## 3 Methods, Application, and Result Discussion
### Proposed Approach  
Figure 8 is a simplified schematic diagram of the proposed CSIDTL. The CSIDTL algorithmic architecture can be simplified in three particular steps. **Step 1:** A set of homogeneous learning models, namely, LSTM networks (refer to [26] for details on the mathematical background of LSTM networks), are trained in a loop on the different subsets (13 subsets related to operating conditions) of the variable-speed and -load dataset. This means that each LSTM network is trained for the classification process of 7 different operating modes. In the first round, \(k=0\), the LSTM networks' learning parameters are randomly initialized from specific probability distribution \(P\) with mean \(\mu\) and standard deviation \(\delta\) as in (1). \[\left\{w_{k}^{i},\ w_{k}^{r},\ b_{k}\right\}=P(\mu,\delta) \tag{1}\]  
In this case, the 3-fold cross-validation technique is used to train each network per condition in different rounds \(k\). Consequently, there is a total of 13 \(\times\) 3 trained LSTM networks in each round. **Step 2:** Among these 13 \(\times\) 3 models, only a few models are selected to accomplish the training process of LSTM networks in upcoming rounds. The selection process involves some specific criteria (i.e., testing accuracy, in our case) to perform the following SCIDTL rounds. These models are then stored, and their parameters, aggregated and transferred to initialize all deep networks in the following rounds, and so on. It should be mentioned that the number of stored learning models is incremented. This means that in each new round, both the old, selected models and the newest ones are used for the aggregation and initialization of the learning parameters using transfer learning. **Step 3:** Weight initialization involves the aggregation of collected weights under different conditions. This can be referred to as a collaborative training, where the global learning parameters combine information across all different conditions. The transfer learning concept, in this case, appears in both cases of aggregation and weight initialization. Figure 8: Schematic diagram of the proposed CSIDTL approach: (**a**) training a set of learning models separately for each operating condition; (**b**) using the cross-validation technique to evaluate model performance; (**c**) selecting and storing the learning parameters of the best learners based on accuracy; (**d**) unleashing the aggregation process only for selected parameters and determine initial parameters of following rounds. This means collaboration across conditions and the fine tuning of the LSTM network for upcoming rounds. So, if we think of global loss in specific round \(k\) (\(l_{G(k)}\)) as our main objective function to be minimized, \(l_{G(k)}\) can be presented as the minimization of loss in both the source domain and the target domain as in (2). \[l_{G(k)}=\|l_{(k-1)+}l_{(k)}\| \tag{2}\]  
In our case of the LSTM network, from the set of input weights \(w^{i}_{(c,f,k)}\) and recurrent weights \(w^{r}_{(c,f,k)}\) and \(b_{(c,f,k)}\), where \(k=1:m\) is the number of rounds, we only select a few models and limit them to a new set of parameters, \(\left\{w^{r}_{s},w^{r}_{s},\ b_{s}\right\}\), where \(s=1:n\) is the number of selected models incremented in each round. \((c,f)\), in this case, refer to the condition label and the cross-validation fold index, respectively. Afterwards, these collected parameters are aggregated using an averaging method as illustrated in Equations (3) and (4) and used as initial parameters \(\left\{w^{i}_{0},w^{r}_{0},\ b_{0}\right\}\) of the all the models in all following rounds. \(S\) refers to the selected model index; \(n\) is the total number of selected models per round; \(c\) is the number of conditions; \(f\) is the number of folds; \(k\) is the index of a round; and \(m\) the is maximum number of rounds. Subscript 0 refers to initial parameters.