## 3 Results and Discussions
### Artificial Neural Network (ANN)
#### 3.4.4 Training Results and Visualization  
The comparative analysis of the three optimization algorithms (LM, BR, and SCG) reveals distinct characteristics in terms of convergence speed, generalization ability, and computational efficiency. The simulation performance and regression results of the LM optimization algorithm are presented in Figure 6.  
The LM algorithm, known for its fast convergence and suitability for small-to-medium-sized datasets, demonstrated rapid error reduction and reasonably good fitting capabilities. However, it showed a tendency to overfit the data slightly, as indicated by a remarkable gap between the training and test losses, suggesting sensitivity to complex data patterns (Figure 6a). \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Value** & **Description** \\ \hline Input Layer & 1 neuron & \begin{tabular}{c} Represents the dose values \\ Number of neurons in the two \\ hidden layers \\ \end{tabular} & \begin{tabular}{c} \\ [8,16] neurons \\ \end{tabular} & \begin{tabular}{c} hidden layers \\ TL glow curves \\ Optimization algorithms \\ \end{tabular} \\ Output Layer & 221 neurons & \begin{tabular}{c} TL glow curves \\ Optimization algorithms \\ \end{tabular} \\ Training Algorithm & LM, BR, and SCG & \begin{tabular}{c} Maximum number of training \\ iterations \\ \end{tabular} \\ Epochs & 1000 & \begin{tabular}{c} Step size for weight updates \\ Stopping criteria to avoid \\ overcomputation \\ \end{tabular} \\ Learning Rate & 0.001 & \
\begin{tabular}{c} Step size for weight updates \\ Stopping criteria to avoid \\ overcomputation \\ \end{tabular} \\ Minimum Gradient & \(10^{-7}\) &
\begin{tabular}{c} Step size for weight updates \\ Stopping criteria to avoid \\ overcomputation \\ \end{tabular} \\ Max Fail & 15 & Early stopping mechanism \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The ANN model training parameters for the plagioclase mineral. Figure 6: ANN simulation results for LM algorithm: (**a**) performance validation; (**b**) the linear regression analysis for the target and output. As shown in Figure 6b, the LM algorithm exhibits a strong correlation between predicted and actual values, with most data points closely aligning with the ideal regression line. The high regression values demonstrate that the model effectively captures the underlying relationships within the data. However, slight deviations observed at higher response values indicate potential overfitting, which could impact the model's generalization to new data. The simulation performance and regression results of the BR optimization algorithm are presented in Figure 7.  
The BR optimization provided the most balanced performance by effectively minimizing overfitting while maintaining good prediction accuracy (Figure 7a). The training and validation loss curves for BR were closely aligned throughout the training process, confirming its good generalization ability even with limited data. As shown in Figure 7b, the regression analysis for BR showed the highest correlation between predicted and actual values, achieving the best performance compared to other methods. The predicted data points tightly cluster around the ideal fit line across the entire range of responses. The regularization incorporated within the BR algorithm effectively prevents overfitting and results in a balanced trade-off between bias and variance. The simulation performance and regression results of the SCG optimization algorithm are presented in Figure 8.  
The SCG algorithm, designed for large-scale datasets with memory efficiency, exhibited slower but steady convergence, with a smooth and gradual decrease in loss values (Figure 8a). Figure 7: ANN simulation results for the BR algorithm: (**a**) performance validation; (**b**) the linear regression analysis for the target and output. The regression performance of the SCG algorithm, as shown in Figure 8b, presents a slightly lower regression (R) value compared to LM and BR, with more noticeable deviations from the ideal line.