## 5 Results and Discussion
### ANN Inputs Relative Importance  
To better understand how the proposed assisted-ANN modeling approach improves the prediction accuracy and enables the network to be trained efficiently using small datasets, it is essential to look at the effect of the added assistance inputs on the internal parameters (i.e., the weights) of the ANN. Specifically, the contribution of each input in the ANN's residual strength predictions can be quantified by resorting to the connection weights (CW) algorithm [64]. The relative weight of any of the ANN inputs basically reflects its relative importance, and therefore the different inputs can be ranked based on the relative weight values [65]. That is, the evolution of the ANN internal parameters considering the different scenarios investigated in this work is embedded in the quantification of the relative weight values. The CW algorithm [64] uses the ultimate inputs-hidden connection weights (\(W_{xy}\)), from \(x\)-th input node (\(x~{}=~{}1,\ldots,X\)) to \(y\)-th hidden node (\(y~{}=~{}1,\ldots,H\)), and the ultimate hidden-output connection weights (\(W_{yz}\)), from \(y\)-th hidden node (\(y~{}=~{}1,\ldots,H\)) to the \(z\)-th output node (\(z~{}=~{}1\)),  
Figure 7: Effect of the assisted modeling approach and training dataset size on the correlation of ANN predictions with experimental residual strength values (for all data points used for training, validation, and testing). The effect of the assisted approach is observed when moving from left to right, while the effect of training dataset size reduction is observed when moving from top to bottom. The shown ANN predictions are for one of the 80 training/validation/testing datasets random combinations. obtained for the optimum ANN models of the different scenarios to assess the Relative Weight (RW\({}_{x}\)) of each \(x\)-th ANN input (\(x\ =\ 1,\ldots,X\)) as follows:  
\[\text{RW}_{x}\ =\ \left|\sum\limits_{y\ =\ 1}^{H}W_{xy}W_{yz}\right| \tag{10}\]  
where RW\({}_{x}\) is the \"absolute\" relative weight of the \(x\)-th input node and \(\sum\limits_{y\ =\ 1}^{H}W_{xy}W_{yz}\) is the sum of the product of the ultimate weights of the connections from the \(x\)-th input node to hidden nodes (\(W_{xy}\)) with the connections from the hidden nodes to the output node (\(W_{yz}\)). It is worth mentioning that the sum of products used here is the average sum of products calculated over the 80 cross-validation combinations of each ANN scenario. The absolute value is taken here because the sign of the relative weight is not considered when ranking the inputs according to their importance. The relative weights are calculated for each of the ANN inputs used in the traditional approach and the four assisted approach scenarios. In order to make the comparison of the relative weight of the inputs across the different scenarios easier, the relative weights are normalized for each scenario. After normalization, the total normalized relative weights for all inputs for any of the different scenarios is equal to 1. The normalized relative weights calculations are also done for the different training dataset sizes, and all the data are graphically represented in Figure 8. The first seven inputs (\"\(a\)\" to \"config ID\") are the direct independent inputs, and they are common for the traditional and the four assisted approach scenarios. The last four inputs are the assistance parameters where one or more of them is used at each of the four assistance levels (see Table 1). By inspecting the different parts of the figure, corresponding to the different training dataset sizes, it can be seen that the overall trends seen in these parts are fairly similar (except for few minor differences in the relative weights of some inputs). The figure shows that for the assisted approach scenarios, starting from assistance level 1 up to 4, the relative weights of the assistance input parameters are always increasing (note that for assistance level 3, two assistance parameters are used and thus, their weights are added together).