## 3 Results and Discussion  
The results of the comparative study are listed in Table 5 for the respective runs. In addition to the presented error measures, the training loss of the corresponding epoch with the lowest validation loss was added. As expected, a larger dataset tends to be associated with an increase in prediction accuracy, which can be seen from the decreasing mean DTW distance of the SOC of the validation set. Furthermore, an increase in the accuracy of the DTW distances of the validation sets is often coupled with an increase in the predictive accuracy of the virtual experimental dataset. However, this ratio does not show a linear  
Figure 5: Schematic illustration of a Bayesian neural network. relationship, since the error measure of the experimental dataset exhibits high variations due to the sample size of unity. This sample size was chosen because the experimental dataset represents the actual use case of the MPI and therefore consisted of only one set of material parameters. This also demonstrates a fundamental disadvantage of the direct inverse MPI compared to the iterative optimization-based method, since hereby the mean error of a training or validation set was minimized rather than the error of the present MP set. This again emphasizes the importance of the network's generalization capability, which brought both errors closer together on average. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Run** & **NN** & **Dataset** & **Loss** & **Stopped** & **Best Val. Loss (-)** & **Cor. ** & **MSE MP** & **Mean** & **Mean** \\  & & & **Epoch** & **Loss (-)** & **Training Loss (-)** & **Val. Set (-)** & **DTW SOC** & **DTW SOC** \\ \hline  
NN\_Yang & 3 & COL & 668 & 0.00144 & 0.00122 & 0.0727 & 12.014 & 10.555 \\
22 & CNN\_[PERSON] & 3 & COL & 132 & 0.00086 & 0.00051 & 0.0481 & 3.908 & 3.835 \\
23 & CNN\Based on the validation loss and the corresponding MSE of the material parameters of runs 5 to 12, it can be seen that for the investigated dataset, the sampling types individually or complete of the selected LHS do not have a significant influence on the prediction accuracy. Using the newly defined custom loss function (COL), the prediction accuracy could be increased for the larger datasets, 3 and 4, concerning the DTW distance of the SOC of the validation set. However, this does not apply to the smaller datasets, and therefore, the performance of COL should be investigated more comprehensively in further studies. In general, across all network types, a tendency can be observed that more epochs are needed for training when larger datasets are available, which may result from the network's ability to adjust its parameters more precisely due to a higher number of epochs. As expected, the MLP hyperparameter optimization also led to an increase in prediction accuracy. In contrast, for the CNN, this only led to slight minimization of the validation loss and the MSE of the MP, along with a simultaneous minor increase in the DTW distances of the SOC. A reason could be that the hyperparameters and network architecture of the Default_CNN are already very well suited for the present dataset. Furthermore, the data augmentation with additional noise amounts was successful, which can be seen in the reductions in the DTW distances of runs 14 and 31 compared to the corresponding runs 7 and 19, which each contained the same settings and no additional noise amounts. This is valid for the DTW distance of both the validation set and the experimental set, providing a significant advantage for the final application of direct inverse MPI on experimental datasets. Run 14, with additional noise amounts and Default_CNN of [PERSON] et al. [58], achieved the best results of the entire study, despite not being trained on the largest dataset, 4. However, data augmentation with empirical mode decomposition did not prove to be effective. The reason might have been the implementation into their MPI framework.