*4.4. Data-Based Modelling*

The die temperatures measured by the extruder thermocouple are all around 200 °C, 4.4.1. Modelling of In-Line Measures with Machine-Learning Methods

while the output temperatures measured with a manual thermocouple are much higher and more scattered. It is often a problem with extruder thermocouples which, being placed on the walls of the die, are influenced by its temperature and do not measure the This part presents the results obtained by Machine-Learning (ML) method on in-line measured parameters, which correspond to the parameters measured directly during the extrusion without needing additional experiments.

actual melt temperature. Concerning the simulation results, the first thing to notice is that all temperatures seem to be overestimated by the software, which matches the fact that polyethylene degradation is not considered. The actual viscosity decreases along with the screws and causes less self-heating than what could be expected without degradation. The temperature is, in fact, closer to the one imposed by the extruder. This viscosity error also causes an overvaluation of the pressure in the die, more accentuated for UHMWPE because of its high viscosity. Concerning the torque and the engine power, and in the case of HDPE, the experimental values match pretty well with the simulation. It can be surprising considering the error between simulated and experimental Figures 10 and 11 show the results obtained for centre and exit temperatures (manual thermocouple), torque, engine power and exit pressure for SVR and sPGD methods, respectively. To obtain these results, the model obtained after training with SVR regression has been applied on inputs and these figures represent the resulting outputs compared with the measured ones. Blue dots correspond to the data used for training and constructing the model, whereas the red star ones represent data that are new for the model as they have not been used for the training. Regarding these results, it appears that both methodologies give acceptable results as the dots are well distributed along the *x* = *y* line and relatively close to it.

viscosities caused by degradation. However, as viscosity decreases along the extruder, we can think that the torque value is mainly ruled by the most viscous part, which is the raw

**Figure 10.** Results of the outputs predicted by the SVR method versus the experimental ones. The blue dots correspond to training data, and the red stars correspond to test data for HDPE and UHMWPE indiscriminately. **Figure 10.** Results of the outputs predicted by the SVR method versus the experimental ones. The blue dots correspond to training data, and the red stars correspond to test data for HDPE and UHMWPE indiscriminately. *Polymers* **2022**, *14*, x FOR PEER REVIEW 16 of 23

polyethylene present in the first screw elements and not yet degraded. The torque is then ruled by the viscosity of raw polyethylene, which is the one implemented in Ludovic® (SC-Consultants, Saint-Etienne, France) The ability of machine learning algorithms to

This part presents the results obtained by Machine-Learning (ML) method on in-line measured parameters, which correspond to the parameters measured directly during the

Figures 10 and 11 show the results obtained for centre and exit temperatures (manual thermocouple), torque, engine power and exit pressure for SVR and sPGD methods, respectively. To obtain these results, the model obtained after training with SVR regression has been applied on inputs and these figures represent the resulting outputs compared with the measured ones. Blue dots correspond to the data used for training and constructing the model, whereas the red star ones represent data that are new for the model as they have not been used for the training. Regarding these results, it appears that both methodologies give acceptable results as the dots are well distributed along the *x = y* line and

make better predictions than classic simulation is studied in what follows.

4.4.1. Modelling of In-Line Measures with Machine-Learning Methods

extrusion without needing additional experiments.

*4.4. Data-Based Modelling* 

relatively close to it.

**Figure 11.** Results of the outputs predicted by the sPGD method versus the experimental ones. The blue dots correspond to training data, and the red stars correspond to test data for HDPE and UHMWPE indiscriminately. **Figure 11.** Results of the outputs predicted by the sPGD method versus the experimental ones. The blue dots correspond to training data, and the red stars correspond to test data for HDPE and UHMWPE indiscriminately.

errors but that the sPGD can be more precise for most parameters, particularly for the die pressure. On the contrary, exit temperature is a little bit more precisely modelled with

**R2 Error Centre Temperature Exit Temperature Torque Engine Power Die Pressure**  SVR train 0.93 0.93 0.91 0.93 0.92 SVR global 0.92 0.88 0.8 0.92 0.75 sPGD train 0.99 0.91 0.82 1 0.98

global 0.99 0.88 0.71 0.99 0.84

Finally, either of these methods gives better results than the classical Ludovic® (SC-Consultants, Saint-Etienne, France) software model and good predictions without the need to understand the physical phenomena involved in the extrusion process. However, if this last point appears to be an advantage in favour of this method, it should be noted that one should be wary of it because the algorithm can model data that are false in the absolute. For example, the measured exit and centre temperature are very probably underestimated. Both algorithms, however, succeed to predict them, which proves that there is a logic between input parameters and these values. Nevertheless, they do not prevent

In order to have more precise comparison tools, R2 scores were calculated on the results and presented in Table 4. The closer the score is to 1, the closer the model is to measures. Whereas obtaining a good score for training data is accessible, obtaining it for

**Table 4.** R2 scores errors for SVR and sPGD methods.

eventual systematic errors in the measurements.

