## 5 Discussion  
Groove plays a significant role in various fields, including motor rehabilitation, enhancement of motor function, and the promotion of physical activity. However, accurately measuring and predicting groove remains a challenging task. This study proposes a differential evolution (DE)-based optimization approach for fractional-order convolutional neural networks (FOCNNs) aimed at enhancing the accuracy of groove classification in music. Experimental results indicate that the Caputo fractional-order gradient method (CFOGM) outperforms traditional integer-order gradient methods. Furthermore, DE-FOCNN demonstrates significant performance improvement over standard FOCNN across various fractional-order values, achieving higher test accuracy and significantly reducing overfitting. Notably, DE-FOCNN achieved its best performance at the fractional-order value of \(v=1.4\). Additional experiments showed that DE-FOCNN not only had superior accuracy compared to other popular evolutionary algorithms (such as GA-FOCNN and PEO-FOCNN) but also exhibited lower variance. The experimental results clearly indicate that, firstly, FOCNN performs better at specific fractional-order values compared to integer-order values (\(v=1\)). This suggests that fractional-order convolutional neural networks have unique advantages in capturing the fine features present in Mel-spectrograms of music [28]. Secondly, DE-FOCNN demonstrates significant advantages in the optimization of fractional-order neural networks. Standard FOCNNs, due to limitations in the selection of initial parameters, tend to fall into local optima, resulting in slower convergence and suboptimal accuracy during training [19]. By contrast, DE, with its ability to effectively explore the global search space, significantly improves the selection of initial weights and biases in FOCNN [21; 23; 38], thereby greatly enhancing model training performance. This is particularly evident in reducing overfitting on the training set and improving accuracy on the test set, which is consistent with previous studies that utilized DE for optimizing neural networks [17; 21; 41]. Figure 6: The training and testing accuracies of different algorithms in evolutionary processes: (**a**) training loss; (**b**) testing loss. When comparing DE-FOCNN with other optimization methods (GA and PEO), DE-FOCNN achieved slightly higher average test accuracy compared to PEO-FOCNN, with noticeably lower variance. This suggests that DE-FOCNN not only has an advantage in classification accuracy but also outperforms other optimization algorithms in terms of robustness. The evolution of the loss function shows that DE-FOCNN experiences less fluctuation on the training set and converges more quickly on the test set compared to GA-FOCNN and PEO-FOCNN, further demonstrating its effectiveness in the optimization process. Previous studies have also found that DE excels in terms of stability and robustness [17]. Despite the promising results, there are several limitations of this study that need to be acknowledged and addressed. This work is an exploratory study in a relatively new research area, specifically the application of fractional-order convolutional neural networks (FOCNNs) in music classification, which is still in its early stages. Our current optimization approach primarily focuses on the selection of initial parameters (weights and biases), while future work should explore further optimizations of the network structure itself. We also limited our experiments to relatively simple CNN architectures. Future research should investigate more complex CNN designs, such as multi-layer and multi-channel convolutional networks, incorporating fractional-order transformations and appropriate optimization techniques to achieve better performance. Additionally, DE-FOCNN faces challenges related to training time, particularly when dealing with large datasets and complex audio data. More efficient optimization algorithms could be explored in the future to reduce training time.