## 3 Application of Model-Agnostic Meta-Learning (MAML) Model
### MAML Training and Test Results  
To ascertain the accuracy of the MAML prediction methodology, database Meta2, which is not involved in the training process, is utilized to evaluate its network performance. This posterior database is partitioned into distinct training and testing sets, and during the fine-tune process these sets are locked. However, throughout every meta-learning epoch, the assignment of tasks to training and testing sets remained dynamic; upon the completion of an epoch, tasks were reallocated, ensuring that both the training and testing sets were consistently resampled at a 4:1 ratio. The hyperparameters \(\alpha\) and \(\beta\) of the model were set to 0.01. In order to mitigate the possibility of incidental outcomes, a total of six repeated predictions were conducted, and the predictive results are illustrated in Table 5. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & 0.18075 & 0.17955 & 0.22624 & 0.12481 & 0.14662 \\ True value & 0.15113 & 0.15639 & 0.0797 & 0.10752 & 0.09624 \\  & 0.0985 & 0.10752 & 0.04812 & 0.06917 & 0.12105 \\  & 0.06842 & 0.07293 & 0.0609 & 0.05338 & 0.04812 \\ \hline \multirow{2}{*}{Test number} & \multicolumn{4}{c}{Comparation between true value and predicted value} \\ \hline  & Prediction & 0.19447 & 0.18574 & 0.21813 & 0.11399 & 0.12917 \\  & Error (\%) & 7.59\% & 3.45\% & 3.58\% & 8.67\% & 11.90\% \\  & Prediction & 0.1315 & 0.13119 & 0.10402 & 0.11414 & 0.11631 \\  & Error (\%) & 12.99\% & 16.11\% & 30.51\% & 6.16\% & 20.85\% \\ \(1\) & Prediction & 0.10779 & 0.10747 & 0.0575 & 0.10267 & 0.13875 \\  & Error (\%) & 9.43\% & 0.05\% & 19.49\% & 48.43\% & 14.62\% \\  & Prediction & 0.0962 & 0.0962 & 0.08221 & 0.07765 & 0.07765 \\  & Error (\%) & 40.60\% & 31.91\% & 34.99\% & 45.47\% & 61.37\% \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Prediction results of MAML. considering the coupling effect of sulfate ion corrosion with the wet-dry cycle environment. As a result, certain data points may lack crucial features, rendering the model unable to fully comprehend the data, thereby leading to significant prediction errors. This is an inevitable bias in deep learning algorithms. However, since these larger biases occur only sporadically, it can still be seen that the network possesses good learning and prediction capabilities. Moreover, there has been no indication of overfitting when the input factors reach 19, which may be attributed to factors such as regularization, cross-validation, and the model architecture within the MAML algorithm. Furthermore, the average errors from these six repeated tests are not consistent, which may be attributed to the variability in the initialization of model weights and biases before each training process. Figure 1 illustrates the evolution of the proximity between predicted values and true values during the training and testing processes of the MAML algorithm. In order to assess the prediction accuracy more comprehensively, we decided to no longer rely solely on simple relative errors but introduce the coefficient of determination, R-squared, to better characterize this relationship. The equation of R-squared is shown in Equation (5):  
\[R^{2}\ =\ \begin{cases}0&R^{2}\leq 1\\ 1-\frac{\sum(y_{i}-\hat{y}_{i})^{2}}{\sum(y_{i}-\overline{y})^{2}}&0\leq R^{2} \leq 1\\ 1&R^{2}\geq 1\end{cases}  \tag{5}\]  
where \(y_{i}\) is the ith test value, \(\hat{y}_{i}\) is the ith predicting value, and \(\overline{y}\) is the average value of the test values. The number of iterations, serving as a special hyperparameter, is also tested, as shown in Figure 1. In MAML, an epoch refers to the number of iterations as well as the operation number of the entire training and testing dataset. To achieve the best results, it is essential to properly set the maximum number of epochs. In this study, epochs in the range of 0 to 5000 are selected.