# Machine Learning Based Surrogate Models for the Thermal Behavior of Multi-Plate Clutches
## 3 Results
### Use Case 1--Axial Force + Rotational Speed  
The RMSE scores of the 10-fold cross-validation for the polynomial regression, decision tree, support vector regression, and Gaussian process are shown in Figure 7. For the PR model, except for the third fold, low RMSE values with a mean of 8.12 were evident without considerable variance. In the case of the decision tree model, an average RMSE score of 17.60 was obtained, with the values of the individual folds ranging from 10 to 23. Similarly, in the case of the SVR, all scores were above 20, and an average RMSE score of 25.97 was achieved. The RMSE values for the cross-validation of the Gaussian process model exhibited the lowest mean value, at 7.41. However, the scores of the individual folds  
Figure 6: Graphical representation of the nested cross-validation procedure. demonstrated a strong variance between themselves. As was the case with the PR, the third fold yielded an RMSE score of over 20, while the first fold, for example, had an RMSE score of only 1.93. Figure 8 presents the training history for the BPNN. The training was performed for 10,000 epochs. Early stopping was used as a threshold to stop the training if the validation loss did not improve for 100 epochs. The learning rate was adjusted during training so that if the validation loss reached a plateau and did not improve for 50 epochs, the learning rate was reduced by half. The plotted curve shows that the training process converged to a RMSE of 3.45 after approximately 1200 epochs. The RMSE for the validation set amounted to 3.61, which indicates that no overfitting occurred. Figure 8: Training and validation loss of the BPNN during training for Use Case 1. Figure 7: RMSE for 10-fold cross-validation of the PR, DT, SVR and GP models for Use Case 1. Prediction on the test data (725 data points) set were performed to verify further the ability of the models to generate accurate predictions based on unused data. The RMSE and the R-squared values for each model for the test dataset are listed in Table 5. The computation time required for the inference is provided as well. As during the training procedure, the BPNN achieved the best RMSE score of 4.29, followed by the GP model, with 6.772. The PR model also achieved a comparatively low RMSE of 9.535, while both the DT and SVR models performed the poorest, with higher RMSE scores of 17.564 and 28.22, respectively. All models yielded MAPE values between 0.51% (BPNN) and 3.69% (PR). The order of performance of the models was analogous to the RMSE values. Concerning the required computing time, the individual algorithms differed slightly. Given its higher complexity, the BPNN required the highest computing time. In this sense, PR and DT were the most efficient algorithms with very short computation times of less than 0.1 s. All inference times, however, fell below the 1.0 s range, but were virtually negligible compared to the more than 1000 s required for the FE simulation. To further illustrate the results and explore the application of the models, the predictions for a selected slip cycle are shown in Figure 9. Overall, all models except the SVR model provided good predictions for the selected circuit. In this particular instance, the GP model almost reproduced the data, whereas the PR and the BPNN exhibited only weak deviations at certain points. In the case of DT, the shape of the curve was correctly mapped, but a weak general shift of the curve to higher temperatures was evident. The model thus provided a higher maximum temperature than the simulated values. Noticeable in the case of the SVR model is that the model has difficulty reproducing the oscillations in temperature. As a result, this model exhibited the worst performance. To investigate the influence of the volume of data, Figure 10 illustrates the obtained RMSE for each model as a function of the number of simulations used. Overall, it became evident that for all algorithms an increase in the amount of data also led to an improvement in performance (even if only a slight one in some cases).