## 3 Computation of Data for ANN Training and Testing
is valid for the support of random design variables \(\Omega_{\mathrm{e}}\), which is higher than the normal support related to the uncertainty of the manufacturing process, which can be denoted by \(\Omega_{\mathrm{n}}\). In the case of this paper, the support of the extended PCE meta-model is about two times higher than \(\Omega_{\mathrm{n}}\). The coefficients of the extended PCE meta-model (Equation (8) in [15]) are calculated using Beta probability distribution with the shape parameters \(\alpha=\beta=2.5\). These values of the shape parameters of the Beta distribution were found to be optimal in terms of the quality and size of the extended PCE meta-model. It should be noted that the corresponding values of these parameters for the case of the work presented in [15] are smaller. In the second step of the method, the author uses the extended PCE meta-models and the analytical Formulas (18), (19), (24), (25), (31), and (32) from [15] to calculate the PCE meta-models for each of N supports \(\Omega_{\mathrm{n}}\) associated with corresponding nominal values of design variables. Each extended PCE meta-model is derived with FDTD-PCE computation using the openEMS Matlab-Based FDTD simulation tool [13] and the Matlab-based package called UQLab [14]. The idea of the implementation of the extended PCE meta-model is to reduce the number of required FDTD-PCE computations. Each such computation contains 11 and 16 FDTD simulation runs for supports \(\Omega_{\mathrm{e}}\) and \(\Omega_{\mathrm{n}}\) considered in this paper, respectively. The main advantage of computing the extended PCE meta-model is that all of the PCE meta-models, whose supports are included in \(\Omega_{\mathrm{e}}\), can be calculated analytically. Consequently, the computation of one extended PCE meta-model lets us calculate the mean and variance of \(\left|\,S_{11}\,\right|\) for more than one set of nominal design variables. Let us note that there is only one design variable whose variation has a significant impact on the change of \(\left|\,S_{11}\,\right|\). It is the design random variable \(\mathbf{y}_{0}\). Having this in mind, the relationship between support \(\Omega_{\mathrm{e}}\) and support \(\Omega_{\mathrm{n}}\) can be presented as in Figure 12.  
For the best quality of the PCE meta-models, it is assumed that the size of \(\Omega_{\mathrm{n}}\) is 0.172 mm, while the size of \(\Omega_{\mathrm{e}}\) is 0.272 mm. The nominal values of the random design variable \(\mathbf{y}_{0}\) are sampled with the step of 0.05 mm. Consequently, each FDTD-PCE computation of the extended PCE meta-model enables the calculation of three PCE meta-models for support \(\Omega_{\mathrm{n}}\). Each FDTD-PCE computation takes about 45 min (16 FDTD simulation runs) using a four-core Intel platform. The corresponding computation time of the PCE meta-model with support \(\Omega_{\mathrm{n}}\) is about 30 min. Computations of K extended PCE meta-models require about half of the time needed to compute N PCE meta-models for support \(\Omega_{\mathrm{n}}\). For example, when N equals 500 and computations are performed using one computer, then about 5 days of computation time are saved. In the next section, the author presents ANNs that model the mean and standard deviation of \(\left|\,S_{11}\,\right|\) of an ordinary patch antenna shown in Figure 10. The data for the training of ANNs are as follows. The minimum and maximum nominal values of \(\mathbf{y}_{0}\) are  
Figure 8:0 Illustration of the relationship between supports \(\Omega_{\mathrm{e}}\) and \(\Omega_{\mathrm{n}}\). 0.3 and 1.2 mm, respectively. They are sampled with a 0.05 mm step, as mentioned in the previous paragraph. The minimum and maximum nominal resonance frequencies are 24.8 GHz and 28.4 GHz, respectively. They are sampled with a 600 MHz step, which is also the approximate bandwidth of the antenna. The nominal values of \(g_{0}\) are 4, 3.9, 3.81, 3.72, 3.64, 3.54, and 3.46 mm, and they correspond to the nominal values of \(f_{0}\): 24.8, 25.4, 26.0, 26.6, 27.2, 27.8, and 28.4 GHz, respectively.