## 3 Results and Discussion
### Artificial Neural Network Prediction Analysis
#### 3.2.4 ANN Model Training and Simulation  
In the present ANN analysis, 35 sets of sample data are taken from the CFD analysis for ellipsoidal hub geometry configuration. Neural Network Analysis is carried out using MATLAB R2021b to predict the performance of the fan. To estimate the performance parameters of the centrifugal fan, the MLP model available in the software is used. The data set is shared for training, testing and validation in the ratio of 70:15:15 on a random basis. Selection of Training Algorithm  
The training is performed using three available training algorithms, namely Levenberg-Marquardt (TRAINILM), Scaled Conjugate Gradient (TRAINSCG) and Bayesian Regular (TRAINBR). The number of neurons in the hidden layer is arbitrarily chosen to be between three and eight in increments of one. This wide range of the number of neurons ensures that the RMSE attains a relatively lower value. The network is trained using all three algorithms for all the cases of the hidden layer. Figure 23 depicts the variation of RMSE for all the training algorithms used for the chosen number of neurons in the hidden layer. It is seen from Figure 23 that RMSE is relatively minimal for the TRAINBR training algorithm irrespective of the number of neurons in the hidden layer. As mentioned earlier, due to the robust and accurate training performance, the error seems to be minimum for TRAINBR, and hence this algorithm is considered throughout the analysis for the prediction of results. Selection of the Number of Neurons in the Hidden Layer  
In an ANN analysis, the number of neurons in the hidden layers is identified based on the trial and error method. There is no clear expression readily available to determine the exact number of neurons to be considered in the hidden layer which would give the optimum performance. However, researchers in the past used many rules of thumb to determine the number of neurons in the hidden layer. Figure 23: Variation of RMSE for various training algorithms. [PERSON] et al. [21] reported a trial-and-error method of determining the number of neurons in the hidden layer with the help of Equation (14):  
\[N_{h}=[0.5\times(I+O)]+\sqrt{T_{R}} \tag{14}\]  
Using this relation for the present analysis, the total number of neurons in the hidden layer is found to be seven. Also, to verify the above fact, the training is performed using TRAINBR as the training algorithm. The number of neurons in the hidden layer is varied from three to eight in increments of 1. For each case, the Correlation Coefficient (R) and Root Mean Square Error (RMSE) are captured, and the variations are depicted in Figure 24.  
It is observed that the value of R is found to be 0.99999 and is the maximum for seven neurons in the hidden layer. For the same neuron number, RMSE is found to be 0.028 and is the minimum among all the other configurations of neurons in the hidden layer. Hence, in the present analysis, the hidden layer with seven neurons is considered. The 2-7-2 architecture of the neuron networks is displayed in Figure 25.  
Figure 24: Plot of R and RMSE for various values of the number of neurons in the hidden layer. Figure 25: Network architecture used in the training process. Results of the Training Process  
Figure 26 depicts the regression graph plotted against the best-fit line. It is seen from this figure that the correlation coefficient values are 1.00000 for the training set, 0.99992 for the test data and 0.99999 for the overall data set used in the study. Also, the results predicted by the ANN training model will fit closely with the results obtained by the numerical analysis. Hence, it is concluded that the data obtained in the CFD analysis are precisely predicted by the chosen ANN model. Figures 27 and 28 depict the comparison of head coefficient and relative theoretical efficiency values obtained through CFD analyses and those predicted by ANN, respectively. It is seen that the highest error of 0.013 in head coefficient and 0.43% in relative theoretical efficiency is obtained.