## 7 Results and Discussion
Regarding the Adamax optimizer (Figures 10b and 11b) with batch sizes of 256 and 512 there was greater variability in the results obtained, and an average of 10.25 was maintained in both cases. With batch sizes set to 96 and 128, a similar consistency was observed, with a mean of approximately 9.6. In Figure 10c,d and Figure 11c,d, corresponding to the GRU architecture, the results present a higher variability, which is more visible with batch sizes of 256 and 512 for both optimizers. With batch sizes of 96 and 128, the Adamax optimizer showed higher repeatability in the results than the Adam optimizer. Finally, Figure 10e,f and Figure 11e,f correspond to the implementation of the LSTM topology. This network presented different behavior between them; in the case of the application of the Adam optimizer there was a greater variation in the results with batch size of 256, with an average of 10.4, but this variation decreased with batch size of 512, obtaining an average of 9.9. In contrast, with batch sizes of 128 and 96, LSTM topology presented a similar behavior. In the case of the Adamax optimizer, a more significant variation is found with a batch size of 256 and 512, which is more visible in the results. The results show that the Adamax optimizer is more visible in the results than the Adam optimizer. Figure 11: RMSE variability obtained from modeling PM\({}_{10}\) 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 Adamax optimizer. (**e**) LSTM neural network using Adam optimizer. (**f**) LSTM neural network using Adamax optimizer. size of 512 and an approximate mean of 10.7 was obtained. For batch sizes of 256 and 96, similar results were obtained, highlighting an outlier for batch size of 256, with a standard for both of them of 9.6. Finally, the 128 batch size showed a low variability of the results obtained after the ten tests performed. Figure 11a,b corresponds to the results of the Elman network implementation, which produced a wide variety of possible results with the application of the Adam optimizer with batch sizes of 96, while larger batch sizes showed higher repeatability and lower variance. However, for all three cases, outliers are shown which seem larger than the general average. Despite the high repetitiveness of the results, the Elman network seemed to have complications modeling the behavior of the particulate matter PM\({}_{10}\) using the Adam optimizer, in the sense that the variance was higher. In the case of the Adamax optimizer implementation, the results show a comparative improvement, with high repetitiveness in the 512 batch size, with a mean of 16.6. In the case of smaller batch sizes, the mean improved as the batch size was reduced, albeit with higher variability. In Figure 11c,d the GRU network showed different behavior for each optimizer, in which the optimizer Adam and the batch sizes of 96, 256, and 512 presented a variation in their results of 0.6 in the RMSE. For the case of the batch size of 128, there was greater repeatability of the results, but with two outliers, one at each side of the boxplot. Using the Adamax optimizer for batch sizes of 96 and 128, there was a variation in the results of 0.2 with a mean of 16.3, in contrast to the variation present with higher batch sizes. Finally, Figure 11e,f indicate that, in the LSTM network, the expected results was observed. This means that the RMSE increased with larger batch sizes. Also, there is a difference between using the Adam and the Adamax optimizers, especially in the results of the batch size of 96. The Adam optimizer obtained results with a mean of 16.1, with a single outlier. In the case of the batch size of 128, the same low variance was observed, but without the outlier.