use of the steepest descent. The issue with the normal downward steepest descent is that often a minimum cannot be quickly found. The use of mini-batches helps to quickly reach a minimum. Figure 3 shows a zoom of the cost maps after 40 epochs during neural network training using SD, SDMB, H, and HMB. The Hessian provides better tuning in comparison with steepest descent, and the Hessian with mini-batches provides better tuning in comparison with steepest descent with mini-batches. The tuning of the neural network with SD, SDMB, H, and HMB during training is shown in Figure 4. During 40 epochs, the neural network trained with steepest descent failed to tune, and its  
Figure 3: A zoom of the cost maps after 40 epochs during training. Figure 2: The cost maps during training, steepest descent (SD), steepest descent with mini-batches (SDMB), Hessian (H), Hessian with mini-batches (HMB). tuning was very slow when compared to the neural network trained with Hessian and mini-batches, which provided better tuning. The Hessian provided better tuning than steepest descent. The neural network tuning using SD, SDMB, H, and HMB during testing is shown in Figure 5. Similar to the training results, the neural network trained using steepest descent did not have the ability to predict. The neural network prediction using the Hessian with mini-batches was better than that using the other methods, as can be seen in Figure 6, which is a zoom of the neural network prediction with SD, SDMB, H, and HMB during testing. Figure 4: The neural network with the algorithms during training. Figure 5: The neural network with the algorithms during testing. Table 1 compares the results of SD, SDMB, H, and HMB during training and testing with 40 epochs in terms of the determination coefficient (\(R^{2}\)) and cost (\(E\)). It should be noted that the neural network had 6 neurons in its hidden layer and each mini-batch had a size of \(y=32\). \(R^{2}\) has values between 0 and 1, where values close to 1 correspond to algorithms with better tuning. Since HMB obtained the biggest value of \(R^{2}\) during training and testing and obtained the smallest value of \(E\) during training, HMB provides the best tuning in comparison with H, SDMB, and SD. Table 2 compares the results of SD, SDMB, H, and HMB during training and testing for 40 epochs in terms of the mean absolute error (\(MAE\)) and the mean absolute percent error (\(MAPE\)).