## 5 Conclusions  
In this paper, a multi-physics and multi-objective optimization strategy for fixing a cubic Fabry-Perot cavity based on data learning is proposed, consisting of the following three items:  
1. Performance indices acquired by multi-physics coupling simulation and key design variables determination: a multi-physics model for the cubic FP cavity is established and the performances under fixing force and random vibrations are obtained via finite element analysis, then the key performance indices (\(V\), \(w_{F}\), \(w_{F}\)) and key design variables (\(d\), \(l\), \(F\)) are determined considering the features of the FP cavity.
2. Training data are obtained by orthogonal experiment and a fitting model is established based on a neural network: 49 sets of data obtained from the orthogonal experiment are used to establish and compare different data learning models (NN, RSF, KRG), with the results indicating that the neural network has the best performance.
3. Finally, an optimized combination of key design variables is obtained using an evolutionary algorithm: NSGA-II is adopted as the optimization algorithm, the Pareto-optimal front is obtained, and the optimal combination of design variables is finally determined as \(\{5,32,250\}\). The performance after optimization demonstrates a great improvement, with the displacement under the fixing force and vibration test both decreasing by more than 60%.  
In conclusion, the method we have proposed here significantly improves the mechanical performance of the FP cavity and shows great application potential. In the future, we anticipate the application of this optimization strategy in various situations and fields.  
Conceptualization, H.Z. and F.M.; methodology, H.Z. and F.M; formal analysis, H.Z.; investigation, Z.W.; data curation, H.Z.; writing--original draft preparation, H.Z.; writing--review and editing, H.Z. and L.M.; visualization, H.Z.; supervision, J.J.; project administration, X.Y.; funding acquisition, X.Y., L.M. and J.J. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the National Key R&D Program of China, grant number 2021 YFC2201804, the National Natural Science Foundation of China, grant number 12103014, and the Center-Initiated Research Project of Zhejiang Lab, grant number K2022 MH0 AL05.  
Not applicable.  
The data presented in this study are available on request from the corresponding author. The data are not publicly available due to privacy.  
The authors would like to express gratitude to [PERSON] for his professional suggestions during the experimental design and implementation.  
The authors declare no conflict of interest.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline  & \(d\)/mm & \(l\)/mm & \(F\)/N & \(V\)/m\({}^{3}\)(\(\times\)10\({}^{-4}\)) & \(w_{F}\)/mm (\(\times\)10\({}^{-7}\)) & \(w_{vib}\)/mm (\(\times\)10\({}^{-21}\)) \\ \hline Before & 10 & 25 & 200 & 9.588 & 14.858 & 7.626 \\ After & 5 & 32 & 250 & 9.524 & 5.658 & 2.852 \\ \(\Delta\) & \(-50\%\) & \(+28\%\) & \(+25\%\) & \(-1\%\) & \(-62\%\) & \(-63\%\) \\ \hline \end{tabular}
\end{table}
Table 4: Performance comparison before and after optimization.