## 2 Methodology
### Active Learning  
Active learning (AL), also known as \"_query learning_\" or \"_optimal experimental design_\", is a machine learning approach that selects the most informative data points to reduce the computational cost of labeling data. In the context of multi-objective optimization (MOO), active learning focuses on reducing the number of expensive constraint function evaluations while ensuring accurate approximations of the Pareto front. This is particularly valuable in engineering design problems where each evaluation might involve costly simulations or experiments. The primary goal of AL is to query only those points where the model exhibits uncertainty, avoiding redundant evaluations and concentrating computational resources on data points that provide the most information. By iteratively querying high-uncertainty points, the model converges more efficiently toward an accurate approximation of the Pareto front. The active learning process begins with a small set of candidate solutions, for which both objective function values and constraint evaluations are available. Based on this training set, a surrogate model is constructed to approximate the feasibility of the candidate solutions, meaning whether they satisfy the constraints or not. In this work, Random Forests (RFs) are employed as the active learner due to their robustness and ability to model complex non-linear relationships [23]. RFs are an ensemble method that aggregates the predictions of multiple decision trees trained on random subsets of the data, providing reliable uncertainty estimates that guide the query process. As new points are queried, the surrogate model is retrained on the updated dataset, improving its predictions and enhancing the search for the Pareto front. The Random Forest model used in this study is specifically designed to perform binary classification for feasibility determination while predicting whether a candidate solution satisfies the constraints or not. This binary classification ensures that only feasible solutions are considered for Pareto front approximation. Uncertainty Quantification for Binary Classification  
The uncertainty in the active learning process is driven by the model's ability to predict whether a candidate solution is feasible, i.e., satisfies the constraints. For each candidate solution, the Random Forest classifier provides a probability distribution over the two classes: feasible (0) and infeasible (1). The uncertainty of a given design point \(x\) can be expressed as  
\[p(1|x) =\frac{\text{Number of trees predicting infeasible for }x}{\text{Total number of trees in the forest}} \tag{1}\] \[p(0|x) =1-p(1|x) \tag{2}\]  
The certainty of the model's prediction is quantified using a confidence parameter \(\gamma\), which measures the difference between the predicted probabilities of the two classes:  
\[\gamma=p(1|x)-p(0|x)=1-2p(0|x) \tag{3}\]  
If the confidence \(\gamma\) is below a predefined threshold \(\beta\), the model considers the prediction uncertain and queries the true feasibility of the candidate solution. A higher threshold \(\beta\) will result in more queries, whereas a lower value will reduce the number of queries but could compromise the accuracy of the predictions. In this study, the balance between exploration and exploitation is controlled by adjusting \(\beta\) and the number of trees in the Random Forest, \(\phi\). By dynamically selecting uncertain points for evaluation, this active learning approach minimizes the number of expensive objective function evaluations while maintaining the diversity and accuracy of the solutions on the Pareto front. An illustrative example is presented to demonstrate the application of the active learning framework. In this case, a dataset comprising 800 instances, equally divided into two classes, is used, as shown in Figure 1. The active learner is first trained on a small subset of these instances.