## 2 Methodology
### System Modeling Based on Machine Learning (ML) Method-Gaussian Process Regression (GPR)
\bigg{)} \tag{9}\]  
where \(\mathbf{K}(\mathbf{x},\mathbf{x}_{*})\), \(\mathbf{K}(\mathbf{x}_{*},\mathbf{x})\) and \(\mathbf{K}(\mathbf{x}_{*},\mathbf{x}_{*})\) are covariance matrix calculated through Equation (7), and here the expressions: \(\mathbf{K}(\mathbf{x},\mathbf{x}_{*})=\mathbf{K}(\mathbf{x}_{*},\mathbf{x})^{T}\). Thus, by deriving the conditional distribution corresponding to Equation (9), the posterior distribution of the test function value \(\mathbf{f}_{*}\) for GPR can be expressed as [35]:  
\[\mathbf{f}_{*}|\mathbf{x},\mathbf{y},\mathbf{x}_{*}\sim N(\overset{-}{\mathbf{ f}_{*}},cov(\mathbf{f}_{*})) \tag{10}\]  
where, the mean value:  
\[\overset{-}{\mathbf{f}_{*}}=\mathbf{K}(\mathbf{x}_{*},\mathbf{x})[\mathbf{K} (\mathbf{x},\mathbf{x})+\sigma_{t}^{2}\mathbf{I}_{t}]^{-1}\mathbf{y} \tag{11}\]  
the variance of \(\mathbf{f}_{*}\):  
\[\text{cov}(\mathbf{f}_{*})=\mathbf{K}(\mathbf{x}_{*},\mathbf{x}_{*})-\mathbf{ K}(\mathbf{x}_{*},\mathbf{x})[\mathbf{K}(\mathbf{x},\mathbf{x})+\sigma_{t}^Search Information Updating Based on Metaheuristic Algorithm Adaptive Weight Multi-Objective Particle Swarm Optimization (AWMOPSO)  
The PSO algorithm proposed by [PERSON] and [PERSON] is a metaheuristic algorithm in view of swarm intelligence inspired by the movement behaviour in a bird flock or fish cluster [36]. PSO algorithms make few assumptions for the optimization problem and can search a large candidate solution space. It has been widely applied in optimization problems because of the simplicity and efficiency for gradient-free optimization algorithms. In PSO algorithm, each particle (bird or fish) searches for food in a cooperative way and learns from the experience of the swarm. Then, global collective behaviour emerges in the swarm to locate the optimal position of food by sharing information among the swarm. PSO algorithms try to improve candidate solutions in the corresponding search space for optimizing a problem by iteration. The PSO process starts with a stochastically initialized particle swarm as possible optimal solutions for a problem and every particle is represented with locations as well as velocity. All position vectors would be assessed by the fitness function, which could evaluate and judge a position [37].  
Considering a swarm with \(p\) particles in an \(n\)-dimensional space, the position and velocity of the \(i\)-th particle at \(t\) iteration are represented \(x_{i}^{t}=(x_{i1},x_{i2},\ldots,x_{in})\) and \(v_{i}^{t}=(v_{i1},v_{i2},\ldots,v_{in})\), respectively. These points of velocity and position of the \(i\)-th particle at \(t\) iteration can be updated and adjusted according to Equations (13) and (14). There are three components for each particle's movement at each iteration in a PSO algorithm, i.e., velocity update, individual cognition, and social learning.  
\[v_{i}^{t+1}=ww_{i}^{t}+c_{1}r_{1}^{t}(\textit{pbest}_{i}-x_{i}^{t})+c_{2}r_{2} ^{t}(\textit{gbest}-x_{i}^{t}) \tag{13}\]  
\[x_{i}^{t+1}=x_{i}^{t}+v_{i}^{t+1} \tag{14}\]  
where \(v_{i}^{t+1}\) and \(x_{i}^{t+1}\) are the velocity and position of the \(i\)-th particle at \((t+1)\) iteration for the \(n\)-th dimension. For the first term of velocity update in Equation (13), \(w\) denotes the inertia weight parameter determining how much the particles' previous motion is preserved. In the individual cognition term, an individual-cognition parameter \(c_{1}\) is a positive constant; \(r_{1}\) is a random value within [0,1], avoiding premature convergences and increasing the most likely global optima; \(\textit{pbest}_{i}\) represents the \(i\)-th particle's own best position. As for the third one (social learning) meaning that all particles in the swarm share the information of the globally best point achieved (_gbest_), a social learning parameter \(c_{2}\) is also a positive constant; \(r_{2}\) plays exactly the same role as \(r_{1}\). In general, set \(c_{1}=c_{2}=2\) for acceleration, \(w=w_{\max}-\textit{iter}\times(w_{\max}-w_{\min})/\textit{iter}_{\max}\), the value of \(w\) decreases with the increase of iteration times.