## 3 Results and Discussion
### Virtual Sample Generation Technology  
In the previous study on active learning, two small datasets and various ML models with MAPE values less than 10% were obtained. Then, virtual samples were generated based on the sample feature distributions in the two updated small datasets to further enhance the precision of these models. The GBDT models trained under two sampling routes both demonstrated the lowest MAPE within their respective active learning processes, so they were used to predict the mass ablation rate of the generated virtual samples in their respective routes. The labeled virtual samples were then mixed with the real samples to form mixed datasets. The reconstructed datasets were trained by each algorithm, and the predictive ability of the new model was still verified by samples in the test set. After training a model with sufficiently high prediction accuracy, using the contents of fillers as inputs, reliable predictions of the mass ablation rate of new samples can be obtained.  
At present, there is no conclusion on the optimal number of virtual samples or the influence of the number of samples on the performance. Therefore, it is necessary to discuss the effect of introducing different numbers of virtual samples. Hence, 50 to 500 virtual samples at 50 intervals for each route were generated, and the performance of models trained on mixed datasets with different sample sizes was evaluated. The MAPE of the models' test sets in these cases is shown in Figure 9.  
For the small dataset resulting from the exploration strategy, it can be observed that after the introduction of 300 virtual samples, the MAPE of the GBDT model has significantly reduced, with the percentage error on the test set decreasing by approximately 25% to 3.1%. The AdaBoost model also achieves certain performance improvements in most cases when virtual samples are introduced, although the improvement is not significant. However, the performance of the RF model decreases after the introduction of a certain number of virtual samples, which may be attributed to the fact that the MAR value of virtual samples is generated by the GBDT model. The understanding of the data by RF differs from that of the boosting model, leading to a decrease in performance. By comparison, the influence of introducing virtual samples is more pronounced in enhancing the performance of models trained on the small dataset generated through EI sampling, as depicted in Figure 9e-h.  
Figure 8: (**a,b**) SHAP-based analysis of the importance of sample features; and (**c**) predicted mass ablation rate for all sample points in feature space.  
After the introduction of 400 virtual samples, the averaged MAPE of repeating modeling by the XGB algorithm decreased to 2.48%. In general, the introduction of virtual samples can indeed further optimize the predictive ability of ML models.  
Figure 9: **(a)** MAPE of the test set for the GBDT model under exploration route, **(c)** MAPE of the test set for the AdaBoost model under exploration route, **(d)** MAPE of the test set for the XGB model under exploration route, **(e)** MAPE of the test set for the GBDT model under EI route, **(f)** MAPE of the test set for the RF model under EI route, **(h)** MAPE of the test set for the AdaBoost model under EI route, and **(h)** MAPE of the test set for the XGB model under EI route.  
It is worth noting that the labelled values of the generated virtual samples are calculated by the resulting model of active learning. The model itself is in error in its prediction of ablation rates, so the generated virtual samples may suffer from the model's misinterpretation of the filler content-mass ablation rate relationship. They eventually deviate from the true sample distribution and become noise in the training set, which may affect the predictive ability of the mass ablation rate prediction model.