## 3 Methodology
### A Transferable Meta-Learning Algorithm Frame
The detailed steps of the proposed frame are as follows:  
Input: \(D_{train}=\left\{(x_{1},y_{1}),(x_{2},y_{2}),\ldots,(x_{n},y_{n})\right\}\) is an imbalanced training dataset. n balanced sub-datasets \(D_{j}^{k}(j=1,2,\ldots,n)\) are constructed from \(D_{train}\) by UnderBagging. The statistical features of \(D_{j}^{k}\) are \(SF^{k}=\left[S_{1},SF_{2},\ldots,SF_{R}\right]_{1\times R}(k=1,2,\ldots,K)\), The meta-features of \(D_{j}^{k}\) are \(MF_{k}=\left[S^{k},P_{k}\right](k=1,2,\ldots,K)\), \(P^{k}=\left[p_{1},p_{2},\ldots,p_{R}\right]_{1\times R}\). \(D_{test}=\left\{(x_{1},y_{1}),(x_{2},y_{2}),\ldots,(x_{n},y_{n})\right\}\) is an imbalanced test dataset. n balanced sub-datasets \(D_{j}^{test}(j=1,2,\ldots,n)\) are constructed from \(D_{test}\) by UnderBagging. The statistical features of \(D_{j}^{test}\) are \(SF^{\prime\prime w}=\left[S_{1}^{t},SF_{2}^{t},\ldots,SF_{R}^{\prime}\right]_{ 1\times R}(w=1,2,\ldots,W)\), the meta-features of \(D_{j}^{test}\) are \(MF_{w}^{t}=\left[S^{F^{\prime\prime w}},P_{w}^{t}\right](w=1,2,\ldots,W)\), \(P_{w}^{t}=\left[p_{1}^{t},p_{2}^{t},\ldots,p_{R}^{\prime}\right]_{1\times R}\). The accuracy of \(SEM_{k}\) are \(acc=\left[acc_{1},acc_{2},\ldots,acc_{k},\ldots,acc_{K}\right]^{T}\). The history library (\(HL\)) stores information on similar historical problems solved by optimization algorithms. \(HL=\left\{(SF_{1},sol_{1}^{\prime}),\left(S_{F2},sol_{2}^{\prime}\right), \ldots,\left(S_{F_{o}},sol_{o}^{\prime}\right),\ldots,\left(S_{Fpn},sol_{pn}^{ \prime}\right)\right\}\), where \(SF_{o}\) and \(sol_{o}^{\prime}\) are the statistical features and the optimal solution set information of the \(o\)-th historical problem, respectively. Output: The phases of HEAs. Historical problems with the same number of alloy components, types of learners, number of learners, and number of parameters corresponding to the learners are selected, and the statistical features of these historical problems along with the optimal solution sets of the learners are added to \(HL\). Step 1: \(k=1\), \(t=0\), \(o=1\). Step 2: A selective ensemble model \(SEM_{k}(k=1,2,\ldots,K)\) is constructed by randomly selecting R \(Cart_{i}\) trees from \(Cart_{i}(i=1,2,\ldots,n)\) trained on the \(D_{j}^{k}\). Step 3: The \(acc_{k}(k=1,2,\ldots,K)\) of \(SEM_{k}\) are obtained on the validation dataset \(D_{validation}\). Step 4: \(k=k+1\), if \(k>K\), go to Step 5. Otherwise, return to Step 2. Step 5: The meta-knowledge database is established based on meta-features \(D_{j}^{k}\) and accuracies of \(SEM_{k}\). The input variable \(MF\) and the output variable acc of meta-learner are the values of meta-features and the accuracies of selective ensemble models. Step 6: The meta-learner is trained based on the input variable \(MF\) and output variable \(acc\) in Step 5. Thus, get a trained meta-learner, also known as the meta-model. Step 7: \(t=t+1\). The meta-features \(MF_{w}^{t}\) of \(D_{j}^{test}\) are encoded to form an initial population. The AMWO algorithm iterates a small number of generations (\(\#\)), inputs the values of meta-features to the meta-model in each iteration, and obtains the accuracy of the ensemble model \(SEM\). Step 8: If \(t=\#\), then the solution set is \(Sol_{t}=\left\{sol_{t,1},sol_{t,2},\dots,sol_{t,N}\right\}\), \(sol_{t,n}\) is a solution to the target problem and represents the parameters of Cart trees. Additionally, the solutions and their corresponding statistical features constitute the target domain sample set \(\text{Tar}_{S}=\left\{\left(\text{SF}_{t,1}^{\prime},\text{sol}_{t,1}\right), \left(\text{SF}_{t,2}^{\prime},\text{sol}_{t,2}\right),\dots,\left(\text{SF}_{ t,N}^{\prime},\text{sol}_{t,N}\right)\right\}\). Otherwise, if \(t<\#\), return to Step 7, if \(t>\#\), go to Step 9.