# Vehicle Interior Noise Prediction Based on Elman Neural Network
## 5 Prediction and Optimization of Noise in the Vehicle
### Elman Neural Network  
A neural network comprises a large number of interconnected neurons. On the basis of the interconnection among neurons in a network, common network structures mainly include feed forward neural networks, feedback neural networks, and self-organizing neural networks. The BP neural network is a mature multi-layer feedforward neural network with strong nonlinear mapping characteristics and good generalization ability. When training this network, the forward transmission of information and the back propagation of errors are used to influence the weight in the network. The value and threshold are adjusted to ensure that the predicted output of the BP neural network constantly approaches the expected output. In the feedback network, the information is transmitted in the forward direction at the same time as the reverse transmission. Feedback of this information can occur between neurons across different network layers or be limited to only a certain layer of neurons. The Elman neural network (ENN) is both a typical feedback network and a dynamic recurrent neural network that is based on the basic structure of the BP neural network with an extra continuation layer added to the hidden layer as a one-step delay operator to achieve the purpose of memory. The system can adapt to time-varying characteristics to enhance the global stability of the network. The Elman neural network has a higher computing power than typical feedforward neural networks and can be used to solve fast optimization problems. The network model established by the Elman neural network can be used in fields with nonlinear time series characteristics given its strong robustness, good generalization ability, strong versatility, and high objectivity. The Elman neural network generally has four layers, namely, the input, hidden, undertaking, and output layers, which are connected similar to a feed forward network. The input layer unit plays a signal transmission role, whereas the output layer unit plays a weighting role. The hidden layer unit has linear and nonlinear activation functions, and the undertaking layer memorizes the output value of the hidden layer unit at the previous moment, which can be regarded as a delay operator with a one-step delay. The output of the hidden layer is self-connected to the input of the hidden layer through the delay and storage of the undertaking layer. However, this self-connection method is sensitive to historical data. The addition of the internal feedback network enhances the ability of the network to process dynamic information. The input, hidden, undertaking, and output layer nodes are denoted by \(i\), \(h\), \(t\), and \(o\), respectively, whereas the number of input, hidden, and output layer nodes are denoted by \(j\), \(k\), and \(m\). The Elman neural network topology is shown in Figure 11. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Variable Name** & **Initial Value/mm** & **Variable Interval/mm** \\ \hline Side wall & 0.9 & [0.7, 1.1] \\ B-pillar inner panel & 1.0 & [0.8, 1.2] \\ C-pillar inner panel & 1.0 & [0.8, 1.2] \\ Middle bottom panel & 1.6 & [1.3, 1.9] \\ Rear bottom panel & 1.6 & [1.3, 1.9] \\ Taillight inner panel & 0.8 & [0.6, 1.0] \\ A-pillar inner panel & 2.0 & [1.7, 2.3] \\ Roof & 1.0 & [0.8, 1.2] \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Value ranges of the panels. where \(z\) and \(a\) denote the input and output of each node layer, respectively, whereas \(b_{h}\) and \(b_{o}\) denote the biases of the hidden and output layers. The output of the input layer can be expressed as:  
\[a_{i}^{(1)}=z_{i}^{(1)}=x_{i} \tag{5}\]  
where \(x_{i}\) denote input variable, \(i\) = 1, 2, 3,..., \(j\). The hidden layer input (input layer output) can be formulated as:  
\[z_{h}^{(2)}=w_{h1}a_{1}^{(1)}+w_{h2}a_{2}^{(1)}+\cdots+b_{h}^{(2)}=\sum_{i=1}^ {j}w_{hi}a_{i}^{(1)}+b_{h}^{(2)} \tag{6}\]  
where \(h\) = 1, 2, 3,..., \(k\), \(w_{hi}\) is the weight from node \(i\) of the input layer to node \(h\) of the hidden layer. Equation 6 can take the following matrix form:  
\[