SVR.

sPGD

In order to have more precise comparison tools, R<sup>2</sup> scores were calculated on the results and presented in Table 4. The closer the score is to 1, the closer the model is to measures. Whereas obtaining a good score for training data is accessible, obtaining it for both training and test data is trickier. This table shows that both methods give acceptable errors but that the sPGD can be more precise for most parameters, particularly for the die pressure. On the contrary, exit temperature is a little bit more precisely modelled with SVR.


**Table 4.** R 2 scores errors for SVR and sPGD methods.

Finally, either of these methods gives better results than the classical Ludovic® (SC-Consultants, Saint-Etienne, France) software model and good predictions without the need to understand the physical phenomena involved in the extrusion process. However, if this last point appears to be an advantage in favour of this method, it should be noted that one should be wary of it because the algorithm can model data that are false in the absolute. For example, the measured exit and centre temperature are very probably underestimated. Both algorithms, however, succeed to predict them, which proves that there is a logic between input parameters and these values. Nevertheless, they do not prevent eventual systematic errors in the measurements.

#### 4.4.2. Modelling of Viscosity and Molecular Weight

One of the main interesting aspects of data-based simulations is that there is no need to understand the physics behind the measurements to obtain predictive models of these results. Consequently, it is a less time and power-consuming way to obtain predictions for any measurements, as long as the correct inputs are given. ML methods can also succeed in predicting values depending on unknown phenomena. However, they work as black boxes and cannot help to understand these phenomena.

For example, in this work, understanding the degradation mechanisms sufficiently to predict the final molecular weights of the material seems out of reach. On the contrary, predicting these values with ML methods seems entirely feasible. Figure 12 presents the results obtained with SVR and sPGD methods when predicting zero-shear viscosity *η*<sup>0</sup> and weight and number average molecular weights *Mw* and *Mn*. Table 5 presents the R<sup>2</sup> scores obtained, indicating the precision of these methods.

Both methods successfully model *Mw* values with reasonably high precision, but *Mn* values predictions are less accurate. *Mw* and *Mn* values implemented in the software are deduced from the viscoelastic behaviours of the melt samples. It has previously been shown that this method is more accurate for predicting *Mw* than *Mn*. This fact can explain the significant error noticeable for some of the data. Concerning the viscosity, the results obtained with the SVR method are pretty bad, and the predicted values seems shifted from the real ones. The cause of this phenomenon is unclear since the algorithm succeeds to predict *Mw* which, as seen previously, can directly be related to viscosity. sPGD algorithm presents similar R<sup>2</sup> scores for viscosity, but less shifted values, which shows that the choice of the regression method is crucial.

4.4.2. Modelling of Viscosity and Molecular Weight

obtained, indicating the precision of these methods.

black boxes and cannot help to understand these phenomena.

One of the main interesting aspects of data-based simulations is that there is no need to understand the physics behind the measurements to obtain predictive models of these results. Consequently, it is a less time and power-consuming way to obtain predictions for any measurements, as long as the correct inputs are given. ML methods can also succeed in predicting values depending on unknown phenomena. However, they work as

For example, in this work, understanding the degradation mechanisms sufficiently to predict the final molecular weights of the material seems out of reach. On the contrary, predicting these values with ML methods seems entirely feasible. Figure 12 presents the results obtained with SVR and sPGD methods when predicting zero-shear viscosity *η0* and weight and number average molecular weights *Mw* and *Mn*. Table 5 presents the R2 scores

**Figure 12.** Results of zero-shear viscosity and weight and number average molecular weights predicted by SVR and sPGD methods versus the experimental ones. The blue dots correspond to training data, and the red stars correspond to test data for HDPE and UHMWPE indiscriminately. **Figure 12.** Results of zero-shear viscosity and weight and number average molecular weights predicted by SVR and sPGD methods versus the experimental ones. The blue dots correspond to training data, and the red stars correspond to test data for HDPE and UHMWPE indiscriminately.


**Table 5.** R2 scores errors for the determination of *η0*, *Mw* and *Mn* with SVR and sPGD methods. **Table 5.** R 2 scores errors for the determination of *η*<sup>0</sup> , *Mw* and *Mn* with SVR and sPGD methods.

#### Both methods successfully model *Mw* values with reasonably high precision, but *Mn* 4.4.3. Stochastic Models

values predictions are less accurate. *Mw* and *Mn* values implemented in the software are deduced from the viscoelastic behaviours of the melt samples. It has previously been One problem with data-driven models is that their accuracy depends on the accuracy of the data, which necessarily includes inaccuracies due to measurement techniques. Stochastic models allow considering probability curves instead of points as data, thus smoothing out these inaccuracies and generally simplifying the model. The SVR method coupled with the stochastic approach was tested on ten data corresponding to HDPE extruded at Tmax = 390 ◦C. The results are shown in Figure 13.

