# Internal Flow Prediction in Arbitrary Shaped Channel Using Stream-Wise Bidirectional LSTM
## 4 Results and Discussion
### Qualitative Evaluations  
The target field predictions of the four different models are visualized in Figures 10-14. The representative data in each figure are for cases with a nearly constant vertical cross-sectional area, converging, diverging, converging-diverging, and diverging-converging, respectively. The colorbar range was determined by calculating the minimum and maximum values of predictions and targets for each case. In this section, the focus is mainly on the differences between the models with and without the SB-LSTM module, as the differences between the models that include the SB-LSTM module are not clearly visible within the range that encompasses the results of all four models. The qualitative comparison between models with SB-LSTM is visualized in Figures A1-A5 of Appendix A. As shown in Figure 10, both the ED and U-Net models encountered a problem with a sudden discontinuity appearing in the \(u\) field at approximately three quarters of the width from the inlet in the x-direction. This indicates that the models struggled to accurately estimate the flow rate near the outlet of the channel shape. The issue with the \(v\) field was not as severe in this interval, instead appearing as a blurry error in the form of a thin line. As for the \(P\) field, as it approached the inlet from the outlet, the predicted value deviated more and there was a sudden change in value at around one quarter of the width from the inlet in the x-direction. Additionally, in Figures 11 and 12, it can be observed that the error in the predicted value increases as \(u\) approaches the outlet and \(P\) approaches the inlet. Figure 9: Training history of four different methods. From top left to bottom right: (**a**) ED, (**b**) U-Net, (**c**) ED with SB-LSTM, (**d**) U-Net with SB-LSTM. These limitations are observed in the convolutional neural network (CNN) components of ED and U-Net. The structure of the CNN is inspired by the visual cortex in animals [41; 42] and it thus has the ability to learn and generate high-level feature maps. However, a drawback of the fully convolutional network is that it only considers information within the receptive field range. This means that if the necessary information for the prediction of the target value is outside of this range, the learning process may be flawed and solutions may be generated from irrelevant features. Consequently, when \(u\) and \(P\) move away from the fixed boundary condition, the results may show tendencies that are not present in the training dataset, as seen in Figure 14, where \(u\) suddenly increases despite there being no possibility for a further speed increase. Figure 11: Qualitative results of four different methods. From top to bottom: \(u\), \(v\), \(P\). From left to right: (**a**) ED, (**b**) U-Net, (**c**) ED with SB-LSTM, (**d**) U-Net with SB-LSTM, (**e**) target. Figure 10: Qualitative results of four different methods. From top to bottom: \(u\), \(v\), \(P\). From left to right: (**a**) ED, (**b**) U-Net, (**c**) ED with SB-LSTM, (**d**) U-Net with SB-LSTM, (**e**) target. The proposed network, when combined with the SB-LSTM module, produced convincing results without encountering these types of issues in all the cases that were included in our test dataset. This confirms that not only errors related to patterns that deviate from fluid physics were successfully addressed, as shown in Figures 10-12 and 14, but also the resolution of the rows in Figure 13 was improved. This result can only be achieved by providing and receiving the appropriate features for estimation through lateral connections during the forward process using the attached SB-LSTM module. In other words, this demonstrates that the neural network that we designed possesses a structure that effectively learns the information to transfer during the backpropagation process. In addition to the above results, a qualitative evaluation related to the direction of LSTM in the SB-LSTM module is also presented. Figure 12: Qualitative results of four different methods.