## 3 Methodology
### Fractional-Order Convolutional Neural Network Architecture
#### 3.1.3 Application of Caputo Derivative in CNN  
In the convolutional neural network (CNN) architecture, the Caputo fractional-order gradient method (CFOGM) is applied to both convolutional and fully connected layers, as these are the layers where parameter updates are required during training. Pooling layers and the flattening layer do not involve learnable parameters, and thus, CFOGM is not applied to them. 1. Forward Propagation  
Forward propagation is used to calculate the output of each layer based on the inputs and current weights and biases. Fully Connected Layer:  
In the fully connected layer, forward propagation is described by Equation (2), which computes the activation values for each neuron. The input from the previous layer \(a^{l-1}\) is linearly combined with the weight vector \(w_{i}^{l}\) and bias \(b_{i}^{l}\) for the \(i\)-th neuron, producing the pre-activation output \(z_{i}^{l}\). An activation function \(f\) is then applied to yield the final output \(a_{i}^{l}\). This process enables the network to extract progressively more abstract features across layers. \[ \begin{array}{l}z_{i}^{l}=w_{i}^{l}\cdot a^{l-1}+b_{i}^{l}\\ a_{i}^{l}=f\Big{(}z_{i}^{l}\Big{)}\end{array}  \tag{2}\]  
Figure 1: The structure of a convolutional neural network. The input to the model is image data, and the output is the corresponding classification of the image into one of the three groove levels. where  
* \(z_{i}^{l}\) is the input to the \(i\)-th neuron in the \(l\)-th layer. * \(a_{i}^{l}\) is the output of the \(i\)-th neuron in the \(l\)-th layer after applying the activation function. * \(w_{i}^{l}\) is the weight vector connecting the \(l-1\)-th layer to the \(i\)-th neuron in the \(l\)-th layer. * \(b_{i}^{l}\) is the bias term for the \(i\)-th neuron in the \(l\)-th layer. * \(f\) is the activation function. Convolutional Layer:  
The forward propagation in the convolutional layer similarly computes the output feature maps via convolution operations. Equation (3) represents the application of the weight filter \(w_{m}^{l}\) to the input features \(a^{l-1}\), adding the bias \(b_{m}^{l}\) to generate the pre-activation output \(z_{m}^{l}\). The final result \(a_{m}^{l}\) is obtained by passing \(z_{m}^{l}\) through the activation function \(f\). This architecture allows the convolutional layer to effectively capture local spatial features, and when combined with the updates from the fractional-order derivative method, significantly enhances the precision of the learning process [33]. \[z_{m}^{l}=conv\Big{(}w_{m}^{l},a^{l-1}\Big{)}+b_{m}^{l} \tag{3}\] \[a_{m}^{l}=f\Big{(}z_{m}^{l}\Big{)}\]  
where  
* \(\mathrm{conv}(\cdot)\) represents the convolution operation. * \(z_{m}^{l}\) is the output feature map before activation for the \(m\)-th filter in the \(l\)-th layer. * \(a_{m}^{l}\) is the activated output of the \(m\)-th filter in the \(l\)-th layer. * \(w_{m}^{l}\) is the weight matrix (filter) for the \(m\)-th filter in the \(l\)-th layer. * \(b_{m}^{l}\) is the bias term associated with the \(m\)-th filter. * Backward Propagation with Fractional-Order Gradients  
Fractional-Order Gradient Calculation  
To enable the network to more flexibly adapt to varying scales of feature changes, we employ the Caputo fractional derivative as a replacement for conventional gradient calculations. Using Equation (4), we can compute the fractional-order derivative \(\mathcal{D}_{v}^{w}L\) of the loss function \(L\) with respect to each weight and bias. This method allows for a more adaptable adjustment process based on the different states of the weights [44]. For both fully connected and convolutional layers, the gradients of the loss function \(L\) with respect to weights and biases are approximated as  
\[ \begin{array}{l}\mathcal{D}_{v}^{w}L\approx\frac{\partial L}{\partial z} \cdot\frac{\partial z}{\partial w}\cdot\frac{1}{\Gamma(2-v)}\cdot w^{1-v}\\ \mathcal{D}_{v}^{b}L\approx\frac{\partial L}{\partial z}\cdot\frac{1}{\Gamma( 2-v)}\cdot b^{1-v}\end{array}