## 2 Theoretical Investigations for Antenna Design and Proposed Algorithm  
The previous section discussed the background and literature of the proposed high-gain antenna design. This section explains the analytical approach for validation of the structure and parameters of the proposed SABNN-based antenna model. At first, the basic principle of currently used techniques such as Gaussian Process (GP) is discussed in this section. Consider the observations \(m\left(u=u^{1},\,\ldots u^{n}\right)\) and \(\left(v=v^{1},\,\ldots v^{n}\right)\), where the \(v(u)\) is the sample of Gaussian-distributed stochastic process with variance \(\sigma\) and mean \(\mu\). Based on the \(m\) observations, the GP predicts the value of \(v(u)\) for next \(u\); this correlation function [25; 26; 27; 28] is written as  
\[Corr\left(u^{a}u^{b}\right)=\exp(-\sum_{i=1}^{j}\theta_{i}\Big{|}u_{i}^{a}-u_ {i}^{b}\Big{|}pi \tag{1}\]where the dimension of \(u\) is represented by \(j\), and \(pi\) and \(\theta_{i}\) are the hyper parameters. The liklihood function is calculated as  
\[\frac{1}{\left(2\pi\sigma^{2}\right)^{m/2}\sqrt{\det(H)}}\exp\left[-\frac{(v- \mu U)^{T}H^{-1}(v-\mu U)}{2\sigma^{2}}\right] \tag{2}\]  
where \(U\) is an \(m\times 1\) vector, and \(H\) is \(m\times m\) covariance matrix. The predicted value and prediction uncertainty \(v(u^{*})\) and \(S(u^{*})\) are discussed [29; 30] as  
\[v(u^{*})=\mu+r^{T}H^{-1}(v-U\mu) \tag{3}\]  
The prediction uncertainty plays a key role when judging the potential of user antenna design. For this operation, the prescreening, low confidence bound (LCB), and probability are widely used. The LCB method is introduced in this paper using the objective function \(v(u)\) with \(N\big{(}v(u),S^{2}(u)\big{)}\) predictive distributions. This is written as  
\[v_{cd}=y(u^{*})-\omega S(u) \tag{4}\]  
where \(\omega\) is constant; this is set for antenna problems in AI algorithms. The training cost and huge computational time of GP are the main drawbacks. Thus, to handle these constraints, the huge computational time, and training cost, in this paper, the SADE method is proposed for antenna design optimization.  
There are two optimization techniques used for antenna design, named local optimization and global optimization. The global optimization method is applied in this paper for antenna design using an evolutionary algorithm (EA). Taking \(\Psi_{pop}\) population, which has \(N_{pop}\) solutions, and each solution is denoted by \(u=u_{1},\ldots\ldots u_{n}\in H^{*}\), the donor vector for creating a child solution \(x=x_{1},\ldots x_{n}\) is calculated as  
\[y^{a}=u^{a}+F.\Big{(}u^{best}-u^{a}\Big{)}+F.\Big{(}u^{r1}-u^{r2}\Big{)} \tag{5}\]  
where \(u^{a}\) means the \(a^{th}\) vector in current \(\Psi_{pop}\), \(u^{best}\) denotes the best user solution in current \(\Psi_{pop}\), \(u^{r1}\) and \(u^{r2}\) are randomly selected mutual solutions from the population. The parameter \(y^{a}\) represents the \(a^{th}\) mutual vector and \(F\) is used for scaling factor. Figure 1 explores the flow chart of the presented algorithm, which shows the basic role of the BNN and LCB mechanisms.  
Figure 1: Flow chart description of presented SADE for antenna design optimization.