## 3 Results
### Personalization of the Model Parameters
\begin{table}
\begin{tabular}{c c c c c} \hline  & \multicolumn{2}{c}{**Proposed Machine Learning**} & \multicolumn{2}{c}{**Nonlinear Least Squares**} \\  & \multicolumn{2}{c}{**Method**} & \multicolumn{2}{c}{**Method**} \\ \hline  & \(\mathbf{T_{s}=1/4}\) & \(\mathbf{T_{s}=1}\) & \(\mathbf{T_{s}=1/4}\) & \(\mathbf{T_{s}=1}\) \\ \hline \(\mathrm{RMSE}_{N}\) & \(2.2891\times 10^{4}\) & \(2.3747\times 10^{4}\) & \(1.6740\times 10^{4}\) & \(1.8361\times 10^{4}\) \\ \hline \(\mathrm{RMSE}_{L}\) & \(4.4498\times 10^{4}\) & \(4.7333\times 10^{4}\) & \(5.9670\times 10^{4}\) & \(5.9306\times 10^{4}\) \\ \hline \(\mathrm{RMSE}_{T}\) & \(5.4686\times 10^{5}\) & \(5.8619\times 10^{5}\) & \(4.2857\times 10^{5}\) & \(4.4625\times 10^{5}\) \\ \hline \end{tabular}
\end{table} 
Table 3: RMSE evaluated across \(M\) entries of the validation dataset for different \(T_{s}\). The estimated model parameters \(\hat{\mathbf{\theta}}\) determined from \(\hat{\mathbf{\xi}}\) according to (8) and their corresponding confidence interval \(\Delta\hat{\mathbf{\theta}}\) determined according to (29) are compared with the actual parameter vector \(\mathbf{\theta}\) in Figure 5.  
The coordinates of the target coexisting state were estimated from the personalized model as  
\[\hat{N}_{0}=2.9029\times 10^{5},\hat{L}_{0}=15.015\times 10^{5},\hat{T}_{0}=8.063 0\times 10^{5}. \tag{42}\]  
The reliability of the personalized models was assessed by comparing their dynamic responses to the reference state responses used during the personalization process, providing validation within the time domain. A comparison of the reference state responses \(N(t),L(t),T(t)\) considering vector \(\mathbf{\theta}\) and the responses \(\hat{N}(t),\hat{L}(t),\hat{T}(t)\) obtained using the estimated vector \(\hat{\mathbf{\theta}}\) can be seen in Figure 6.  
Figure 4: State responses and their parametric approximations by the exponential function for \(T_{s}=1/4\). Figure 5: Elements of the actual vector \(\mathbf{\theta}\) and of the estimated vector \(\hat{\mathbf{\theta}}\) together with the corresponding confidence interval \(\Delta\hat{\mathbf{\theta}}\) obtained using the MLP NN with exponential function parametrization of the state responses for \(T_{s}=1/4\). Considering the identical state responses as in the previous scenarios, numerical minimization of (32) yields  
\[\hat{\xi}=\Big{(}0.178\quad 0.681\quad 0.005\quad-1.165\quad-0.600\quad-1.822 \quad 0.188\quad-0.816\quad 0.575\Big{)}^{\text{T}}. \tag{43}\]  
A corresponding visual comparison of the estimated parameters can be seen in Figure 7.  
The coordinates of the target coexisting state were estimated from the personalized model as  
\[\hat{N}_{0}=2.9478\times 10^{5},\hat{L}_{0}=14.677\times 10^{5},\hat{T}_{0}=7.0081 \times 10^{5}. \tag{44}\]  
A comparison of the reference state responses \(N(t),L(t),T(t)\) considering vector \(\xi\) and the simulated responses \(\hat{N}(t),\hat{L}(t),\hat{T}(t)\) obtained using the estimated vector \(\hat{\xi}\) can be seen in Figure 8.  
Figure 5.0 Comparison of the reference state responses with the simulated responses considering the personalized model for \(T_{s}=1/4\). Figure 5.1 Elements of the actual vector \(\theta\) and of the estimated vector \(\hat{\theta}\) personalized using the least squares method for model response fitting.