## 3 Materials and Methods
### Deep Residual Network Connections  
The terms \"residual connections\" and \"skip connections\" are interchangeable. They are utilized to allow gradients to flow directly through a network, bypassing non-linear activation functions. The non-linear character of non-linear activation functions causes gradients to erupt or vanish (depending on the weights). Skip connections resemble a 'bus' that travels the length of the network, with gradients flowing backwards.  
The residual link, also known as a skipped connection, skips the two or three tiers of the network. Figure 6 depicts the DL network's solitary remaining connection block. As shown in Figure 5, there are three residual blocks in our proposed CNN model. The benefit of using residual connectivity in a DL model is that the function from the previous layer is added to the next layer by the preceding levels of the model network. A shortcut link, as shown in Figure 6, defines the residual network by transforming the network building block into its residual counterpart. The identity mapping shortcuts mentioned in Equation (5) can be used directly when the input and output dimensions are the same.  
\[y=F(x,\ \{W_{i}\})+x \tag{5}\]  
The building block is changed to a bottleneck building block for computational reasons. Instead of two layers, a stack of three layers is employed for each residual function F, as  
Figure 6: ResNet-50 bottleneck building block.  
shown in Figure 6. The three layers are 1 \(\times\) 1, 3 \(\times\) 3, and 1 \(\times\) 1 convolutions, with the 1 \(\times\) 1 layer lowering and then raising (restoring) dimensions, and the 3 \(\times\) 3 layer acting as a bottleneck with reduced input/output dimensions. Practical concerns have led to the use of the bottleneck building block. Furthermore, the bottleneck construction block is caused by the deterioration problem of plain networks. The architectural layers of ResNet-50 are depicted in Table 3.  
The genuine output value is H(x), and the residual learning of layers in the network input x is F(x).  
The input size (32, 32, 32, B) represents the initial image size with B being the batch size. h is the number of output channels from the patch embedding layer, and n/p is the resulting spatial dimension after patch splitting. The ConvMixer blocks consist of alternating convolutional layers applied channel-wise and space-wise. Skip connections are added after each ConvMixer block to directly add the output of the block to its input. This helps avoid the vanishing gradient problem and allows the model to go deeper effectively.  
Global average pooling is applied to obtain a global representation of the feature map.  
The final Dense layer is used for classification, and the SoftMax activation function is applied to produce the probabilities for each class. The output size is (num_classes, B), where num_classes is the number of classes in the classification task. Please note that the actual values of h, p, e, and num_classes depend on the specific configuration and requirements of the ConvMixer model and the ResNet-50 architecture being used. The table provides a general outline of how skip connections can be incorporated into the ConvMixer model to make it deeper and more powerful, similar to ResNet-50.