## 3 Results and Discussion
### Active Learning  
There are usually a few labeled samples and many unlabeled samples with known features when faced with an ML task. Labeling all unlabeled samples for modeling is resource-intensive and time-consuming. Instead, choosing and labeling just the samples that significantly contribute to the building of a model that meets the requirements is a more efficient approach [27; 28; 29]. Based on this concept, the pool-based active learning process is illustrated in Figure 3. Initially, a basic model is built with a few labeled samples to predict the MAR. The model is then employed to choose candidates from the pool consisting of unlabeled samples. These candidates are subsequently labeled and added to the training set, then utilized to update the model. This process is repeated until a satisfactory model is obtained. In this paper, the final mean absolute percentage error (MAPE) of the model on the validation set needs to be less than 10%. In the above process, the key is in the selection of the appropriate samples to be prepared. The selection criterion is usually based on the uncertainty of the unlabeled samples. Typically, the higher the uncertainty of a sample, the more information it might encompass [30; 31; 32]. There are two main directions for uncertainty-based sampling: exploitation and exploration. Exploitation involves using the established model to predict samples in the pool and selecting the sample with the highest predicted value for the next round of labeling. The exploration strategy samples by choosing the sample with the highest prediction error from the current model. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Variables** & **Description** & **Type** & **Unit** \\ \hline SiO\({}_{2}\) & Content of SiO\({}_{2}\) whose values & & \\  & range from 5 to 50 & & \\ SiC & Content of SiC whose values & & \\  & range from 2 to 20 & & \\ CF & Content of CF whose values & & \\  & range from 2 to 20 & & \\  & Mass ablation rate of samples & & \\ MAR & after exposure for 30 s under the & & \\  & heat flux of 4 MW/m\({}^{2}\) & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: List of input and output variables for machine learning. In this work, sampling based on uncertainty was carried out as follows [33]. During each round of sampling, the currently labeled dataset is resampled by a bootstrap method several times, equal to the number of labeled samples, forming a sample subset. This subset is then trained, and the resulting model is used to predict the MAR of all unlabeled samples in the pool. Repeating the above steps 1000 times will result in 1000 subsets and then produce 1000 models by training these subsets. Every model predicts for all unlabeled samples and 1000 values are assigned to each unlabeled sample. Hence, each sample has 1000 predicted MARs, and the averaged MAR \(\upmu\) and standard deviation \(\sigma\) of these 1000 predictions are calculated. The exploitation strategy chooses the one with the maximum \(\upmu\), but it is used in this paper to select the samples composing the validation set because of its defect in local optimization. The exploration strategy acquires the sample with the largest \(\sigma\). Expected improvement (EI), a strategy balancing exploitation and exploration, is introduced as another sampling method, whose acquisition is based on the biggest U:  
\[U=\sigma[\varphi(z)+z\Phi(z)] \tag{2}\]  
\[z=\frac{\mu_{max}-\mu}{\sigma} \tag{3}\]  
where \(\varphi(z)\) and \(\Phi(z)\) are the probability density function and the cumulative distribution function of the standard normal distribution, respectively, while \(\mu_{max}\) is the maximum value among 1000 predictions for each sample. Two sampling strategies of exploration and expected improvement were carried out in parallel, with each iteration adding two new samples to the initial dataset. The prediction model of the MAR was updated until the MAPE of the verification samples was less than 10%.