## 5 Transfer Learning-Assisted Dynamic Evolutionary Algorithm
### Transfer Learning-Assisted Initial Population Generation  
We take the objective space in the current environment as the target domain, and then we select the objective space in the historical environments with the slightest discrepancy in the marginal and conditional distributions from the target domain as the source domain. Our idea is to use BDA to find an optimal feature transformation so the marginal and conditional distribution discrepancies between domains are as small as possible in the latent space obtained by the transformation. Furthermore, each nondominated solution in the OPF of the source domain is re-represented in the latent space. Therefore, the corresponding solution found in the target domain can be as close to it as possible after being transferred to the latent space. These solutions are used to construct the initial population. The diversity of samples in the source and target domains is important to reduce the probability of negative transfer and improve the quality of the initial population. Therefore, unlike traditional random sampling, we perform sample presearch based on MOEAs to increase the diversity of the samples. The sample distribution can represent the actual distribution of domains as much as possible [47]. BDA is adopted to learn the feature transformation once the domain sampling is completed, inspired by [PERSON] et al. [46]. In the D-HRDLB problem, the source and target domains have different marginal and conditional distributions, but the distributions between the domains are correlated. In addition, the marginal and conditional distributions are not equally important for knowledge transfer. Our goal is to find an optimal feature transformation that makes the difference in the interdomain weighted distribution based on the marginal and conditional distributions as small as possible. A nonparametric distance estimation method called maximum mean discrepancy (MMD) differentiates distributions in the reproducing kernel Hilbert space (RKHS). The weighted distribution discrepancy based on MMD is defined as follows. \[ \begin{split}\text{MMD}(D_{s},D_{t})&\doteq\ (1-\mu)\cdot\left|\left|\frac{1}{m}\sum_{x_{i}\in D_{s}} \phi(x_{i})-\frac{1}{n}\sum_{x_{j}\in D_{t}}\phi(x_{j})\right|\right|^{2}\\ &+\mu\cdot\sum\limits_{c=0}^{1}\left|\left|\frac{1}{m_{c}}\sum_{ x_{i}\in D_{s}^{c}}\phi(x_{i})-\frac{1}{n_{c}}\sum_{x_{j}\in D_{t}^{c}}\phi(x_{j}) \right|\right|^{2}_{\mathcal{H}}\end{split}  \tag{24}\]  
where \(\phi(. ):\mathcal{D}\rightarrow\mathcal{H}\) represents the mapping from the objective space (\(\mathcal{D}\)) to the RKHS (\(\mathcal{H}\)); \(m\) and \(n\) represent the number of samples in the sample sets of the source domain (\(D_{s}=\{x_{1},\,\ldots,\,x_{m}\}\)) and the target domain (\(D_{t}=\{x_{1},\,\ldots,\,x_{n}\}\)), respectively; \(c\in\{0,1\}\) represents the category of solutions, where class 0 and class 1 represent promising solutions and wrong solutions, respectively; \(D_{s}^{c}\) is the sample set belonging to class \(c\) in \(D_{s}\); \(D_{t}^{c}\) is the sample set belonging to class \(c\) in \(D_{t}\); \(m_{c}\) and \(n_{c}\) denote the number of samples in \(D_{s}^{c}\) and \(D_{t}^{c}\), respectively; and \(\mu\in[0,\;1]\) denotes a balance factor that adjusts the importance of the marginal and conditional distributions. In Formula (24), the first and second terms represent the marginal distribution discrepancy and the conditional distribution discrepancy between the domains, respectively. By further taking advantage of the kernel method and matrix transformation, Formula (24) can be rewritten as:  
\[\text{MMD}(D_{s},D_{t})\ \doteq\ \textbf{tr}\Bigg{(}\textbf{W}^{\text{T}}\textbf{K} \Bigg{(}(1-\mu)\textbf{M}_{0}+\mu\sum_{c=0}^{1}\textbf{M}_{c}\Bigg{)}\textbf{K} ^{\text{T}}\textbf{W}\Bigg{)} \tag{25}\]  
where **tr**(.)