# Research on Remaining Useful Life Prediction of Bearings Based on MBCNN-BiLSTM
## 2 Theoretical Background  
### CNN Network  
CNN is a representative feed-forward multilayer neural network [25] which has the role of feature extraction. The convolutional layer, pooling layer and fully connected layer are its main components. The convolutional layer performs convolutional operations on the input data, and the pooling layer performs dimensionality reduction on the input data. The convolutional layer and pooling layer work together to form a convolutional neural network, which performs layer-by-layer feature extraction on the input data and finally implements the regression or classification task by the fully connecting layer. The primary structure of CNN is displayed in Figure 1.  
The core of a CNN is the convolutional layer. The convolution layer contains multiple convolution kernels, which convolve the input data using multiple specific weights. The convolution operation is a linear transformation that is implemented by locally weighting  
Figure 1: CNN network structure.  
the input data. Since the vibration data of the bearing are one-dimensional data in this paper, we use one-dimensional convolution and the equation is shown below:  
\[y_{i}=\sum_{n=1}^{N}x_{n}*w_{i,n}+b_{i} \tag{1}\]  
where \(x_{n}\) is the \(n\)-th partial input connected to the convolution kernel in the \(i\)-th channel; \(N\) is the number of this input; \(w_{i,n}\) is the weight of the \(n\)-th input part of the \(i\)-th channel; \(*\) indicates the convolution operation; \(b_{i}\) is the bias that in the \(i\)-th channel; \(y_{i}\) represents the output that in the \(i\)-th channel.  
Pooling is another important operation in CNN, by which the network output at a specific location is replaced with the summary statistics at a local location. Its main function is to raise the effectiveness of feature extraction and decrease the size of feature dimensions. From a certain point of view, it also plays a role in suppressing overfitting. Pooling mainly has maximum pooling and average pooling. Since the bearings' degradation information is not obvious, where maximum pooling has the property of amplifying the information of finely differentiated features, maximum pooling is used in this paper. The calculation equation is illustrated below:  
\[p^{l(i,j)}=\max_{(j-1)V+1\leq t\leq V}\Big{\{}a^{l(i,t)}\Big{\}} \tag{2}\]  
where \(V\) denotes the range of the pooling; \(a^{l(i,t)}\) is the activation value; \(p^{l(i,j)}\) is the output after pooling.  
Multi-branch neural network is a deep learning architecture based on CNN network, which contains multiple branches in the network, each with its own computational path. This architecture is able to learn and process multiple tasks simultaneously, thus increasing the efficiency and performance of the network. The bearings' time domain information contains the vibration characteristics of the signal over time, and the frequency domain information contains the characteristics of the signal in terms of frequency, but does not contain the temporal time variation. So, only one domain or multiple domains combined for analysis cannot fully describe the degradation information of the bearing. Therefore, this paper uses a multi-branch branching structure with time and frequency domains as two branching inputs for bearings' RUL prediction.