## 2 Materials and Methods
### Regression Analysis
#### 2.3.2 Segmented Nonlinear Regression
We report the duration for both cases in h:mm:ss format, where h is hours, mm is minutes, and ss is seconds. The fastest time is in bold. Figure 8: Long-term prediction of all models on Crack A2. (**a**) Ten-year prediction of each model on Crack A2 from 1 August 2019 to 1 August 2029, and (**b**) displays eight-month prediction of each model on Crack A2 from 1 August 2019 to 1 April 2020.

assumption where it shows that the model produced linearly decreasing values, while the ground truth abruptly increases. Thus, we consider that the linear regression model may be inadequate for long-term prediction as well. We observe that the segmented nonlinear regression model generated incremental values, while the other models produced either constant values or decremental outputs in Figure 8a,b. Moreover, Table 1 shows that the model produced predictions that had comparable performance to the Seq2 Seq LSTM model on each sensor, despite several abrupt changes in presence. Above all, the segmented nonlinear regression model required a short period of training and inference processes that are comparable to those of the linear regression model. Considering the factors as mentioned earlier, it is reasonable to conclude that the segmented nonlinear regression model is an acceptable application for the long-term prediction of crack growth when there are multiple abrupt changes in the data. ## 5 Conclusions and Future Work

In this study, we exhaustively explore several methods for the long-term prediction of crack growth in two separate facilities that are practically in use. We evaluate the performance of LSTM, Seq2 Seq LSTM, linear regression, and segmented nonlinear regression models on the test set, with RMSE as our evaluation metric. We also analyze 10-year and eight-month predictions of the models on each crack sensor. Our experiment results show that the LSTM model has the worst performance among the models in terms of RMSE score. On the other hand, the Seq2 Seq LSTM model achieves the highest performance in the same test sets. While the linear regression model is comparable to the segmented nonlinear regression model on the Crack A1 and Crack A2 test set, the segmented nonlinear regression model obtains more satisfactory results when there are abrupt changes apparent in the data, such as Crack B1 and Crack B2. The long-term prediction results show that the LSTM model and Seq2 Seq LSTM model are ineligible as the predictor, due to a failure to emulate the crack growth, where both models give a constant output. Additionally, both models require a tremendous amount of time for training and inference. Finally, our analysis of long-term prediction points to segmented nonlinear regression being the ideal candidate as the predictor, which is computationally efficient and exhibits comparably high performance. In future work, we would like to consider other research directions for more robust prediction. The first one is to incorporate temperature as the independent variable of prediction, which we assume causes rapid expansion and shrinkage of the materials that lead to the growth of the crack. The next one to regard in the regression analysis are the external shocks that cause abrupt changes in crack sensor reading. Finally, we plan to extend our machine learning and deep learning-based research to accomodate the influence on the stress distribution in the cracked region using the geometry of the structure and loading cases [16; 17; 18; 19]. Conceptualization, D.-K.K. and S.M.I. ; methodology, D.-K.K. ; software, S.M.I. ; validation, D.-K.K., J.-R.P., K.-I.J. and J.-S.L. ; formal analysis, S.M.I. and D.-K.K. ; investigation, J.-R.P. ; resources, J.-R.P. ; data curation, J.-R.P. ; writing--original draft preparation, S.M.I. ; writing--review and editing, D.-K.K. ; visualization, S.M.I. ; supervision, D.-K.K. ; project administration, D.-K.K., K.-I.J., and J.-S.L. ; funding acquisition, D.-K.K., K.-I.J. and J.-S.L. All authors have read and agreed to the published version of the manuscript.