## 7 Results and Discussion  
Comparison between the behavior of the real data (solid blue line) and the neural network model (solid orange line) for both PM\({}_{10}\) and PM\({}_{2.5}\) is shown in Figure 7. This comparison was made using an LSTM neural network in 2019. Furthermore, Figures 8 and 9 shows the repeatability and the range corresponding to the CC, obtained with different batch sizes (96,128,256, and 512) and optimizers (Adam and Adamax), using the three network architectures evaluated for the behavior of the PM\({}_{2.5}\) and PM\({}_{10}\) particulate material, respectively. Figure 8 shows the variability of the CC when using the Adam optimizer (on the right) and the Adamax optimizer (on the left) in PM\({}_{2.5}\) particle modeling using the three network architectures. Figure 8a,b correspond to the results of the Elman network architecture, where the best results shown were using the Adamax optimizer, especially using a batch size of 512, with an average CC of 0.757. In contrast, with the use of the Adam optimizer for a batch size of 96, atypical data were generated, demonstrating that the network had problems in cases where the behavior of the data was highly chaotic and that for this neural network, it is better to use the Adamax optimizer. In the case of the use of the GRU (Figure 8c,d) and LSTM (Figure 8e,f) networks, it was observed that the behavior was very similar; batch sizes of 96 obtained the best results, regardless of the optimizer used. Highlighting the following differences, in the use of the GRU network, errors were observed when the batch sizes were greater than 256, regardless of the optimizer implemented. The high repeatability obtained by the Adamax optimizer is notable. In the case of the LSTM network in conjunction with the Adamax optimizer, for tests with batch sizes of 128 and 256, it was observed that the results obtained are consistent while maintaining high repeatability with values of approximately 0.76, compared to those obtained with the use of the Adamax optimizer. Figure 7: Behavioral comparison of real data and data modeled by the LSTM network. 0.002. Finally, items Figure 9e,f, corresponding to the LSTM architecture, show that with batch sizes of 96, 128, and 256, they had average CC values of 0.79 for both optimizers, with greater compactness observed with the Adam optimizer, while for a batch size of 512 there was a considerable difference. In the LSTM architecture, the optimizer does not show a representative change for this application. In Figures 10 and 11, the variability obtained from the RMSE is shown. These Figures show greater variability in the results obtained compared to those obtained from the correlation coefficient. In Figures 8a and 10a, it is observed that, for the Elman network, there was an outlier point when using the Adam optimizer and a batch size of 96, indicating that when the Elman network was modeling the behavior of PM\({}_{2.5}\) many of the results were below the expected performance. Conversely, with the rest of the batch sizes evaluated,  
Figure 9: CC variability in PM\({}_{10}\) particle behavior modeling. (**a**) Elman neural network using Adam optimizer. (**b**) Elman neural network using Adam optimizer. (**c**) GRU neural network using Adam optimizer. (**d**) GRU neural network using Adamx optimizer. (**e**) LSTM neural network using Adam optimizer. (**f**) LSTM neural network using Adam optimizer. high repeatability of approximately 9.9 was maintained with a 512 batch size, showing the best possible results for this topology. Figure 10: RMSE variability obtained from modeling PM\({}_{2.5}\) particles. (**a**) Elman neural network using Adam optimizer. (**b**) Elman neural network using Adamax optimizer. (**c**) GRU neural network using Adam optimizer. (**d**) GRU neural network using Adam optimizer. (**e**) LSTM neural network using Adam optimizer. (**f**) LSTM neural network using Adamax optimizer.