## 3 Data Experiment
### Ablation Experiment  
Both the low- and high-fidelity data were randomly split into training and test sets. The data split was 80:20 training:test for low-fidelity data and 20:80 training:test for high-fidelity data to ensure rigorous data experiments where the high-fidelity data for training is scarce. To avoid scale effects, all the data were normalised into the range of \(-1\) to 1 for machine learning computation. Two ablation experiments (denoted as E1 and E2) were performed to investigate the performance of the proposed multi-fidelity machine learning. The objective of E1 was to validate the efficacy of the multi-fidelity framework so the data fidelity was varied, as listed in Table 4. E1-1 was performed to provide a basic benchmark for machine learning in the studied inverse problem. The other experiments in E1 were designed to validate the efficacy of multi-fidelity machine learning. MLP was adopted as the algorithm in the multi-fidelity framework for E1. The MLP used in this study consisted of three hidden layers given the limited features in the data. For E1, the neuron numbers of each hidden layer were 128, 256, and 128, the activation function was ReLU, and the L2 regularization weight was \(1\times 10^{-3}\). Further, the influences of the typical hyperparameters, namely the neuron number, activation function, and L2 regularization weight, were investigated in E2-1 (see Table 5). During training, the \begin{table}
\begin{tabular}{c c c c} \hline
**No.** & **Model** & **Training Set** & **Test Set** \\ \hline E1-1 & Low-fidelity model & Low-fidelity data & Low-fidelity data \\ E1-2 & Low-fidelity model & Low-fidelity data & High-fidelity data \\ E1-3 & High-fidelity model & High-fidelity data & High-fidelity data \\ E1-4 & Low- and high-fidelity models (multi-fidelity model) & Low- and high-fidelity data & High-fidelity data \\ \hline \end{tabular}
\end{table} 
Table 4: Setup of E1. Figure 9: Comparison of low- and high-fidelity data. learning rate was scheduled by an adaptive moment estimation optimiser with an initial learning rate of \(1\times 10^{-3}\). The maximum number of training steps was set to 1,000,000 and the tolerance for error was defined as \(1\times 10^{-8}\). Other hyperparameters and settings followed the default values provided by the scikit-learn Library [45]. The objective of E2 also included the comparison of different algorithms. The results of multi-fidelity machine learning using RF and XGBoost (denoted as E2-2 and E2-3, respectively) were compared. Key hyperparameters of RF and XGBoost were varied to present comprehensive comparison results, as shown in Table 5. Other settings followed the default values in the scikit-learn Library [45] and XGBoost Library [43]. The data experiments were performed using AMD Ryzen 7 5800H with Radeon Graphics as the computing device. To evaluate the performance of the trained models, the coefficient of determination (\(R2\)) and root mean square error (\(RMSE\)) were used as metrics, measuring the differences between ground truth and prediction. The perlite settlement that was input into the numerical simulation is referred to as the ground truth, for validating the model performance in solving the inverse problem. \(R2\) measures the proportion of variance in the output that is explained by the input of the model. Its value ranges from 0 to 1, where 1 indicates that the model perfectly explains the variability in the data, and 0 means that the model explains none of it. \(RMSE\) measures the average prediction error by taking the square root of the mean squared differences between predicted and ground truth. \(RMSE\) has the same unit as the predicted variable, making it easy to interpret. Since it squares the errors, large deviations have a stronger impact on \(RMSE\), which makes it more sensitive to outliers. A smaller \(RMSE\) indicates more accurate predictions. \(RMSE\) is commonly used to compare different models. \begin{table}
\begin{tabular}{c c c c} \hline
**No.