## 2 Materials and Methods
### Model Performance and Uncertainty Metrics  
Various statistical measures were used to evaluate the performance of the ML models. The mean absolute error (\(MAE\)) is the averaged absolute difference between the actual and the predicted values. The equation for the \(MAE\) is the following:  
\[MAE=\ \frac{\sum_{i=1}^{n}\lvert y_{i}-\hat{y}_{i}\rvert}{n} \tag{12}\]  
where \(y_{i}\) and \(\hat{y}_{i}\) are the actual and the predicted value of the \(i\)-th sample, respectively, for a total of \(n\) samples.  
The root-mean-squared error (\(RMSE\)) is calculated as the square root of the average squared errors. The \(RMSE\) is computed as follows:  
\[RMSE=\ \sqrt{\frac{1}{n}\sum_{i=1}^{n}\left(y_{i}-\hat{y}_{i}\right)^{2}} \tag{13}\]  
The coefficient of variation of the root mean squared error (\(cvRMSE\)), or the scatter index (\(SI\)), is the ratio of the \(RMSE\) to the mean of the actual values \(\overline{y}=\sum_{i=1}^{n}y_{i}\). It represents the percentage of \(RMSE\) with respect to the mean of observations and gives the expected error. It can be calculated as follows:  
\[cvRMSE=\ \frac{RMSE}{\overline{y}} \tag{14}\]  
The coefficient of determination (\(R^{2}\)) represents the proportion of variation in the dependent variable that is predicted by the independent variables in the model. It is an indicator of goodness of fit and a measure of how well the model predicts the outcome. When the \(R^{2}\) value is 1, the predicted and the true values are perfectly aligned. \(R^{2}\) has the following mathematical representation:  
\[R^{2}=\ 1-\frac{\sum_{i=1}^{n}\left(y_{i}-\hat{y}_{i}\right)^{2}}{\sum_{i=1}^{n} \left(y_{i}-\overline{y}\right)^{2}} \tag{15}\]  
The mean absolute percentage error (\(MAPE\)) is an evaluation metric sensitive to relative errors. The smaller its value, the better. It is given by:  
\[MAPE=\frac{1}{n}\ \sum_{i=1}^{n}\frac{\lvert y_{i}-\hat{y}_{i}\rvert}{y_{i}} \tag{16}\]  
The root-relative-squared error (\(RRSE\)) is the square root of the sum of squared errors of a predictive model normalized by the sum of squared errors of a simple model. The lower its value, the better the model. It can be calculated as:  
\[RRSE=\sqrt{\frac{\sum_{i=1}^{n}\left(y_{i}-\hat{y}_{i}\right)^{2}}{\sum_{i=1}^ {n}\left(y_{i}-\overline{y}\right)^{2}}} \tag{17}\]Another performance metric used in this study is the variance accounted for (_VAF_), with the best possible score of 1. The following equation was used to compute this metric.  
\[VAF=1-\frac{VAR(y_{i}-\hat{y_{i}})}{VAR(y_{i})} \tag{18}\]  
The model uncertainty is defined as the inability of the model to effectively express the torsional capacity. In this research, the predictive model uncertainty related to beam \(i\) is equal to the ratio between the experimental and the predicted torsional capacity:  
\[M_{i}=\frac{T_{R\_Exp_{i}}}{T_{R\_Pred_{i}}} \tag{19}\]  
where \(M_{i}\) is the model uncertainty of the _i-th_ beam sample. The mean and standard deviation of the model uncertainty are represented by \(\sigma_{M}\) and \(\mu_{M}\), respectively. It is better to obtain a model with a mean \(\mu_{M}\) close to 1 and a standard deviation \(\sigma_{M}\) close to 0.