## 2 RBFNN Approximation Method
### Theoretical Background of RBFNN and SPP  
A brain is composed of networks of neurons. A typical neuron receives an input, which is either excitation or inhabitation, from many other neurons. As its net excitation reaches a certain level, the neuron fires. The firing is propagated through a branching axon to many other neurons, where it in turn acts as input to those neurons. The neuron's firing is thought of its output as a binary or continuous valued quantity. The neural network models are algorithms for cognitive tasks, such as learning and optimization, which are based on concepts derived from research in the nature of the brain. In the neural network model, the neurons are generally organized into layers. The inputs to the neural network model are presented to the input layer, and the outputs are generated as signals of the output layer. Since the numeric activations and signals comprise vectors, the input vector applied to the input layer generates an output signals vector across the output layer. These signals pass through one or more intermediate or hidden layers which transform the signals depending upon the neuron signal functions [26; 27; 28]. In this study, RBFNN was used to generate a meta-model that can be applied to approximate design optimization, and the performances of the generated meta-model were evaluated according to the variation of the learning parameters such as the SSP value, the number of training data, and the number of hidden layers. RBFNN has been developed for scattered multivariate data interpolation [1; 8]. This technique uses linear combinations of a radial symmetric function based on Euclidean distance to approximate response functions. Let \(x_{1},\cdots,x_{n}\in\Omega\subset R^{n}\) be a given set of nodes, and a set of any RBF basis function, \(g(x)\), is to be:  
\[g_{j}(x)=\phi\big{(}||x-x_{j}||\big{)}\in R,\ j=1,\cdots,n \tag{1}\]  
In Equation (1), \(\phi\) is power spline basis function and the norm \(||x-x_{j}||\) is Euclidean distance. By using of Euclidean distance, the data of input layer are classified as homogeneous cluster. Power spline basis function is expressed as follows:  
\[\phi=r^{c_{j}} \tag{2}\]  
where \(c\) is a shape function variable with positive constant value. Given interpolation data \(y_{1},\cdots,y_{n}\in R\) at data points \(x_{1},\cdots,x_{n}\in\Omega\subset R^{n}\), RBFNN interpolant \(f^{\prime}(x)\) through the network learning process as shown in Figure 1 is defined as follows:  
\[f^{\prime}(x)=\sum_{j=1}^{n}\alpha_{j}g_{j}(x)+\alpha_{n+1} \tag{3}\]  
This study adopted one layer, two layers, and three layers respectively in the neural network architecture in order to evaluate the effect of the number of hidden layers on the performance of RBFNN for the rotor blade design approximation. The interpolant is obtained by solving the system of \(n+1\) linear equation as follows:  
\[\sum_{j=1}^{n}\alpha_{j}g_{j}(x_{i})+\alpha_{n+1}=y_{i},\ i=1,\cdots,n \tag{4}\]  
for \(n+1\) unknown expansion coefficients \(\alpha_{j}\). The SSP is a parameter value used to relax the over-fitting feature that happens in a meta-model approximated via the RBFNN in case of passing through every single data point over whole design space. Its primary purpose is to smooth out noisy data. By not going through every data point, meta-model can be effectively fitted as a smoothing function and employed in an approximate optimization that may be easier to solve. The value specified by the SSP averages the output values of data points that are clustered in the normalized filter domain. The SSP trades smoothness of the fitted function against fidelity to the data and can be expressed by the following equation:  
\[\min\rho||s||^{2}+\frac{1}{n}\sum_{i=1}^{n}\bigl{(}s(x_{i})-f^{\prime}(x_{i}) \bigr{)}^{2} \tag{5}\]  
where \(\rho\) is positive smoothing control parameter and the norm \(||s||\) is smoothness penalty. From Equation (5), larger value of \(\rho\) increases the amount of smoothness relative to the average deviation from the data squared.