## 4 Results and Discussion
### Ann Performance Measures  
Table 2 presents the numerical outcomes related to the training, testing, and validation phases, along with performance metrics, total epochs, and parameters governing backpropagation, including the magnetic number (\(M\)) (Scenario 1). These results illuminate their influence on the axial velocity profile, \(F(\xi)\), within the framework of TRHNF-DC. Meanwhile, Table 2 also illustrates the fluctuations of cone velocity, \(Re_{\Omega}\), and angular velocity, \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Property** & \(\mathbf{H_{2}O}\) & \(\mathbf{Cu}\) & \(\mathbf{Al_{2}O_{3}}\) \\ \hline _Thermal Conductivity_ (W/mK) & 0.6071 & 400 & 40 \\ \hline _Density_ (kg/m\({}^{3}\)) & 997 & 8933 & 3970 \\ \hline _Specific Heat_ (J/kgK) & 4180 & 385 & 765 \\ \hline _Electrical Conductivity_ (s/m) & 5.5 \(\times\) 10\({}^{-6}\) & 59.6 \(\times\) 10\({}^{6}\) & 35 \(\times\) 10\({}^{6}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The thermal properties of alumina, copper, and water [43, 44, 45, 46, 47]. \(Re_{\omega}\), on the radial profile, \(G(\zeta)\) (Scenario 2). Similarly, it demonstrates the variations of the magnetic number \((M)\) on the temperature profile, \(\theta(\zeta)\) (Scenario 3). Table 2 shows the following measures:  
* MSE (Mean Squared Error) in neural networks quantifies the average squared difference between predicted and actual values. It serves as a key metric for assessing the model's performance in regression tasks. Lower MSE values indicate better accuracy and closer alignment between predictions and true outcomes. * \"Performance\" refers to the overall effectiveness of the model in achieving its intended task, whether it is classification, regression, or another objective. Performance metrics such as accuracy, precision, recall, F1-score, and others provide quantitative measures of how well the model performs on a given dataset. Evaluating performance is crucial for assessing the model's reliability and suitability for practical applications. * The gradient represents the rate of change of the loss function with respect to the model's parameters. It guides the optimization process, indicating the direction and magnitude of adjustments required to minimize the loss and improve the model's performance during training. * Mu often represents the learning rate, determining the size of the steps taken during optimization. Adjusting mu impacts the convergence speed and stability of the training process, influencing the network's ability to learn effectively. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Case** & \multicolumn{2}{c}{**MSE**} & \multicolumn{1}{c}{**Performance**} & **Grad** & **Mu** & **Epochs** \\ \hline \multicolumn{5}{c}{**Training**} & **Validation** & **Testing** & & \\ \hline \multicolumn{5}{c}{**Scenario 1, Influence of \(M\) on \(F(\zeta)\)**} \\ \hline  
\hline \end{tabular}
\end{table} 
Table 2: Metrics from the neural network training and testing for the three considered scenarios. * Epoch refers to a single pass of the entire training dataset through the model. Multiple epochs are typically required to iteratively adjust the model's parameters to minimize the loss function and improve performance. Increasing the number of epochs allows the model to learn more complex patterns in the data. Figure 3 depicts the variation in Mean Squared Error (MSE) and the regression analysis for Scenario 1: case 1. It is observed that the optimal training efficiency is \(3.2275\times 10^{-10}\) at epoch 98, with a gradient of \(9.75\times 10^{-9}\) and a learning rate (\(\mu\)) of \(1\times 10^{-9}\) at epoch 98. Regression plots assess the relationship between output and desired output; if R is close to 1, the interaction is close; if R is close to 0, a stochastic relationship is implied. The lower the MSE value, the more dependable the solution.