## Appendix B Interpolation-Based Data-Augmentation Protocol for Expanding Sparse Process Parameters
\] (A18)  
where \(X_{i}\) is the input feature value at the interpolation point \(i\), \(X_{k}\) is the coordinate vector (input feature values) of the \(k\)-th known point, \(Y_{k}\) is the dependent/output variable value at the \(k\)-th known point, and \(Y_{i}\) is the interpolated output feature value at the interpolation point. It is worth mentioning that interpolation methods still have certain limitations and drawbacks. First, nearly all the interpolation techniques are highly sensitive to noise in the input data. Even small errors in the known data points can be amplified in the interpolation results [38]. Additionally, linear interpolation and cubic spline interpolation typically assume smooth transitions between data points, which may fail to capture abrupt changes or nonlinear behavior present in the actual data [39]. For IDW, unevenly distributed data can introduce bias, as distant points may have disproportionately high weights [40]. To mitigate these issues, an active learning strategy is implemented to filter and select interpolated data points with minimal noise for further processing. Before executing active learning, all the data is divided into a sample pool, validation set, and test set. The validation and test sets consist of the original 48 simulations (40 CFD simulation data for model validation and 8 CFD simulation data for final testing), while the sample pool is composed entirely of the interpolated data. To ensure the interpolated data remains sufficiently distinct from the validation and test sets, two constraints are applied during the data generation. Firstly, the X (input feature) value for interpolation is determined based on a uniform step size, which is set such that each interpolated \(\mathrm{x_{m}}\) (the m-th element in the input feature) is at a 2% difference from the original boundary input feature values. This step size is used to sample all the X-values. Secondly, during the interpolation process, Gaussian noise is added to the corresponding y-values with a probability of 5% (Equation (A19)). These measures introduce meaningful variability into the interpolated data, thereby reducing the risk of overfitting and data leakage. \[ \begin{array}{c}\delta\sim N\big{(}0,\sigma^{2}\big{)}\\ \sigma=|Y_{end}-Y_{start}|\times 5\%\\ Y_{i^{\prime}}=Y_{i}+\delta\end{array} \] (A19)  
where \(\delta\) is a normally distributed random variable, \(\sigma\) is the standard deviation, \(\sigma^{2}\) is the variance, \(Y_{i}\) is the interpolated data before adding noise, and \(Y_{i^{\prime}}\) is the interpolated data after adding noise. Based on these constraints, a total of 3040 interpolated datasets were generated. The predicted temperature distribution of particles on the substrate, considering a mask with a wire diameter of 0.89 mm, an opening size of 1.22 mm, and an open area of 33%, along with a substrate SOD of 10 mm and a mask SOD of 4 mm, under inlet conditions of 2.73 MPa and 619 \({}^{\circ}\)C, as well as the endpoints used for the interpolations, are presented in Figure A4. The process of refining and extracting interpolated data points with minimal noise through an active learning strategy involves several stages. The detailed workflow is illustrated in Figure A5. First, during the initialization phase, a validation set and an unlabeled data pool (candidate sample pool) are prepared. Then, a random subset of samples is selected from the sample pool to construct the initial labeled dataset for model training. To ensure randomness, the candidate sample pool is shuffled before selection. Next, the model is trained using the initialized labeled dataset, and the initial baseline error, measured as the Mean Squared Error (MSE), is calculated using the validation set. After the initialization phase, the process enters an iterative active learning stage. In each iteration, a small batch of candidate samples is temporarily added to the labeled dataset, and the model is retrained using this expanded set.