*5.2. Comparison to Polynomial Regression, Support Vector Machines and Neural Network Models*

For the purpose of comparing our results and trained models with the other models described previously, Table 5 shows the different predictions generated by the models for the previously unknown dataset in our test study.


**Table 5.** Comparison of the predictions of the different models used in this contribution.

It is shown that only the GPR model is capable of producing meaningful outputs, while the other models are not able to achieve a prediction quality close to the GPR model. When comparing the training results on root mean squared error, mean absolute error and coefficient of prognosis set, the story becomes even more clearer (see Table 6).

**Table 6.** Comparison of the prediction qualities of the models on the unknown data set.


The results show that the GPR model was the best model compared to PR, SVM and NN. It is worth noting that we have used polynomial degree of 2 for the PR models, as this produced the best prediction results, a higher polynomial degree of 3 to 5 led to a decrease in RMSE, MAE and CoP. This also shows that especially the SVM and NN are not capable of producing meaningful prediction output. The PR fitting overall shows acceptable prediction quality of around 70%, however the GPR has better RMSE and MAE values, so it would be selected for further consideration. Furthermore, GPR provided better results on the training dataset. It is important to always evaluate RMSE, MAE and CoP together, as all three values allow a thorough evaluation of the prediction model. In brief, RMSE and MAE characterize the spread predictions better than the CoP, the CoP returns an overall performance score of the model. The weak performance of SVM can possibly be explained by the small dataset used for training, since SVM need way more training data, as the model only scores around 30% CoP on the training dataset. For extrapolation on the test dataset the trained SVM model was not feasible. The same could be the case for the NN, as NN rely on big datasets for training and show weaker extrapolation capabilities.

#### **6. Conclusions**

This contribution evaluated the use of Gaussian processes and advanced data visualization in the design of amorphous carbon coatings on UHMWPE. This study focused on elaborating an overview of the required experimental setup for data generation and

the concepts of ML, and also provided the corresponding ML algorithms. Afterwards, the deposition and characterization of amorphous carbon coatings were presented.

The use of ML in coating technology and tribology represents a very promising approach for the selective optimization of coating process parameters and coating properties. In particular, this could be demonstrated by the GPR models used to optimize the mechanical properties of the coatings and, consequently, the tribological behavior, by increasing the hardness and thus the abrasive wear resistance. However, further experimental studies and parameter tuning are needed to obtain better predictive models and better process maps. The initial results of these visualizations and the GPR models provide a good basis for further studies. For our approach the following conclusions could be drawn:


For our use case we implemented a four-step process, mainly consisting of data generation via design of experiments to create the initial dataset. This initial dataset was then analyzed via Python-based scripting tools, to create meaningful prediction models via GPR. Those GPR models are then used for the presented visualization approach. To put it all together one Python script was created to lead through the process. This Python script can be configured to look into different values, however we focused on indentation hardness.

Based on this work, further experimental studies will be conducted, and the proposed models will then be re-trained using the available data. The dataset generated for this article was considered as a starting point for the ML algorithms used and will be supplemented with future experimental data and thus grow. When more data is available, maybe different ML models like neural networks will come into perspective.

**Author Contributions:** Conceptualization, C.S. and B.R.; methodology, C.S. and B.R.; specimen preparation B.R.; coating deposition, B.R.; mechanical experiments, B.R.; data analysis, C.S., B.R. and N.P.; writing—original draft preparation, C.S. and B.R.; writing—review and editing, N.P., M.B., B.S. and S.W.; visualization, C.S. and B.R.; supervision, M.B., B.S. and S.W. All authors have read and agreed to the published version of the manuscript.

**Funding:** We acknowledge financial support by Deutsche Forschungsgemeinschaft and Friedrich-Alexander-Universität Erlangen-Nürnberg within the funding programme "Open Access Publication Funding".

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** For tinkering with the visualization a similar model can be found under http://csmfk.pythonanywhere.com/ (accessed on 1 February 2022). The related data generated and analyzed for the contribution is available from the corresponding author on request.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**

