## 4 Experiments
### Experimental Results  
We conducted experiments to compare the proposed DE-FOCNN with the standard FOCNN across a range of different fractional-order values. As shown in Figure 4, the DE-FOCNN consistently achieved higher average test accuracy than the standard FOCNN and exhibited less overfitting on the training set. This demonstrates the effectiveness of the DE algorithm in optimizing the performance of FOCNN. Additionally, for both DE-FOCNN and FOCNN, the Caputo fractional-order gradient method (CFOGM) outperformed traditional integer-order gradient methods at specific fractional orders, such as \(v=0.8\), \(v=0.9\), and \(v=1.3\). Notably, when the fractional order \(v=1.4\), DE-FOCNN achieved its best performance. When \(v=1\), both FOCNN and DE-FOCNN function as integer-order neural networks, equivalent to traditional CNNs and DE-optimized CNNs, respectively. Our results indicate that the testing accuracy of FOCNN shows a slight downward trend compared to its performance at \(v=0.7\), \(v=0.9\), and \(v=0.9\), although the decline is not significant. In contrast, DE-FOCNN exhibits a more noticeable drop in performance at \(v=1\). Specifically, the testing accuracy at fractional orders \(0.9\) and \(1.1\) is significantly higher than at \(v=1\), further emphasizing the superiority of fractional-order models over integer-order ones. A horizontal comparison between FOCNN and DE-FOCNN at \(v=1\) reveals a testing accuracy of \(0.69\) vs. \(0.73\), respectively, demonstrating that the DE-optimized CNN outperforms the traditional CNN. To further validate the superiority of DE-FOCNN over other popular evolutionary algorithm-based FOCNN models (such as GA-FOCNN and PEO-FOCNN), multiple experiments were conducted on the Janata dataset. The upper and lower bounds of the initial biases and weights were set to \(0.5\). The parameters for DE-FOCNN, GA-FOCNN, and PEO-FOCNN are listed in Table 2.  
The results for test accuracy across DE-FOCNN, GA-FOCNN, and PEO-FOCNN are presented in Table 3. It is evident from Table 3 that DE-FOCNN exhibited higher average \begin{table}
\begin{tabular}{c c} \hline
**Algorithm** & **Parameters** \\ \hline DE-FOCNN & \(v=1.4\), crossover probability \(CR=0.7\), scaling factor \(F=0.8\) \\ GA-FOCNN & \(v=1.4\), crossover probability \(CP=0.6\), mutation probability \(m_{\text{p}}=0.05\) \\ PEO-FOCNN & \(v=1.4\), mutation parameter \(\lambda=0.7\) \\ \hline \end{tabular}
\end{table} 
Table 2: The parameters of DE-FOCNN, GA-FOCNN, and PEO-FOCNN. Figure 4: The average training and testing accuracies of FOCNN and DE-FOCNN with different fractional orders. (**a**) Fractional-order convolutional neural network; (**b**) differential evolution—fractional-order convolutional neural network. test accuracy compared to the other algorithms, and its variance was lower than that of GA-FOCNN and PEO-FOCNN, further demonstrating its stability. The confusion matrix of the 10 experiments for DE-FOCNN is shown in Figure 5. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Algorithm** & **Average** & **Standard Deviation** \\ \hline DE-FOCNN & 0.859 & 0.027 \\ GA-FOCNN & 0.843 & 0.073 \\ PEO-FOCNN & 0.855 & 0.034 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The statistical results of testing accuracy of DE-FOCNN, GA-FOCNN, and PEO-FOCNN. Figure 5: The confusion matrix of the DE-FOCNN model with a fractional order of 1.4 over 10 experimental trials. Figure 6 illustrates the evolution of the loss functions for DE-FOCNN, GA-FOCNN, and PEO-FOCNN on both the training and test sets. It is clearly visible that DE-FOCNN achieves a lower loss value compared to the other three algorithms, indicating its superior optimization performance. Additionally, to verify the generalizability of DE-FOCNN and its effectiveness on a new dataset, we applied the model to classify the groove levels of music from the new dataset. We also conducted a cycling task to measure the synchronization between movement and music to validate the effectiveness of DE-FOCNN.