## 5 Results and Discussion
### Results  
Next, two Artificial Neural Networks (ANNs) have been trained, using as input parameters the \(R\) and \(L\) properties of the shunted piezoelectric beam, as well as the graphene nanoplatelets volume content \(V_{GPL}\), the fibre volume content \(V_{F}\), and the fibre angle \(\theta\). In the first ANN output is the vibration reduction (\(PI\)) that can be achieved by the shunted piezoelectric beam. This output is provided as the absolute value of the difference between the maximum response from the short-circuit condition (no damping) and the open-circuit condition (damping) (see Equation (35)). In the second ANN, 5 output parameters are considered, namely, the first and second natural frequencies for the short-circuit, the first and second natural frequencies for the open-circuit, as well as the shunt damping performance \(PI\), also considered in the first ANN. For the parametric simulations, which are considered to develop the dataset that is then used for the training, 16,807 simulations leading to respective dataset points are considered. For the input parameters, the following range of values have been considered in the parametric simulations:  
\[1000 \leq R\leq 37000\] \[33 \leq L\leq 39\] \[0 \leq V_{GPL}\leq 0.036\] \[0.1 \leq V_{F}\leq 0.70\] \[0 \leq\theta\leq 90\]  
After the dataset points are generated from the parametric finite element simulations, which are conducted in Matlab, all input parameters are normalised to be in the range \([0,1]\). All datasets generated by the aforementioned procedure are available upon request (the dataset format is described in the Appendix A). Next, the two ANNs are developed, trained, and validated. The data are split to 75% of the overall dataset points for training, 15% for validation and 15% for testing. The Levenberg-Marquardt backpropagation training function is adopted in Matlab. Before concluding the final ANN architecture, some numerical tests are conducted in the first ANN, the one with five input and one output parameter. Thus, to investigate the optimal number of neurons per hidden layer, a parametric training is implemented varying between 1 and 20 neurons per hidden layer, and the corresponding Root Mean Squared Error (RMSE,  
Figure 4: Vibration suppression for varying graphene nanoplatelets \((X)\), and fibre volume \((Y)\) contents. %) derived from each training is recorded. The process is repeated for one and two hidden layers, and the corresponding results are shown in Figure 5.  
Both figures show that the RMSE tends to decrease as the number of neurons increases. A low number of neurons in both cases leads to a significant error attributed to underfitting. Though some error picks are shown for an increased number of neurons per hidden layer, mainly for the case of two hidden layers (Figure 5b), the error is less than 10% for more than 5 neurons and occasionally is close to zero, for the case with two hidden layers (Figure 5b). It is noted that though it is not clearly depicted for the tested number of neurons per hidden layer, it is expected that for a significant increase in the neurons number, overfitting resulting in poor performance and high RMSE may also arise. As shown in Figure 5, the lowest values of error arise in the ANN with two hidden layers and various numbers of neurons, such as 8, 9, 10, 12, 13, 15, 16, 18, 19 (Figure 5b). Thus, for the subsequent results, training of an ANN with two hidden layers and 13 neurons per hidden layer is considered. For the first ANN, thus the one with five input parameters and one output (the percentage of vibration reduction achieved by the shunted piezoelectric system _PI_), the error derived from the training, validation, and testing of the model with two hidden layers and 13 neurons per layer, is shown in Figure 6.  
Figure 5: Parametric investigation on the influence of the number of neurons per hidden layer for the ANN model (**a**), one hidden layer (**b**), and two hidden layers.