## 1 Introduction
### Long Short-Term Memory (LSTM) Neural Network  
The architecture of LSTM neural networks is intricately designed, comprising an input layer, a hidden layer, a recurrent layer, and an output layer. Addressing the challenges of gradient vanishing and explosion inherent in recurrent neural networks (RNN), LSTM neural networks incorporate memory unit states within the hidden layer. This modification fosters enhanced computational efficiency and learning precision. Within this hidden layer, control units are distinctly categorized into the input gate, forget gate, and output gate. The input gate is tasked with the selective recording of new information into the cell state, ensuring that only relevant data are assimilated. Conversely, the forget gate is instrumental in selectively discarding redundant or irrelevant information from the cell, optimizing the storage efficiency. The output gate then meticulously channels the retained information to the succeeding neuron. This selective retention and omission of information endow the LSTM neural network with the capacity for long-term memory, enabling it to adeptly extract temporal features. By meticulously curating and processing data, the LSTM neural network stands as a robust model for managing complex sequential and time series data, ensuring precision and reliability in predictions and analyses.  
Initially, vibration signals indicative of gearbox faults were meticulously collected. Based on the distinct characteristics of these signals, an LSTM model was thoughtfully designed and calibrated. Subsequently, an illustrative analysis was undertaken wherein the original vibration signals, obtained from the gearbox, were decomposed utilizing the VMD method. This allowed for an intricate examination and processing of the signals, ensuring that nuanced features were not overlooked. The sorted dataset, enriched with comprehensive insights, was then subjected to temporal information fusion. This process was facilitated by the rigorously established long short-term memory neural network model, ensuring that the resultant data were both holistic and precise, ready for further analysis and interpretation.  
\[f^{(t)}=\sigma\Big{(}W_{f}h^{(t-1)}+W_{f}x^{(t)}+b_{f}\Big{)} \tag{14}\]where, \(\sigma\) is the Sigmid activation function. \(b_{f}\) is the threshold of the forgetting gate. \(W_{f}\) is the weight of the forgetting gate.  
Update the two-part output of the input gate. The equation is  
\[i^{(t)}=\sigma\Big{(}W_{i}h^{(i-1)}+W_{i}x^{(i)}+b_{i}\Big{)} \tag{15}\]  
\[c^{\prime(t)}=\cdot\text{anh}\Big{(}W_{c}h^{(t-1)}+W_{c}x^{(t)}+b_{c}\Big{)} \tag{16}\]  
Update the cell state. The formula is  
\[C^{(t)}=C^{(t-1)}*f^{(t)}+i^{(t)}*C^{\prime(t)} \tag{17}\]  
where \(C^{(t-1)}\) is the memory unit at moment \(t-1\).  
Update the output gate output. The formula is  
\[o^{(t)}=\sigma\Big{(}W_{0}h^{(t-1)}+W_{0}x^{(i)}+b_{\circ}\Big{)} \tag{18}\]  
\[h^{(t)}=o^{(t)}*\text{tanh}\Big{(}C^{(t)}\Big{)} \tag{19}\]  
where, \(W_{\circ}\), \(b_{\circ}\) are the weights and thresholds corresponding to the output gates, respectively. \(h^{(t)}\) is the output vector of the hidden layer.  
Update the forecast output at the current moment. The formula is  
\[\widetilde{y}^{(t)}=\sigma\Big{(}Vh^{(t)}+c\Big{)} \tag{20}\]  
where \(V\) and c are the weights and thresholds of the implicit to-output layer connections, respectively. Equations (14)\(\sim\)(20) comprise the process of LSTM forward propagation, and then the error between the predicted and actual values is back-calculated to update the weights and thresholds until the maximum number of iterations is satisfied.