## 2 Principle
### Principle of Two-Dimensional Convolutional Neural Networks  
A 2D-CNN is a deep learning model, mainly made up of convolutional layers, pooling layers, and fully connected layers [27]. The convolutional layer utilizes the feature maps from the preceding layer as input for convolution processes, subsequently generating the output feature map via an activation function [28; 29]. The output feature map can be represented as:  
\[x_{n}^{l}=f(\sum_{k=1}^{K}x_{k}^{l-1}*W_{nk}^{l}+b_{n}^{l}) \tag{25}\]  
Here, \(W_{nk}^{l}\) denotes the kernel connecting the \(n\)-th and \(k\)-th features of the \(l\)-th layer, while \(x_{n}^{l}\) and \(x_{k}^{l-1}\) signify the \(n\)-th and \(k\)-th feature maps of the \(l\)-th and \((l-1)\)-th layers, respectively. Additionally, \(b_{n}^{l}\) represents the associated bias. In executing intricate convolution processes, we typically split them into real-valued operations. The 2D-CNN works on the real and imaginary parts of the complex signal separately before combining them, which allows it to perform the convolution operation on the complex signal. Activation functions are crucial in neural networks because they introduce nonlinear properties to the network. After a linear transformation in the convolutional layer, the results are mixed with a nonlinear activation function before moving on to the next layer. In 2D-CNN, there are two common types of activation functions: traditional S-type functions like sigmoid and tanh, and non-saturated functions like ReLU, Leaky ReLU, ELU, PReLU, and RReLU [30]. However, traditional S-type functions can cause the problem of vanishing gradients as the model gets deeper. However, with traditional S-type functions, the problem of vanishing gradients starts to occur as the model gets deeper. Non-saturated activation functions have been proposed to address this issue [31; 32]. The ReLU function is predominantly utilized in convolutional layers. The mathematical representation of ReLU is as follows:  
\[\text{ReLU}(x)=\{ \begin{subarray}{c}x,x\geq 0\\ x,x<0\end{subarray} =\text{max}(0,x) \tag{26}\]  
When \(x\) is non-negative, ReLU outputs the input, whereas if \(x\) is negative, it outputs 0. This simple process gives the ReLU function nonlinear features and remarkable computing efficiency. Sigmoid functional representation:  
\[Sigmoid(x)=\frac{1}{1+e^{-x}} \tag{27}\]  
The range is [0, 1], making it ideal for model output functions that generate probability values inside (0, 1). This method works for models that produce projected probabilities, such as binary classification categories or confidence levels. However, above a certain range, this function's derivative approaches 0. This will cause neurons to become saturated during training, slowing weight updates during backpropagation and complicating model training. We call this the vanishing gradient issue. The tanh function formula is:  
\[\text{tanh}(x)=\frac{e^{x}-e^{-x}}{e^{x}+e^{-x}} \tag{28}\]The hidden layer uses the tanh function, whereas the output layer uses the sigmoid function in normal binary classification tasks. In 2019, [PERSON] announced the Mish activation function [33]. This research used it:  
\[Mish(x)=x\text{tanh}(\ln(1+e^{x})) \tag{29}\]  
Mish accepts negative inputs and prevents ReLU from rejecting negative gradients using ReLU's straight component and tanh's curves. Mish is smoother and has fewer vanishing gradients than ReLU. Dropout [34], a regularization technique for neural network training, mitigated this issue in the 2D-CNN. Dropout randomly deactivates neurons with a probability \(p\), resulting in an output of 0. Neuron output is maintained throughout training and testing by retaining and scaling all neurons. A loss function quantifies the discrepancy between expected and real labels. The loss function modifies the parameters of network training. Changing model parameters lowers training loss function.