## 2 Methodology
### Construction Method for the Multi-Input ResNet  
In the field of artificial intelligence, ANNs are widely used in fault diagnosis due to their strong expression ability and learning ability. However, with increasing ANN network depth, the training accuracy gradually increased to saturation and then began to degrade rapidly. The deep residual network (ResNet) proposed by [PERSON] in 2016 can be used to resolve the problem of degradation [24]. ResNet is based on the following concepts:  
* The features obtained by shallow neural networks are not sufficiently abstract. By increasing the layers of the neural network, the abstraction of features can be improved and the training effect for some specific tasks can be improved to some extent.  
With the increase in neural network layers, the training accuracy of the neural network gradually becomes saturated and then rapidly decreases. This degradation problem limits the improvement of neural network training accuracy.
* In theory, adding identity mapping to the optimal shallow neural network can not only increase the depth of the neural network, but also retain the same training errors as the shallow neural network so that the deeper network can perform better than the shallow network.  
Based on these ideas, the problem of gradient degradation can be handled by adding identity mapping to the shallow neural network. The implementation of ResNet is equivalent to fitting an identity map.  
The structure of the deep residual learning block is shown in Figure 4. The residual block is realised by a shortcut connection, which is equal to adding the input and output of the block directly and activating the block by the activation function. Therefore, the introduction of the identity shortcut does not increase the computational complexity, nor does it require additional parameters. In fact, identity mapping is ideal and impossible to realise. And it's a desired underlying mapping. Therefore, the deep residual learning block can only be used to infinitely close the identity mapping by underlying mapping.  
Suppose that the underlying mapping realised by the deep residual learning block is H(x), where x represents the input of the deep residual block. F(x):= H(x) - x is defined as a residual mapping, and then H(x) = F(x) + x. The deep residual block is infinitely close to H(x), which is equivalent to stacking multiple nonlinear layers in the residual block to approximate F(x). It is easier to implement residual mapping with multiple nonlinear layers to approximate F(x) to 0 than to approximate identity mapping with multiple nonlinear layers.  
Figure 5 presents the structure of the multi-input ResNet based on the deep residual learning block. The AM-FM components of the response of the CUT obtained by the improved EWT are input from different input terminals of the multi-input ResNet, and the feature vector of each AM-FM component is cascaded. Finally, the result is output through the fully connected (FC) layer and softmax function. The multi-input ResNet model is obtained by combining the input layer, multiple ResNet units, concat layer, and FC layer. There are N ResNet units with the same structure in Figure 5. The main purpose of this structure is to use a ResNet unit to learn the characteristics of an AM-FM component and sufficiently learn the characteristics of each component. Although the structure of each ResNet unit is the same, their weights and biases obtained by training are different to obtain more abundant fault characteristics under different CUT fault types.  
Figure 4: Deep residual learning block.  
Each ResNet unit consists of a convolution block, 4 residual blocks and an average pooling layer. The convolution block consists of a one-dimensional convolution layer, batch normal (BN) layer, RELU layer, and average pooling layer. The residual block is constructed in the manner described in Figure 4, and the details are shown in [24].