## 3 LSTM-Based Method for Performance Degradation Prognosis of Components of Brake Systems  
### Long Short-Term Memory Network  
As a deep learning method, the long short-term memory (LSTM) network is designed to overcome the gradient disappearance problem of the recurrent neural network (RNN) when processing long-term data, and it is suitable for processing time series data with long-term dependencies. In addition, the LSTM network is enhanced by several layers of non-linear transformations and has a long-term information memory function, making it ideal for predicting important events with relatively long intervals and delays in the time series. Similar to back propagation (BP), RNN, and other neural networks, the LSTM network also consists of three layers, namely, the input layer, the hidden layer, and the output layer, but the hidden layer adds an input gate, a forget gate, and an output gate, which are used to determine whether to retain existing information. Specifically, the hidden layer of the LSTM network consists of a cell, an input gate, an output gate, and a forget gate. The cell is responsible for remembering the value at any time interval, and the three gates are responsible for regulating the flow of information in and out of the cell. This improved design of the LSTM network not only takes into account the correlation between data points in the time series, but it also allows for the discarding of unwanted information and enables good long-distance transmission of useful information from previous data. The basic structure of the LSTM network is illustrated in Figure 2.  
In Figure 2, \(c_{t-1}\) and \(h_{t-1}\) denote the state of the cell and the output of the hidden layer at time \(t-1\), respectively; \(x_{t}\), \(c_{t}\), and \(h_{t}\) respectively denote the input data, the state of the cell, and the output of the hidden layer at time \(t\); \(f_{t}\), \(i_{t}\), and \(o_{t}\) respectively denote the output of the forget gate, the input gate, and the output gate at time \(t\); \(\sigma\) denotes the sigmoid activation function, and the range of its output is \([0,1]\); _tanh_ denotes the hyperbolic tangent activation function, and the range of its output is \([-1,\,1]\); and \(\otimes\) denotes the Hadamard product, and the output is \(0\) or \(1\), indicating the amount of information allowed to be delivered. The message forwarding transfer process of the LSTM network consists of four steps, i.e., the forget gate, update input information, update cell state, and network output information, and the details are as follows:  
(1) Forget gate  
Firstly, the \(f_{t}\) can be obtained based on the output \(h_{t-1}\) of the hidden layer at the previous moment and input data \(x_{t}\) at the current moment, and the historical state information \(c_{t-1}\) of the cell at the previous moment can be determined to pass or not. The forget gate adopts the sigmoid activation function and outputs the number from the range of [0, 1]; among them, \"0\" denotes completely abandoned, while \"1\" denotes completely reserved. The calculation formula is as follows:  
\[f_{t}=\sigma(w_{fh}\cdot h_{t-1}+w_{fx}\cdot x_{t}+b_{f}) \tag{5}\]  
(2) Update input information  
Secondly, deciding what new information is stored in the cell is carried out. The input gate uses a sigmoid activation function to generate the data \(i_{t}\) used to determine which values are to be updated, and _tanh_ is used to generate new candidate values \(\widetilde{c}_{t}\).The calculation formula is as follows:  
\[i_{t}=\sigma(w_{ih}\cdot h_{t-1}+w_{ix}\cdot x_{t}+b_{i}) \tag{6}\]  
\[\widetilde{c}_{t}=\tanh(w_{ch}\cdot h_{t-1}+w_{cx}\cdot x_{t}+b_{c}) \tag{7}\]  
(3) Update cell state  
Then, the state of the cell \(c_{t-1}\) from the previous moment multiplies \(f_{t}\) element by element to discard unwanted information, and then adds \(i_{t}\otimes\widetilde{c}_{t}\) to obtain the candidate value \(c_{t}\) of the cell to thus achieve the state updating of the cell.