## 4 Results and Discussion  
The LMS-BPNN is established to scrutinize the outcomes for various pertained parameters of desired interest, \(\beta\), \(M\), \(Rd\) and concentration of nanoparticles (\(\phi_{1}\), \(\phi_{2}\), \(\phi_{3}\)) for Casson ternary hybrid nanofluid over a porous lessening sheet. There are five scenarios each with three cases. The pertained parameters of interest are shown in Table 3. Figures 2 and 3 show the performance and error analysis with histograms for scenarios 1-5 for the proposed LMS-BPNN. However, Figure 4 shows the regression analysis for scenarios 1-5 using case 1 with LMS-BPNN. Table 4 shows convergence in terms of MSE for training, testing and validation, performance, epochs/iterations, time and \(Mu\) is calculated for the performance of artificial neural networking. Figure 2a-e depicts a convergence plot of MSE for case 1 of all five scenarios of ternary hybrid nanofluid for training, testing and validation. The best curve is attained at 1000, 710, 332, 345, and 721 epochs, while \(10^{-11}\), \(10^{-9}\), \(10^{-9}\), \(10^{-6}\), and \(10^{-8}\), respectively. The numerical values of \(Mu\) and the gradient of LMS-BPNN are shown in Table 4. These values are [1.0 \(\times\)\(10^{-7}\), 1.0 \(\times\)\(10^{-7}\), 1.0 \(\times\)\(10^{-8}\), 1.0 \(\times\)\(10^{-6}\) and 1.0 \(\times\)\(10^{-8}\)] and [1.09 \(\times\)\(10^{-7}\), 3.66 \(\times\)\(10^{-6}\), 9.61 \(\times\)\(10^{-6}\), 1.32 \(\times\)\(10^{-5}\), and 9.97 \(\times\)\(10^{-8}\)]. These results have demonstrated the validity and effective convergence of LMS-BPNN for each scenario of ternary hybrid nanofluid. The examination of regression analysis can look into the information concerning correlation. Regression plots are shown in Figure 4a-e for scenarios 1-5 of ternary hybrid nanofluid. Regression analysis research can be used to investigate the facts concerning correlation. The value of correlation (\(R\)) is unity in Figure 4a-d, this shows that the model is best fitted. In terms of data analysis of the numerical data sheet, the value of correlation (\(R\)) is close to unity supporting the usefulness of LMS-BPNN for creating a ternary hybrid nanofluid. These values examine the goodness of fit and demonstrate the accuracy of modelling. The comparison of LMS-BPNN results with numerical data is shown in Figures 5a-f and 6a-d for scenarios 1-5, and error analysis is used to further confirm the accuracy of results. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Scenario**} & \multirow{2}{*}{**Case**} & \multicolumn{5}{c}{**Input Parameters**} \\ \cline{3-6}  & & \(\beta\) & \(M\) & \(S\) & \(\phi_{1}\) = \(\phi_{2}\) = \(\phi_{3}\) & \(Rd\) \\ \hline \multirow{3}{*}{1} & 1 & 1 & 0.1 & 0.05 & 0.1 & 0.1 \\  & 2 & 2 & 0.1 & 0.05 & 0.1 & 0.1 \\  & 3 & 3 & 0.1 & 0.05 & 0.1 & 0.1 \\ \hline \multirow{3}{*}{2} & 1 & 1 & 0.1 & 0.05 & 0.1 & 0.1 \\  & 2 & 1 & 0.5 & 0.05 & 0.1 & 0.1 \\  & 3 & 1 & 0.9 & 0.05 & 0.1 & 0.1 \\ \hline \multirow{3}{*}{3} & 1 & 1 & 0.1 & 0.05 & 0.1 & 0.1 \\  & 2 & 1 & 0.1 & 0.1 & 0.1 & 0.1 \\  & 3 & 1 & 0.1 & 0.5 & 0.1 & 0.1 \\ \hline \multirow{3}{*}{4} & 1 & 1 & 0.1 & 0.05 & 0.1 & 0.1 \\  & 2 & 1 & 0.1 & 0.1 & 0.05 & 0.1 \\  & 3 & 1 & 0.1 & 0.5 & 0.01 & 0.1 \\ \hline \multirow{3}{*}{5} & 1 & 1 & 0.1 & 0.05 & 0.1 & 0.1 \\  & 2 & 1 & 0.1 & 0.1 & 0.1 & 0.1 \\  & 3 & 1 & 0.1 & 0.5 & 0.1 & 0.1 \\ \hline \end{tabular}
\end{table} 
Table 3: Portrayal for scenarios of ternary hybrid nanofluids (for details see, refs. [42, 43]). Figure 2: The mean squared error (MSE) for ternary hybrid nanofluid. (**a**) MSE of case 1 of Scenario 1. (**b**) MSE of case 2 of Scenario 1. (**c**) MSE of case 3 of Scenario 1. (**d**) MSE of case 4 of Scenario 1. (**e**) MSE of case 5 of Scenario 1. Figure 3: Error histogram analysis for the intended LMS–BPNN of case 1 of scenarios 1–5. (**a**) Error Histogram of case 1 of scenario 1. (**b**) Error Histogram of case 1 of scenario 2. (**c**) Error Histogram of case 1 of scenario 3. (**d**) Error Histogram of case 1 of scenario 4.