## 4 Conclusions  
In conclusion, a novel transformer with a tranquilized time-series decomposition method was proposed for the fault diagnosis and fault forecasting of sensor arrays. The Gaussian-linearized transformer (GLTrans.) method was used as the proposed transformer's attention module and normalized both query and key before the dot product with only O (L) memory cost in the fault forecasting prediction task, thereby improving forecasting accuracy. The accuracy of fault diagnosis reached 99% with the proposed method, which is superior to that of AT-LSTM and other methods. Both the mean-square-error loss (0.04) and mean absolute error loss (0.15) of fault forecasting using the proposed approach are very low compared with traditional time-series prediction models and other deep-learning models. The proposed model GLTrans. has only O (N) time complexity compared with the original transformer model's O (n\({}^{2}\)) time complexity. Results show that the Gaussian-linearized transformer model provides a good solution for fault forecasting. In the next scientific experiments, we will try to test sensor failure and predict the time of failure in more complex engineering environments. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Models** & **Complexity** & **Decode** & **Class** \\ \hline Trans.-XL ([PERSON] et al., 2019) [(43)] & O (n\({}^{2}\)) & \(\surd\) & RC \\ Sparse Trans. ([PERSON] et al., 2019) [(44)] & O (n \(\sqrt{n}\)) & \(\surd\) & FP \\ Reformer ([PERSON] et al., 2020) [(45)] & O (n log n) & \(\surd\) & LP \\ GLTrans. & O (n) & \(\times\) & KR+LR \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Summary of efficient transformer models. Class abbreviations include: FP = Fixed Patterns or Combinations of Fixed Patterns, LP = Learnable Pattern, LR = Low Rank, KR = Kernel, and RC = Recurrence. Figure 9: Attention visualization for fault forecast task, prediction period I = {2900, 4700}, prediction length O = 1000 points, (**a**) Epoch 10, MSE 0.15, MAE 0.34; (**b**) Epoch 20, MSE 0.08, MAE 0.26; (**c**) Epoch 50, MSE 0.05, MAE 0.2; (**d**) Epoch 100, MSE 0.04, MAE 0.15. **Supplementary Materials:** The following supporting information can be downloaded at: [[https://www.mdpi.com/article/10.3390/app14010218/s1](https://www.mdpi.com/article/10.3390/app14010218/s1)]([https://www.mdpi.com/article/10.3390/app14010218/s1](https://www.mdpi.com/article/10.3390/app14010218/s1)). * [1]**Author Contributions:** Conceptualization, K.Z., W.N. and Y.Z. ; formal analysis, Z.L. and T.W. ; methodology, W.J. ; writing\(-\)original draft preparation, K.Z. ; writing\(-\)review and editing, K.Z., M.Z. and Z.Y. ; funding acquisition, M.Z. and Z.Y. All authors have read and agreed to the published version of the manuscript. * [2]**Funding:** This work was supported by the National Key Research and Development Program of China (2022 YFC3104700), the National Natural Science Foundation of China (62371299, 62301314, and 62101329), the China Postdoctoral Science Foundation (2023M732198), the Oceanic Interdisciplinary Program of Shanghai Jiao Tong University (SL2020 ZD203, SL2021 MS006 and SL2020 MS031), the Scientific Research Fund of Second Institute of Oceanography, Ministry of Natural Resources of China (SL2003), and the Startup Fund for Youngman Research at Shanghai Jiao Tong University. We also acknowledge analysis support from the Instrumental Analysis Center of Shanghai Jiao Tong University and the Center for Advanced Electronic Materials and Devices of Shanghai Jiao Tong University. The computations in this paper were run on the \(\pi\) 2.0 cluster supported by the Center for High Performance Computing at Shanghai Jiao Tong University. **Institutional Review Board Statement:** Not applicable. **Infrared Consent Statement:** Not applicable. **Data Availability Statement:** For privacy reasons, given the sensitive nature of the data, the aggregated data analyzed in this study will not be publicly disclosed but might be available on request from the corresponding author.