As there are only two inputs for these data (Screw rotation speed and Flow rate), the results can be plotted in 3D graphs. The middle surface corresponds to the predictions, and the translucent ones correspond to the superior and inferior acceptation boundaries. Despite the limited amount of data, the method gave satisfactory results. Therefore, this method is promising and needs to be tested with the different polymers and temperatures as inputs and other outputs such as viscosities or molecular weights to see if the stochastic approach can make an improvement.

truded at *Tmax* = 390 °C. The results are shown in Figure 13.

choice of the regression method is crucial.

4.4.3. Stochastic Models

**Figure 13.** Results obtained with the stochastic method coupled to SVR. The dots correspond to training values, and the stars correspond to test ones.

shown that this method is more accurate for predicting *Mw* than *Mn*. This fact can explain the significant error noticeable for some of the data. Concerning the viscosity, the results obtained with the SVR method are pretty bad, and the predicted values seems shifted from the real ones. The cause of this phenomenon is unclear since the algorithm succeeds to predict *Mw* which, as seen previously, can directly be related to viscosity. sPGD algorithm presents similar R2 scores for viscosity, but less shifted values, which shows that the

One problem with data-driven models is that their accuracy depends on the accuracy of the data, which necessarily includes inaccuracies due to measurement techniques. Stochastic models allow considering probability curves instead of points as data, thus smoothing out these inaccuracies and generally simplifying the model. The SVR method coupled with the stochastic approach was tested on ten data corresponding to HDPE ex-

#### **5. Conclusions**

HDPE and UHMWPE were degraded by twin-screw extrusion under different high temperatures (320 < *T* ◦C < 420) and for different process conditions (flow rate and screw rotation speed), leading to numerous different extrusion configurations. Several parameters were measured for each configuration, creating a dataset with four different inputs, five outputs, and thirty-eight data.

The shear viscosity curves of the extruded materials were estimated from the measured die pressure and temperature. Their comparison with frequency sweep measurements showed that despite the numerous simplifications, the results were accurate. This fact shows that this method can be used to rapidly obtain an approximation of the final zeroshear viscosity of extruded materials.

Two methods were tested to estimate the molecular weight of extruded polyethylene. One was based on the viscoelastic behaviour of the material, and the other was deduced from die pressures and temperatures. The results showed that the average molecular weight *Mw* values were similar for both methods and similar to those obtained by SEC for the five samples tested. The method determining *Mw* only from measured die pressure and temperature thus seems more advantageous because it does not involve offline characterizations. However, this method is not sufficient to obtain the complete molecular weight distribution. In contrast, the other method based on viscoelastic measurements determined the complete molecular weight distribution. The results were good for HDPE but with some inaccuracies for UHMWPE samples. Although, as with the SEC, it requires offline characterizations, it is faster and is an interesting alternative.

The Ludovic® (SC-Consultants, Saint-Etienne, France) twin-screw extrusion simulation software was used as a classical model of the extrusion experiments. Since the degradation mechanisms occurring in the extruder are unknown, the simulation was performed considering the viscosities of the raw materials, which led to overestimated pressures and temperatures. Consequently, SVR and sPGD Machine-Learning methods were applied to the dataset and succeeded in modelling the extrusions' torque, engine power, die pressure, and die and centre temperatures. They also gave good results for the predictions of *Mw*. *Mn* has also been successfully predicted but with more inaccuracies, probably caused by its method of determination. Besides, whereas the SVR method gave inaccurate results for zero-shear viscosity modelling, sPGD's results were more acceptable. Finally, stochastic methods were tested on ten of the data giving promising results.

Machine-Learning seems to be a valuable tool for extrusion simulation as it is possible to obtain quickly accurate models. However, it is essential to keep in mind that ML methods cannot be used as predictive tools and also that the accuracy of the results depends on the accuracy of the data. In perspective, it could be interesting to think about the scale-up of this process and about how machine-learning could be a helpful tool for this purpose. Few experiments on a larger scale could then be necessary to adapt the whole model to it.

**Author Contributions:** Conceptualization, P.C. and F.C. (Francisco Chinesta); methodology, F.C. (Fanny Castéran) and K.D., software, F.C. (Fanny Castéran), N.H. and A.A.; validation, F.C. (Fanny Castéran). All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Data Availability Statement:** Data is contained within the article.

**Acknowledgments:** Authors acknowledge ESI Group by its research chair at Arts et Metiers Institute of Technology and the French ANR through the DataBEST project.

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

#### **Appendix A. Detail of Data's**


**Table A1.** In-line measurements for the different tested extrusion parameters.


**Table A1.** *Cont.*

**Table A2.** Ludovic® results obtained for the different tested extrusion parameters.



**Table A2.** *Cont.*

**Table A3.** Viscosities and molecular weights for the different tested extrusion parameters.



**Table A3.** *Cont.*

#### **References**

