## 5 Conclusions  
In this paper, a ventilated labyrinth acoustic metamaterial plate is proposed. The corresponding analytical and numerical calculation models are established, and the validity of the model is verified through experiments. At the same time, the influence mechanism of thermo-viscous losses and the impedance-change rule at the entrance of the LC is revealed, and its noise-reduction mechanism is elaborated. By constructing the BP-ISSA optimization model, the ventilation capacity of the WU at a specific frequency is optimized. The main conclusions are as follows:  
1. The STL and complex frequency planes of the WU were obtained by solving an analytical model based on the impedance series principle and a numerical computational model. The results show that the thermo-viscous losses caused the structure to generate a frequency shift and a reduction in the resonance frequency. At the same time, the maximum value of the STL was changed from an infinite value to a predictable value. Meanwhile, the thermo-viscous loss was responsible for the sound absorption coefficient.
2. By calculating the impedance at the entrance of the waveguide cell, it is found that the impedance was perfectly matched when the labyrinth cell was in resonance mode. When thermo-viscous losses were introduced, the impedance matching state was affected and presented an imperfect match.
3. When the actual noise frequency was 378.5 Hz, the optimized structural parameters were \(l_{1}\)* = 213.59 mm, \(l_{2}\)* = 42 mm, and \(a\)* = 31.8 mm, with a resonance frequency of 378.9 Hz. It was verified that structural parameter combinations could be effectively optimized through the BP-ISSA optimization model to improve the ventilation capacity of the LC.  
Conceptualization, D.Z., X.S. (Xianming Sun) and W.T.; methodology, D.Z. and W.T.; software, D.Z. and X.S. (Xianming Sun); validation, D.Z., Y.S. and C.C.; formal analysis, W.T.; investigation, X.S. (Xiaoming Su) and Y.S.; resources, X.S. (Xiaoming Su) and X.S. (Xianming Sun); data curation, Y.S.; writing--original draft preparation, D.Z.; writing--review and editing, W.T. and X.S. ([PERSON]); visualization, D.Z.; supervision, C.C.; project administration, C.C. and X.S. ([PERSON]); funding acquisition, C.C. and X.S. ([PERSON]). All authors have read and agreed to the published version of the manuscript.  
This research was supported by the National Natural Science Foundation of China under Grant No. 51675350, the State Key Laboratory for Joint Open Fund under Grant No. 2022-KF-2-11, the Natural Science Foundation of Jilin Province under Grant No. YDZJ202401603 ZYTS, and the Jilin Department of Education Science and Technology Research Project under Grant No. JJKH20220011 KJ.  
Figure 14: Comparison of original and optimized WUs. (**a**) Transmission coefficients, (**b**) original structural SPL distribution, and (**c**) original structural SPL distribution.  
**Institutional Review Board Statement:** Not applicable.  
**Informed Consent Statement:** Not applicable.  
**Data Availability Statement:** The original contributions presented in the study are included in the article, further inquiries can be directed to the corresponding author.  
**Acknowledgments:** The authors are grateful for the discussions with colleagues that led to the development of this study.  
**Conflicts of Interest:** The authors declare no conflicts of interest.