## 3 Theory and Methods
### Convolutional LSTM Networks
In addition to station elevation data, we suggest incorporating physics-guided auxiliary variables that capture various state variables. Let \(X_{t}\) represent the spatial data on the \(t_{\text{th}}\) day, which includes climatic variables. Each \(X_{t}\) represents a different climate variable, with an average of data points per unique coordinate over the observation period. Since the available data are limited for forecasting specific locations and times, we utilize them to interpret data from sensors and other sources that provide historical weather information. \[X_{t}=[X_{t-T},X_{t-(T-1)},X_{t-(T-2)},\ldots\ldots\ldots,X_{t-1},X_{t}] \tag{12}\]We introduce the problem that we aim to investigate and present the notations used in this paper. We are given \(N\) time-series, denoted by \(X=\left[X^{1},\ X^{2},\ldots\ldots,\ X^{N}\right]^{T}\in\mathbb{R}^{N\times T_{x}}\), representing the smallest form of the time-series data, as shown in Figure 3.  
Here, \(T_{x}\) represents the length of the entire input sequence. Each \(X^{i}=[x_{1}^{i},x_{2}^{i},\ldots\ldots\ldots,x_{T_{x}}^{i}]^{T}\in\mathbb{R}^ {T_{x}},i\in\{1,2,\ldots\ldots,N\}\) corresponds to a time-series for each input variable \(t\in\{1,2,\ldots\ldots,\ T_{x}\}\),To represent the input variables for each time step \(t\in\{1,2,\ldots\ldots,\ T_{x}\}\), we use the notation \(X_{t}=\left[x_{1}^{i},x_{1}^{2},\ldots\ldots\ldots\ldots,x_{1}^{N}\right]^{T} \in\mathbb{R}^{N}\). Thus, the compact form of all the time-series can alternatively be represented as \(X=\left[X_{1},\ X_{2},\ldots\ldots\ldots,\ X_{T_{x}}\right]^{T}\). Similarly, we denote \(y\in\mathbb{R}^{T_{y}}\) as a time-series where \(y_{j}\in\mathbb{R}\) represents the output produced at time step \(j\). In the context of future time-series prediction, our goal is to develop a (non-)linear mapping, represented by a sequence model, to forecast \(T_{y}\) future values of the output (univariate) time-series given the historical data for \(T_{x}\) input (multivariate) time-steps. Mathematically, we define \(F(. )\) as the mapping to be learned to obtain the forecasted solution (\(\hat{y_{j}}\)) at output time-step \(j\). \[i_{t}=\sigma(W_{i}\times[h_{t-1},x_{t}]+b_{i}) \tag{7}\]0  
This study aims to develop unique mapping functions \(F\) in Equation (13) that offer highly comparable or superior prediction accuracy while revealing the temporal and geographical relationships between input and output. Our approach enhances precise spatiotemporal interpretability, which is crucial in time-series prediction problems. This distinguishes our research from previous studies mentioned in the previous section. We utilize the spatial attention mechanism to assess the relative contributions of various input variables in multivariate time-series prediction. The inclusion of spatial attention in the encoding process has recently been proposed [58]. The spatial attention \(\beta_{t}^{i}\) at time-step \(t\) is computed as follows given the \(i\)-th attribute time-series \(x^{i}\) of length \(T_{x}\):  
\[i_{t}=\sigma(W_{i}\times[h_{t-1},x_{t}]+b_{i}) \tag{7}\]1  
\[i_{t}=\sigma(W_{i}\times[h_{t-1},x_{t}]+b_{i}) \tag{7}\]2  
\[i_{t}=\sigma(W_{i}\times[h_{t-1},x_{t}]+b_{i}) \tag{7}\]3 Using a time-series, the suggested model RMSDC is illustrated trying to compute the result. The weighted input time-series at time \(t,\ x_{t}\) is then substituted for the raw input time-series at time \(t,\ \hat{x}_{t}\), and \(\hat{x}_{t}\) is used as input to the encoder LSTM (function \(f_{1}\)) to compute the new states \(h_{t}\) and \(c_{t}\). \[i_{t}=\sigma(W_{i}\times[h_{t-1},x_{t}]+b_{i}) \tag{7}\]4  
\[i_{t}=\sigma(W_{i}\times[h_{t-1},x_{t}]+b_{i}) \tag{7}\]5  
Following the encoder, it has been proposed to utilize the initial temporal attention method [59]. The attention weight for each hidden state of the encoder is determined by Equation (17) at the output time-step \(j\) of the decoder.