## 3 Experimental Confirmation of SCN's Universal Approximation Property
### Experimental Results  
After conducting 50 independent experiments on each model, we obtained the average performance of these models on the two datasets, as shown in Table 2. The performance of the SVR model is based on the optimal parameters obtained by the grid search method. Table 2 presents the average performance of various models across two datasets, obtained after 50 independent experiments, with the performance values for the SVR model derived from the optimal parameters obtained through the grid search method. The experimental results demonstrate that on the Stock dataset, the SCN model achieves the best performance in both RMSE and MAPE metrics, being 0.101 and 0.166% lower, respectively, than those of the second-best model. In terms of the MAE metric, the SCN ranks second, with a value only 0.054 higher than that of the top-performing model. On the Concrete dataset, the SCN achieves the best performance in terms of the MAE and MAPE metrics. In conclusion, the superior forecasting performance of the SCN demonstrates the rationality and innovativeness of the LSTM-SCN model proposed in this paper. To visually highlight the advantages of the SCN in forecasting, we present the prediction plots, performance indicator boxplots, and loss function plots for the SCN and other \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Dataset** & **Model** & **RMSE** & **MAE** & **MAPE** \\ \hline \multirow{8}{*}{Stock} & SVR & 0.641 & 0.486 & 1.042\% \\  & RVFL & 32.291 & 30.521 & 65.303\% \\  & FC & 1.645 & 1.645 & 2.759\% \\  & LR & 2.335 & 1.795 & 3.857\% \\  & XGBoost & 0.783 & 0.585 & 1.256\% \\  & GBRT & 0.900 & 0.685 & 1.461\% \\  & SCN(our) & 0.540 & 0.540 & 0.876\% \\ \hline \multirow{8}{*}{Concrete} & SVR & 9.610 & 6.245 & 22.595\% \\  & RVFL & 10.793 & 8.345 & 30.080\% \\ \cline{1-1}  & FC & 6.612 & 4.925 & 16.943\% \\ \cline{1-1}  & LR & 2.615 & 2.032 & 4.411\% \\ \cline{1-1}  & XGBoost & 0.893 & 0.655 & 1.412\% \\ \cline{1-1}  & GBRT & 0.963 & 0.742 & 1.599\% \\ \cline{1-1}  & SCN(our) & 1.698 & 0.194 & 0.568\% \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Evaluation indicator values for each model. models on the Stock dataset. Figure 5 compares the predicted and actual values of the SCN with those of the second-best performing model, SVR, in the test set. It is evident that the SCN forecasts the data more accurately than SVR. Figure 6 presents the box plots of the RMSE, MAE, and MAPE results for each model over 50 experiments, as shown in Figure 6a-c. Due to poor performance, the RVFL results are not displayed. The plots show that the SCN consistently achieves the lowest values across all metrics, indicating superior predictive accuracy and strong stability with an approximately symmetric distribution of its values. Figure 5: Forecasting plots for each model on the stock dataset. Figure 6: Boxplots of model performance indicators. (**a**) Box plot of RMSE metrics for each model. (**b**) Box plot of MAE metrics for each model. (**c**) Box plot of MAPE metrics for each model. Among all the models, the SCN, FC, and RVFL require iterative training to achieve optimal performance, and these three models are commonly used in hybrid electric load forecasting models. Therefore, we compared the training errors of the SCN with those of FC and RVFL. Figure 7 illustrates the error plots for the SCN, FC, and RVFL on the training and testing sets. The SCN model optimizes its network structure by progressively increasing the number of hidden layer nodes, where each addition of a hidden node corresponds to a training iteration. Therefore, for the SCN, the x-axis represents the number of hidden layer nodes. The experimental results indicate that the SCN and FC models exhibit roughly comparable convergence speeds during training, but the SCN achieves a lower loss function value at the end of training. This suggests that the SCN is more effective in finding the optimal network structure.