# FFNN-TabNet: An Enhanced Stellar Age Determination Method Based on TabNet
## 5 Experiments and Results
### Feature Correlation and Analysis  
The interpretability of the TabNet model is quantified using the encoder output termed \"feature attributes\" [12], as depicted in Figure 2. The formula is delineated in Equation (13).  
\[\eta_{b}[i]=\sum\limits_{c=1}^{N_{d}}\text{PReLU}(d_{b,c}[i]) \tag{13}\]  
Equation (13) delineates the contribution of the \(i\)-th decision to the model's ultimate warning outcome [12]. The computational methodology for ascertaining the global contribution of features is delineated in Equation (14).  
\[M_{agg-b,j}=\sum\limits_{i=1}^{Nsteps}\eta_{b}[i]\cdot M_{b,j}[i] \tag{14}\]  
After normalization, the global contribution of features is delineated in Equation (15).  
\[M_{agg-b,j}=\sum\limits_{i=1}^{Nsteps}\eta_{b}\frac{[i]M_{b,j}[i]}{\sum\limits _{j=1}^{D}\sum\limits_{i=1}^{Nsteps}\eta_{b}[i]M_{b,j}[i]} \tag{15}\]  
Here, \(M_{agg}\) reflects the global contribution of the feature, while \(M[i]\) represents the feature contribution at the \(i\)-th decision step.  
Figure 10: Comparison of the actual ages of red giants and the determined results by (**a**) LSTM, (**b**) RF, (**c**) BPNN, (**d**) RNN, (**e**) CNN, and (**f**) TCN models. The red solid line denotes the optimal linear fit for the data points. This experiment utilized the test dataset.  
Figure 11 compares the feature contribution between TabNet 1 and FFNN-TabNet. The trend line in Figure 10 is smoother than that in (a), indicating an enhanced ability of the FFNN-TabNet model to capture complex nonlinear relationships between features. This enhancement allows for a more effective utilization of feature information in the determination of stellar ages. Specifically, within the FFNN-TabNet model, mass has the highest contribution among all features, indicating the critical importance of a stellar's mass in determining its age. This finding aligns with astrophysical knowledge, where mass is indeed a key factor in determining a stellar's evolutionary stage. Subsequently, chemical abundances such as N_H and Ca_H also occupy high positions in the feature contribution chart, highlighting their significance in stellar age determination. These chemical abundances reflect the star's internal nuclear synthesis history, and the abundance of different elements is related to the stellar's age. The contribution of \(\log\ g\) is also high, as surface gravity is related to the mass and radius of the stellar, serving as an indirect indicator of its age. The contribution of \(\alpha\) elements is not the highest but is still significant; \(\alpha\) elements are produced in stars through nuclear fusion processes, and their abundance is closely linked to the star's chemical evolutionary history. \(T_{\text{eff}}\) has a moderate contribution, indicating that temperature is also an important factor in determining the age of stellar.  
Observations from Figure 11 also indicate that the contributions of certain chemical abundance features, specifically C_H, K_H, Co_H, Ti_H, Al_H, and Mn_H, are exceedingly minimal, to the extent of being almost negligible. Consequently, this study delved into  
Figure 11: Comparison of feature contributions. (**a**) TabNet feature contributions; (**b**) FFNN–TabNet feature contributions.  
the critical question of whether the exclusion of these features would affect the model's accuracy in determination. The results after removing these features are shown in Figure 12.  
An analysis of Figure 12 reveals that the exclusion of certain features results in a negligible impact on the performance capabilities of the model proposed in this study, with MAE = 0.542, RMSE = 0.716, and R\({}^{2}\) = 0.939. This observation underscores the model's substantial robustness, demonstrating its ability to rely on other more significant features for effective determination.