## 6 Conclusions  
A dynamic multi-population mutation architecture-based equilibrium optimizer (DMMAEO) is developed for strengthening the population diversity and search capability of the canonical equilibrium optimizer (EO). In the DMMAEO, a dynamic multi-population guidance mechanism is introduced to construct a sub-population structure and corresponding information interaction mode, which can increase the population diversity. Moreover, a dynamic Gaussian mutation-based sub-population concentration updating mechanism is integrated to use the neighborhood space information of sub-population particle concentrations, which can strengthen the exploitation ability. Finally, a dynamic Cauchy mutation-based sub-population equilibrium candidate generation mechanism is constructed in order to employ the promising space information of sub-population equilibrium candidates, which can improve the exploration ability.  
For investigating the effectiveness of the three enhancement mechanisms from DMMAEO, the DMMAEO and different EO variants constructed by the three enhancement mechanisms are conducted on the 29 representative test functions. The test experiment results indicate that all three enhancement mechanisms are effective in enhancing the performance of the canonical EO, and the DMMAEO performs best among the different EO variants. Then, the DMMAEO is further compared with several recent promising algorithms on 58 test functions to analyze the superiority of DMMAEO. According to the comparative results of the performance assessment, the DMMAEO performs better optimization performance than the compared algorithms and achieves the top rank. In addition, the DMMAEO is utilized for tackling six engineering design problems and a UGV multi-target path planning problem, which are also solved by several recent promising algorithms for comparison. The results of the engineering application experiments denote that the DMMAEO has practicability and competitiveness in solving engineering application problems. To summarize, the dynamic multi-population mutation architecture in the DMMAEO enhances the optimization ability of the canonical EO via effectively constructing sub-population structure and using sub-population information, which makes it more effective than other algorithms. Moreover, the DMMAEO offers an effective method for tackling complex numerical optimization and engineering application challenges. Meanwhile, the average running time of DMMAEO is slightly higher than other compared algorithms, but better than the top algorithm EBOwithCMAR. Therefore, if accuracy is the main objective of solving optimization problems, the DMMAEO is a competitive and acceptable high performance optimizer.  
Regarding future work, the DMMAEO will be further developed to reduce its running time while maintaining its superior optimization performance. Moreover, designing the extended forms of DMMAEO to deal with the discrete and multi-objective optimization problems is also a noteworthy investigation. Additionally, the DMMAEO is expected to handle more types of practical application problems, such as medical diagnosis, financial stress prediction, structural damage identification, wireless sensor network coverage, and so on.  
Conceptualization, X.W.; methodology, X.W.; validation, X.W. and S.S.; investigation, Y.D.; writing--original draft preparation, X.W.; writing--review and editing, K.H. and Y.D.; supervision, K.H. and Y.D.; project administration, K.H and S.S. All authors have read and agreed to the published version of the manuscript.  
This research was funded in part by the National Natural Science Foundation of China under Grant No.82201753 and in part by the Systematic Major Project of China Railway Group under Grant No. P2021T002.  
Not applicable.  
The original contributions presented in the study are included in the article; further inquiries can be directed to the corresponding author.  
The authors declare no conflicts of interest.