## 3 Original DE Algorithm  
Due to the disaster of dimension, the task becomes more challenging. When the dimension is increased, the search space expands exponentially and the distribution of good solutions is sparse. Differential evolution is an effective population-based meta heuristic algorithm, which has been successfully used in various applications [15]. DE is based on three main operators, mutation, crossover, and selection. Mutation is the core operator in DE, which generates mutation vectors based on the linear combination of different candidate solutions. The role of the crossover operator is to combine the mutant vector with its target vector to produce a test vector. Finally, the selection operator adopts a greedy strategy to select and enter the formula here. Choose the better one of the test vector and the target vector. Differential Evolution (DE) is a meta-heuristic algorithm proposed by [PERSON] and [PERSON] in 1995. The algorithm has been used to solve multiple applications in various engineering fields. It is similar to other evolutionary algorithms (EA) used for optimization problems, and DE is a population-based algorithm that evolves through differences between individuals and achieves evolution through three operators: mutation, crossover, and selection. The execution of the three operators also contains three control parameters: population size NP, scaling factor F, and crossover rate CR. Initialization: starting from the candidate solution randomly generated by NP, usually, a DE population is composed of NP individuals, denoted as:  
\[x_{i,G}=(x_{1,G},x_{2,G},\ldots,x_{D,G})\qquad i=1,2,...,Np, \tag{11}\]  
where \(Np\) is the population size. \(Np\) test individuals are generated from the current population through mutation, crossover, and selection of each generation. This step forms a loop until the termination condition is met. The initial population is randomly generated:  
\[x_{i,j,G}=x_{i,j,G}^{L}+rand\times(x_{i,j,G}^{LI}-x_{i,j,G}^{L}), \tag{12}\]  
where \(x_{i,j}\) represents the jth component of the ith individual; L and U in \(x_{i,j}^{L}\) and \(x_{i,j}^{LI}\) represent the lower and upper bounds of the jth component respectively. \(rand\) represents a random decimal in the range of [0, 1]. Mutation operator: a mutant individual is generated through the difference between two random individuals and F (scaling factor) and a target vector, DE achieves individual mutation through a differential strategy. This is also the embodiment of DE's central idea, which is based on the current individual in the population and randomly selects several vectors for differential operation and generates a differential v vector. Several commonly used differential mutation strategies are as follows:  
1. DE/rand/1 \[v_{i,G}=x_{r1,G}+F\times(x_{r2,G}-x_{r3,G})\] (13)
2. DE/best/1 \[v_{i,G}=x_{best,G}+F\times(x_{r1,G}-x_{r2,G})\] (14)
3. DE/current-to-best/1 \[v_{i,G}=x_{i,G}+F\times(x_{best,G}-x_{i,G})+F\times(x_{r1,G}-x_{r2,G})\] (15)
4. DE/current-to-rand/1 \[v_{i,G}=x_{i,G}+rand\times(x_{r1,G}-x_{i,G})+F\times(x_{r2,G}-x_{r3,G})\] (16)  
where \(r_{1}\), \(r_{2}\), and \(r_{3}\) represent the index subscripts of three different individuals randomly selected from the population. \(F\) represents the scaling factor, which controls the strength of the difference. \(x_{best}\) represents the individual with the best fitness value. Crossover operator: The purpose of the crossover is to generate a new candidate solution based on the combination of the mutation vector and the parent vector to generate a test vector. The crossover operation between the individual \(x_{g,i}\) of the gth generation and the mutated intermediate individual \(v_{g,i}\) can be denoted as:\[u_{ij,G}= \begin{cases}v_{i,j,G},&if\ rand\leq CR\ or\ j=j_{rand}\\ x_{ij,G},&otherwise,\end{cases}  \tag{17}\]  
where \(j_{rand}\) is an integer randomly selected from 1, 2,..., D to ensure that at least one individual is from a variant individual.