# Machine Learning Based Surrogate Models for the Thermal Behavior of Multi-Plate Clutches
## 3 Results
### Use Case 2--Axial Force + Rotational Speed + Lining Thickness  
The results of the 10-fold cross-validation for Use Case 2 are shown in Figure 11. The PR and SVR showed very similar behavior and performance. The RMSE values of the individual folds for both models were similar, and the average RMSE values amounted to 26.67 and 27.59, respectively. The DT performed better, with an average RMSE score of 20.20. However, there was a large variance between the individual folds values, ranging from 13 to 27. This higher variance was also evident in the case of the Gaussian process, in which the values varied between 11 and 26. However, among these four models, the GP achieved the best performance during cross-validation with an average RMSE score of 18.03. Figure 12 presents the training history of the BPNN for Use Case 2. The training was performed with the same settings as in Use Case 1. As in the first use case, the RMSE for both the training set and the validation set converged. After about 850 epochs, an RMSE of 6.94 was achieved for the training set. The validation loss (RMSE) amounted to 6.25. As was similar to Use Case 1, the BPNN did not demonstrate any overfitting issue. Table 6 shows the performance of the created methods on the test set. The performance of the models follows the order in cross-validation on the training dataset. Both polynomial regression and SVR showed the worst results, with RMSE values of approximately 24. The decision tree model performed slightly better, with an RMSE score of 19.71. Similar to the first use case, the Gaussian process and backpropagation neural network performed best, with RMSE values of 14.43 and 6.31, respectively. Whereas the SVR had the worst RMSE value, a better MAPE value than that of the PR was achieved. Although the GP performed better than the DT in terms of RMSE, both achieved similar MAPE values of about 2.2%. In terms of both RMSE and MAPE, the BPNN achieved the best performance. Figure 11: RMSE for 10-fold cross-validation of the PR, DT, SVR and GP models for Use Case 2. In the example of individual slip cycles in Figure 13, the same behavior was seen for the PR and the SVR as for the SVR in Use Case 1. The general increase in temperature was modeled, but the temperature fluctuations were not captured. This effect was also present in a weakened form in the case of the GP. The DT modeled properly models the oscillatory temperature rises, but exhibited a deviation in the range between 5 and 10 s. As in the first use case, the predicted and true temperatures matched best in the case of the BPNN. Figure 14 illustrates the model's performance dependence as a function of the amount of data for Use Case 2. For the PR model it was evident that almost zero correlation was present. Only small fluctuations in the range between RMSE scores of 25 and 30 took place. The SVR modeled performs worse than the PR model for up to 50 simulations. Above 50 simulations, the performance of the two models was very similar. The decision tree model showed the worst performance at a low number of simulations but improved continuously when increasing the amount of data. Starting with about 100 simulations, an RMSE score of about 20 was achieved. Larger amounts of data did not exhibit any further significant improvement. As with Use Case 1, the Gaussian process showed a high dependence on the amount of data. A steady improvement of the results was observed with an increase in the data volume. The best performance was consistently achieved for all data volumes by the BPNN models. Better results than those for the PR and SVR were already achieved with only 10 simulations. From about 60 simulations onwards, there was a further significant improvement in performance. Starting from 100 simulations, a further improvement took place, but only to a small extent.