## 2 Proposed Methodology
### Capturing Spatiotemporal Features through the DED-ConvLSTM Module  
The DED-ConvLSTM model implements two significant improvements to the standard ConvLSTM. First, it uses an encoder-decoder structure to efficiently process long-term sequence data. The encoder part encodes the hidden and cellular states of the entire input sequence and then passes them to the decoder. Its main work can be expressed as follows:  
\[h_{t}^{enc}=f\big{(}W_{xh}^{enc}x_{t}+W_{hh}^{enc}h_{t-1}^{enc}+b_{h}^{enc} \big{)} \tag{1}\]  
\[c_{t}^{enc}=g\big{(}W_{xc}^{enc}x_{t}+W_{hc}^{enc}h_{t-1}^{enc}+b_{c}^{enc} \big{)} \tag{2}\]  
where \(x_{t}\) is the input at the current time step of the input sequence, \(h_{t}\) and \(c_{t}\) are the hidden and cellular states at time step \(t\), \(W\) and \(b\) are the network parameters, and \(f\) and \(g\) are the activation functions. The decoder uses these states as starting points to generate the output sequence, and the work of the decoder can be expressed as follows:  
\[h_{t}^{dec}=f(W_{yh}^{dec}y_{t-1}+W_{hh}^{dec}h_{t-1}^{dec}+b_{h}^{dec}) \tag{3}\]  
\[y_{t}=softmax(W_{hy}^{dec}h_{t}^{dec}+b_{y}^{dec}) \tag{4}\]  
where \(y_{t-1}\) is the output of the previous time step and \(y_{t}\) is the output of the current time step. This structure helps to maintain long-term time dependence and solve the time dependence problem in time-series forecasting. Second, DED-ConvLSTM introduces pyramidal dilated convolution based on ED-ConvLSTM. Dilated convolution is characterized by the application of its dilation rate. The dilation rate defines the spacing between elements within the convolution kernel, i.e., a certain number of intervals are inserted between each element of the convolution kernel to expand the input region it covers. Assuming that the dilation rate is denoted by r, the actual spacing of elements in the convolution kernel is r \(-\) 1 units. If the size of the standard convolutional kernel is k \(\times\) k, then the actual size of  
Figure 1: Architecture of PDED-ConvLSTM. **(a)** Overall architecture of PDED-ConvLSTM. **(b)** Architecture of PC module. (c) Description of DED-ConvLSTM module. the region covered by the convolutional kernel becomes \(\mathrm{k}+(\mathrm{k}-1)(\mathrm{r}-1)\) in the dilated convolution. This structure allows the convolution kernel to achieve a wider sampling range of the input features without increasing the number of parameters. It effectively enhances the model's ability to recognize spatial variability in sea ice concentration. Before delving into the structure of DED-ConvLSTM, it is essential to first introduce the ConvLSTM model. Compared to the traditional LSTM network, the core concept of a ConvLSTM unit is to embody spatial correlations through shared weights, transforming the fully connected operations of the input, forget, and output gates in LSTM units into convolutional operations. This approach also takes into account the relationships between adjacent pixels and spatial context to model sequences [31]. The ConvLSTM model tends to \"remember\" only the short-term past of its sequence, as it cannot fully accumulate information about the entire sequence in its memory cells. However, in the task of sea ice sequence prediction, information from the long-term historical sea ice concentration is crucial for accurate forecasting. As shown in Figure 1c, the encoder considers information from all previous time points when processing the input sequence, while the decoder takes into account the overall contextual information provided by the encoder when generating predictions for each time point. Therefore, a ConvLSTM with an encoder-decoder structure should be employed to capture these long-term temporal dependencies. To extract more comprehensively the spatiotemporal information, we have further expanded the ED-ConvLSTM model into a pyramid dilated convolutional structure. Specifically, this involves replacing the convolution operations in ED-ConvLSTM with pyramid dilated convolutions.