## 3 Computation of Data for ANN Training and Testing  
This section presents an application of the author's method from [15] to the FDTD-PCE computations of data for training and testing of ANNs that are used to model the stochastic behavior of \(\mid S_{11}\mid\) of an ordinary patch antenna. The following notation is used in the paper. Random variables are denoted with bold font, vectors with capital letters, the values of variables or constants with a regular font, and variables with an italic font. An exception from this rule is the \(S_{11}\) parameter, which is denoted in the literature by a capital letter. Before the presentation of the novel application of the method from [15] to the FDTD-PCE computations of training and testing data for the ANNs, let us remind the design parameters of an ordinary patch antenna, whose arrays are shown in Figures 1 and 2. These parameters are denoted by \(g_{0}\), \(w_{0}\), and \(y_{0}\) and recalled in Figure 10. The width of each inset, denoted by \(u\), is not considered as a design variable. Its variation has a marginal effect on the \(\mid S_{11}\mid\) changes. It is considered as a constant equal to 0.5 mm. The author uses the technical data of the RT/Duroid 5880 LZ with substrate thickness equal to 0.254 mm and relative permittivity equal to 2. The width of the microstrip is 0.58 mm to obtain the 50 \(\Omega\) characteristic impedance of the feeding line. The three aforementioned parameters are also not treated as design variables. They remain constants for all the considered designs. Figure 8: Histogram of \(\mid S_{11}\mid\) obtained from the simulation results for \(y_{0}\) having Uniform and Beta distribution with the shape parameters \(\alpha=\beta=6\) for antenna design from Figure 1.  
Figure 9: Histogram of \(\mid S_{11}\mid\) obtained from the simulation results for \(y_{0}\) having a Beta distribution with the shape parameters \(\alpha=4\), \(\beta=6\) and \(\alpha=6\), \(\beta=4\) for antenna design from Figure 1.  
Hence, the input data for training and testing of ANNs include three variables. The author conducted a sensitivity analysis of \(|\,S_{11}\,|\) on the variations of these three aforementioned design variables in the limits corresponding to the geometrical uncertainty associated with the manufacturing process given in the previous section. The Sobol indices up to the second order were applied to quantify these impacts, as in [16]. The conclusion of the analysis is as follows. Within the considered uncertainty, variable \(g_{0}\) has a marginal impact on \(|\,S_{11}\,|\). A little higher impact is associated with variable \(w_{0}\). However, the variations of them both have an overall influence on \(|\,S_{11}\,|\), which corresponds to less than 3% of the whole variance of \(|\,S_{11}\,|\). This observation is reflected in the details of the presentation of the novel application of the method from [15] to the FDTD-PCE computations in the following paragraphs. The author takes advantage of the fact that each nominal value of \(g_{0}\) corresponds to a nominal resonance frequency (\(f_{0}\)) of the antenna. Hence, the author uses vector \(\mathbf{P}=\{f_{0},\,y_{0},w_{0}\}\) of random design variables as an input of the ANNs. Let us assume that N nominal values of the vector of random design variables \(\mathbf{P}\) are planned to be used as input data for the ANNs. Then, the method for the FDTD-PCE computations of the moments of \(|\,S_{11}\,|\) for training and testing of the ANNs can be presented using a diagram shown in Figure 11. By analogy to the work presented in [15], the method can be divided into two steps. In the first step, the extended PCE meta-models of \(|\,S_{11}\,|\) for K nominal values of \(\mathbf{P}\) are computed. The term \"extended PCE meta-model\" is used because this PCE meta-model  
Figure 11: Diagram of a method for FDTD-PCE computations of the mean and standard deviation of \(|\,S_{11}\,|\) for N nominal values of random design variables \(\mathbf{P}\). Figure 10: Ordinary patch antenna parameters.