## 4 Conclusions  
Multilayer insulation (MLI) in a high-vacuum condition can present excellent thermal insulation performance for cryogen storage. However, the ideal high-vacuum condition of MLIs in industrial scale cryogen containers has not been achieved. In fact, the interlayer pressure of the MLIs deviates from the ideal high vacuum by varying degrees. Therefore, the prediction of the thermal insulation performance of a practical cryogen container is more difficult, although the classic layer-by-layer (LBL) model could predict the insulation performance precisely for an ideal vacuum situation. Revealing the interlayer pressure distribution in MLIs for cryogen containers is very important to improve the insulation-performance-predicting quality. This paper reported an inversion method to reconstruct the interlayer pressure of MLIs on the basis of experimentally measuring the reflectors' temperatures. It was found that the thermal insulation performance of the MLI could be precisely predicted. The LBL model was modified by considering the interlayer pressure distribution in MLIs to calculate the reflector temperatures. Based on 20 groups of pre-given pressure distributions for MLI, the corresponding temperature distributions could be calculated by solving the modified LBL model. Then, the interlayer pressure and the corresponding temperature were used to train an extreme learning machine (ELM) model and determine the neuron connection weights. Then, the trained ELM model was used to reconstruct the pressure distribution based on the measured reflector temperature. The proposed ELM-based algorithm framework was tested with four different forms of interlayer pressure distributions, and its adaptability was verified. The validation results showed that the proposed algorithm was accurate in reconstructing the four interlayer pressures. The experimentally measured temperature distributions of a 60-layer MLI example were used as input data. The abovementioned inversion method was adopted, and a reasonable interlayer pressure distribution was obtained. Then, the thermal insulation performance of the MLI was calculated considering the reconstructed interlayer pressure distribution. Conclusively, the predicted heat flux of the MLI deviated from the experimental results by only 2.77%, while the error of the classical LBL model ignoring the non-ideal vacuum condition was as high as 89%. Meanwhile, the predicted corresponding temperature distribution deviated from the tested value by less than 1.13 K.  
The proposed method can be applied to assess the interlayer pressure distribution of industrial cryogen containers and precisely predict the thermal insulation performance of a practical multilayer insulation structure. In future research, the present method can also be used for variable-density MLIs (VD-MLIs). Multi-objective optimization of the optimal interlayer pressure and design parameters can be conducted on the basis of design parameterization. Moreover, it may be a promising and challenging method to predict interlayer pressure distributions in MLIs at liquid hydrogen temperature based on reflector temperature data at liquid nitrogen temperature by inversion models. Methodology, H.W. ; Writing--original draft, H.W. ; Writing--review & editing, H.T. ; Supervision, H.T. ; Project administration, H.T. ; Funding acquisition, H.T. All authors have read and agreed to the published version of the manuscript. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Prediction Methods** & **Q/W**\(\cdot\)**m\({}^{-2}\) & **Deviation/\%** \\ \hline Lockheed [22] & 0.144 & 45.04 \\ Modified Lockheed [22] & 0.156 & 40.46 \\ New Lockheed [22] & 0.170 & 35.11 \\ Classical LBL (present study) & 0.029 & 88.93 \\ Modified LBL (present study) & 0.269 & 2.77 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: The comparison of different insulation prediction methods. **Funding:** This research was funded by National Key R&D Program of China, (Grant No. 2020 YFB1506203). **Institutional Review Board Statement:** Not applicable.