## 2 Materials and Methods
### Non-Destructive Measurement Models
#### 2.2.2 Multi-Scale Short Cut Convolutional Neural Network Model  
The gradient may disappear or explode when training the network with many layers. To avoid this, a theory of residual learning [17] was applied to improve the proposed MS-CNN structure as shown in Figure 3. The working principle of residual learning is to insert a short cut into the basic structure of a traditional linear neural network, which is connected with the main path of the network by skipping one or more layers. The introduction of the short cut enables the gradient in the network to be transmitted directly from the bottom layer to the former layer, which avoids the problem of the gradient vanishing. In Figure 3, two short cuts were introduced to construct a multi-scale short cut convolutional neural network (MS-SC-CNN) model. The multi-scale convolutional layer and the multi-scale downsampling layer were alternately connected, and after passing through the final downsampling layer, the feature was sent to the fully connected layer, ignoring part of the data to some extent. Therefore, the short cut method was used, and the feature data  
Figure 3: The structure of the multi-scale short cut convolutional neural network (MS-SC-CNN) model. Figure 2: The structure of the proposed multi-scale convolutional neural network model, where Conv is short for a convolutional layer, MS Conv is multi-scale convolutional layer, MS Sub is multi-scale subsampling or downsampling layer, FC is fully connected layer. of the multi-scale downsampling layer one and the multi-scale downsampling layer two were preserved; a \"jump\" approach was adopted to cross the multi-layer network into the fully connected layer, which is different from reference [17]. Its structure can be expressed as follows:  
\[\left\{ \begin{array}{c}S_{i}=F(x_{n-n_{i}},w_{n-n_{i}})\\ y_{n-2}=(S_{1},S_{2},S_{3})\\ x_{n-1}=y_{n-2}\end{array} \right., \tag{1}\]  
\[x_{n}=\big{(}\hat{S_{1}},\hat{S_{2}},\hat{S_{3}}\big{)}\cdot\textit{W}_{n}+b_{n}. \tag{2}\]  
When the structure was trained, the gradient of backpropagation was:  
\[ \begin{array}{c}\frac{\partial loss}{\partial x_{n}-x_{n}}=\frac{\partial loss }{\partial x_{n}}.\frac{\partial x_{n}}{\partial x_{n}-x_{n}}\\ =\frac{\partial loss}{\partial x_{n}}.\Big{(}\frac{\partial S_{1}}{\partial x _{n}-x_{n}}+\frac{\partial(S_{2}S_{3})}{\partial x_{n}-x_{n}}\Big{)},\\ =\frac{\partial loss}{\partial x_{n}}.\frac{\partial S_{1}}{\partial x_{n}-x_{ n}}\Big{(}1+\frac{\partial(S_{2}S_{3})}{\partial S_{1}}\Big{)},\end{array}  \tag{3}\]  
where \(\frac{\partial loss}{\partial x_{n}}\) is the partial derivative of the spanning \(n\)th layer variable with respect to the loss function. The layer is located at the end of the network structure in the proposed method, so it retains a larger gradient. \(\frac{\partial S_{1}}{\partial x_{n}-x_{1}}\) only passes through a single layer of the neural network and only reaches the feature extraction layer from the feature, so the gradient cannot die out easily. Although \(\frac{\partial(S_{2}S_{3})}{\partial S_{1}}+1\) has undergone multi-layer chain derivation, and it is typical for the gradient to disappear a traditional network structure which only contains \(\frac{\partial(S_{2}S_{3})}{\partial S_{1}}\), due to the addition of the short cut structure, it is attached with 1, which stops the gradient from disappearing due to the chain rule in the \(n-n_{1}\) layer. Thus, no matter how small \(\frac{\partial(S_{2}S_{3})}{\partial S_{1}}\) is and how deep the network goes, it can be appropriately trained. This approach not only improves the utilization of data but also limits the number of output channels in each layer and reduces the number of parameters and the risk of overfitting the model. The model uses the mean square error (MSE) as the loss function, a function of the average error for a batch.