## 4 Surrogate-Assisted Uncertain Optimization for Pavement Strategy
### GRNN-Based Surrogate Model Generation  
GRNN is a deformation form of the radial basis function (RBF) neural network designed for regression and function approximation [39]. GRNN takes the sample data as the posterior probability to perform Parzen nonparametric estimation and computes the network output according to the maximum probability. It approximates any mapping function between input and output vectors, extracting the function estimate directly from the training samples. In contrast to the RBF neural network, the GRNN performs better in terms of nonlinear approximation, quick learning, fault tolerance, and robustness, even in the case of insufficient samples [40]. Additionally, the GRNN does not require an initial artificial assignment of the network parameters, which minimizes the impact of human subjective assumptions on the prediction results. Similar to the RBF neural network, the GRNN has four layers: an input layer, a pattern layer, a summation layer, and an output layer. The specific process of building the GRNN model is shown in Figure 7, which is described in detail as follows:  
Figure 6: Flowchart of the surrogate-assisted RBDO.  
(1) In the input layer, the training input and output samples are obtained through the finite element model in Section 5.1. The input layer receives information and stores an input vector \(X_{i}\), and the number of input units equals the dimension of the input vector in the training samples. Then, the neurons of the input layer directly transfer the input data to the pattern layer.  
(2) In the pattern layer, the number of neurons consists of an exactly equal number of input training datasets. The pattern Gaussian function of \(\phi_{i}\) is expressed in Formula (11).  
\[\begin{split}&\phi_{i}=\exp\left(-\frac{D_{i}^{2}}{2\sigma^{2}} \right)(i=1,\;2,\ldots,\;n)\\ & D_{i}^{2}=(X-X_{i})^{T}(X-X_{i})\end{split} \tag{11}\]  
where the output of neuron \(i\) is the exponential square of \(D_{i}\), \(D_{i}\) is the Euclidean distance between \(X\) and \(X_{i}\), \(X\) is the input variable of the network, \(X_{i}\) is the specific training sample of neuron \(i\) in the pattern layer, and \(\sigma\) is the smoothing parameter.  
(3) The neurons in the summation layer consist of arithmetic summation (\(S_{A}\)) neurons and weighted summation (\(S_{W}\)) neurons. \(S_{A}\) is one of the neurons in the summation layer, which makes an arithmetic summation of the output of all the pattern layer neurons. \(S_{Wj}\) is the other k neuron in the summation layer, which makes a weighted summation of the output of all the pattern layer neurons. The output vector can be calculated by Formula (12), where \(y_{ij}\) is the connection weight between the \(i\)th neuron in the pattern layer and the \(j^{th}\) neuron in the summation layer.  
\[\left\{\begin{array}{l}S_{A}=\sum\limits_{i=1}^{n}\phi_{i}\\ S_{Wj}=\sum\limits_{i=1}^{n}y_{ij}\cdot\phi_{i}\end{array}\right.\;(j=1,\;2, \ldots,\;k) \tag{12}\]  
(4) After the summations of the summation layer neurons are transferred into the output layer, the output results \(y_{j}\) of the output neuron can be calculated by Formula (13).  
\[y_{j}=\frac{S_{Wj}}{S_{A}}(j=1,\;2,\ldots,\;k) \tag{13}\]  
Particularly, the value of the smoothing factor \(\sigma\) has a great influence on the prediction accuracy and the generalization ability of the GRNN model. In this paper, the optimal smoothing factor with a minimum cross-validation error was determined using a one-dimensional optimization method.  
Figure 7: GRNN model building process.