## 3 The Cyclic Evolutionary Network (CENS)
### Model Framework
The prediction data can be obtained through following the data prediction model

\[D^{\prime\prime}\;=\;f\big{(}D^{\prime}\big{)} \tag{1}\]

where \(f\left(\cdot\right)\) represents the data prediction model, \(D^{\prime}\) denotes the original data processed, and \(D^{\prime\prime}\) denotes the prediction data. We choose the GED (GRU encoder-decoder) model as the data prediction model \(f\left(\cdot\right)\). The GED is a model of GRU encoder-decoder with SST code and dynamic influence link (DIL) [12]. It has a good effect in predicting sea temperature, but it cannot provide spatial feature information. In comparison, the evolutionary network framework can cluster points with different spatial features. Therefore, the GED model was chosen as the data prediction model to capture the historical temporal features of the SST and the spatial feature information by the evolutionary mechanism of CENS. The absolute error is calculated by Equation (2)

\[\Delta X\;=\;\big{|}D^{\prime\prime}-D^{\prime}\big{|} \tag{2}\]

After calculating the absolute error, we use these absolute error data to train an error prediction network

\[\Delta X^{\prime}\;=\;e(\Delta X) \tag{3}\]

where \(\Delta X^{\prime}\) denotes the error prediction data. \(e(\cdot)\) is the error prediction model, and we also use the GED as the error prediction model. If the error prediction of the error prediction network is small, the numerical prediction network is considered as able to predict accurately. If the error prediction is large, the numerical prediction network is considered as able to accurately predict. Therefore, the errors of the data provide the basis for the dataset partition mechanism. The structure of the single network is shown in Figure 2.

Figure 2: Structure of a single network unit. First, the time series data are trained by a GED network [20], and the value predicted by the network is subtracted from the original time series data to obtain the residual data. Then, the residual data are trained by another GED network to obtain the final residual prediction result. ### Dataset Partition Mechanism

The dataset partition mechanism is applied to split the dataset into evolution and elimination sets at each level and then move the elimination set into the next level of the network for continued partitioning. The elimination set consists of those data elements whose characteristics do not match those of the majority of the data, as indicated by large prediction errors in that network. Thus, these data need to be processed using other networks. In this way, the dataset partition mechanism divides the raw data into evolutionary and elimination sets based on the error values obtained in the single network units. After training on a single network, the dataset is divided according to the data partition criteria

\[\text{Data }=\ g\big{(}t_{e},\Delta X^{\prime}\big{)} \tag{4}\]

where \(t_{e}\) denotes the dividing standard line, \(g(\cdot)\) indicates the data partitioning rule that is based on the error mean value. \(\Delta X^{\prime}\) refers to the set of error datasets, and Data denotes the evolution set and elimination set of the original data. Data can be divided into two parts, which can be expressed as

\[\text{Data }=\ \{E,W\} \tag{5}\]

where \(E\cup W=\text{Data}\), \(E\) is the evolution set and \(W\) is the elimination set. The data partitioning rule can be expressed as Equations (6) and (7)

\[t_{e}\ =\ average(pred\_error) \tag{6}\]

\[g\big{(}t_{e},\ \Delta X^{\prime}\big{)}\ =\ \begin{cases}x\in E&\Delta x<t_{e} \\ x\in W&\Delta x>t_{e}\end{cases}  \tag{7}\]

Considering that the GED model has a small variance in predicting each point in practical application, and the mean value is more reliable and stable, we choose the average value as the data partitioning rule. Then, the original dataset can be automatically divided into the evolution set and the elimination set, thus distinguishing the different characteristics of data and dividing the dataset accordingly.