# Shape Optimization of the Streamlined Train Head for Reducing Aerodynamic Resistance and Noise
## 2 Multi-Objective Optimization Process
### Radial Basis Function Neural Network  
After obtaining the experimental design points of the input variables and corresponding output variables using OLHS, the approximate calculation model between input variables and output variables can be established for the optimization design. The neural network is a mathematical model that simulates the essential property of human brains or natural neural networks. It is composed of a large number of nodes (also known as neurons) that are connected to each other. The neural network achieves the purpose of information processing through the interconnection between a large number of nodes. Neural network models have good learning ability, associative storage ability, and high-speed optimization seeking ability and they have been widely used in complex function approximation, pattern recognition, dynamical systems, artificial intelligence, and function optimization. The RBF neural network model has excellent properties in complex function approximation and requires relatively fewer neurons to obtain good approximation results [16].  
RBF neural network is a three-layer forward network including input layer, intermediate layer and output layer. The input signals are received by the input layer, and the signals are output by the output layer. The intermediate layer has no direct relationship with the input and output. The RBF neural network takes the Euclid norm of the points to be measured and the sample points as the independent variables, and the radial function as basis function. The neural network model is then established through the linear superposition method.  
Suppose that the input layer contains N units, and the input signal \(\mathbf{x}\) enters the RBF neural network through the input layer; the intermediate layer contains \(p\) cells, and input of the \(p\)-th unit is represented as \(h_{p}=\|\mathbf{x}-\mathbf{c}_{p}\|\); the output layer contains one unit. Then, the input of the RBF neural network model can be calculated by:  
\[g(\mathbf{x})=\sum_{p=1}^{p}\lambda_{p}\varphi_{p}(\mathbf{x})+\theta=\sum_{p= 1}^{p}\lambda_{p}\varphi(\big{\|}\mathbf{x}-\mathbf{c}_{p}\big{\|})+\theta \tag{4}\]  
where \(\big{\{}\mathbf{c}_{p}\big{\}}_{p=1}^{p}\subset R^{N}\) denotes the center of the basis function, \(c_{p}\) denotes the center of the \(p\)-th basis function, \(\lambda\) denotes the weight coefficient, \(\varphi\) denotes the nonlinear basis function, and \(\theta\) denotes the threshold.  
The input layer to the intermediate layer is a fixed nonlinear transformation, through which the input signal is mapped to a new space. This mapping relationship depends on the central point of RBF. The intermediate layer to the output layer is a linear transformation.  
The output layer is combined in a new linear space with linear weights, where the weights are adjustable parameters of the network.  
The commonly used nonlinear basis function is the Gaussian basis function, which has the following expression:  
\[\varphi_{p}(\mathbf{x})=\exp\left(\frac{\|\mathbf{x}-\mathbf{c}_{p}\|^{2}}{2 \sigma_{p}^{2}}\right) \tag{5}\]  
where the parameter \(\sigma_{p}\) denotes the \"width\" or \"flatness\" of the \(p\)-th Gaussian basis function \(R_{p}(\mathbf{x})\), and its value is  
\[\sigma_{p}=\frac{1}{M_{p}}{\sum\|\mathbf{x}-\mathbf{c}_{p}\|^{2}} \tag{6}\]  
where \(M_{p}\) denotes the number of samples in the \(p\)-th unit.