## 3 Adaptive Surrogate-Assisted Particle Swarm Optimization (ASAPSO)  
In real-world applications, the challenges often arise from small datasets. This requires swift problem-solving strategies due to the constraints of limited data. To achieve this, a harmonious blend of precision and expeditious optimization processes is necessary. The recent methodology of cherry-picking the optimal fitness value for integration into the local surrogate model raises concerns. The benefits of the integration between surrogate models and PSO in optimization problems are widely pointed out. Firstly, in terms of simplicity and ease of implementation, PSO is known for its straightforward implementation compared to other evolutionary algorithms like genetic algorithms (GAs). It requires fewer parameters to tune, which makes it easier to adapt and apply to different problems. Secondly, in terms of fast convergence speed, PSO tends to converge faster to optimal or near-optimal solutions, especially in high-dimensional search spaces. This can be particularly advantageous in applications where computational efficiency is crucial. Finally, in terms of flexibility and adaptability, PSO is highly flexible and can be easily adapted to different optimization problems by adjusting the velocity and position update rules. This flexibility allows for customization to specific problem domains without requiring significant modifications to the core algorithm. Specifically, there is a risk of neglecting areas of uncertainty where multiple solutions exist but the optimal solution remains elusive. This vulnerability may inadvertently direct the algorithm towards a local optimum. The proposed ASAPSO algorithm in this paper introduces two key innovations. Firstly, an amalgamated surrogate model is implemented within the overarching surrogate model, endorsing the dual evaluation paradigm that emphasizes uncertainty. During the transition interplay between the global and local spheres, adaptive parameters that follow a normal distribution are introduced. The ensemble surrogate model emerges by melding RBF, PR, and KRG surrogates under the aeg of the OWS management paradigm. The evaluation matrix of uncertain solutions hinges on computing the root mean square error disseminated across the triad of surrogate models, as illustrated in Figure 1.  
As illustrated in Figure 1, the red region represents an area of uncertainty, hinting at a more extensive unexplored zone nearby. Within this zone is the domain tied to the optimal fitness value. The label of red arrow points to an uncertain solution guiding the algorithm in pursuit of the best result. This method encourages the algorithm to traverse both the landscape of uncertain solutions and the area containing the peak fitness value, prompting it to delve into and refine these areas after each assessment. Such a repetitive approach amplifies the algorithm's local probing prowess. The ensemble surrogate model is the aggregate result obtained through a weighted summation of the outputs generated by all individual members. This relationship is formulated as follows:  
\[f_{en}=w_{1}\hat{f}_{1}+w_{2}\hat{f}_{2}+w_{3}\hat{f}_{3} \tag{7}\]  
where \(\hat{f}_{i}\) stands for the i output of each member, \(1\leq\mathrm{i}\leq 3\), and \(w_{i}\) represents the weight assigned to the i output, as defined:  
\[w_{i}=0.5-\frac{e_{i}}{2(e_{1}+e_{2}+e_{3})} \tag{8}\]  
where \(e_{i}\) signifies the root mean square error of the i model and the weighted aggregation method is employed [32]. In transitioning between the global and local surrogate models, the specific procedure is delineated as follows. A new k-value emerges from a normal distribution with a mean derived from the step and a standard deviation set at 0.1 during each assessment. This is realized through the given equation. \[\mathrm{k}=\mathrm{rand}_{\mathrm{i}}(\mathrm{ustep},0.1) \tag{9}\]  
For the first half of the surrogate model, we utilize its incremental function curve.