# A Hybrid DSCNN-GRU Based Surrogate Model for Transient Groundwater Flow Prediction
## 2 Methodology
### Deep Learning-Based Surrogate Models
#### 2.2.1 Cnn  
The viability of substituting the conventional convolutional neural network (CNN) model is investigated. The specific design is as follows. Initially, the input images are all 100 \(\times\) 100 pixels in size and contain four color channels: red (R), green (G), blue (B), and transparency (A). Each pixel denotes the water level of the study area at the corresponding position. Following Figure 6, a convolutional layer and a pooling layer are combined into a unit, resulting in a total of five such units in the model. A time-distributed convolutional layer is used to process the image input in the time dimension, performing maximum pooling at each time step. The pooled output is further processed across the time dimension before and then flattened to fit into the fully connected layer. Next, the hydraulic conductivity data are input, with each set of coefficients matched one-to-one with the convolved image datasets. A fully connected layer integrates the hydraulic conductivity data with the corresponding image data for further processing. To simplify prediction evaluation, only output images from six time points are retained, each representing the groundwater head information of the entire study area at that specific time point. The image size remains 100 \(\times\) 100 pixels with four color channels, consistent with the input images. Thus, the output size is defined as 6 \(\times\) 100 \(\times\) 100 \(\times\) 4, resulting in a total of  
Figure 5: Simplified schematic summary of selecting the optimal deep learning based surrogate model in this work.  
240,000 elements. Finally, a shaping layer is utilized to convert the output of the fully connected layer into the desired predicted image data for preservation.  
This study uses convolutional operations to extract temporal features from groundwater head field data, while fully connected layers establish relationships between groundwater head field data and hydraulic conductivity data. The ReLU activation function is consistently applied in the network architecture due to its simplicity and computational efficiency. It solves the vanishing gradients issue, enhances convergence of deep neural networks, and accelerates training (Figure 6). For model optimization, the Adam optimizer is employed, which combines the advantages of the AdaGrad and RMSProp algorithms, along with momentum. The default Adam optimizer parameters in Keras are a learning rate of 0.001, beta1 of 0.9, beta2 of 0.999, and epsilon of \(1\times 10^{-7}\). These parameters regulate the decay rates of the momentum variable (m) and second-order momentum variable (v) and prevent division by zero. The model is trained for 200 epochs with a batch size of 23 to ensure thorough training.  
Case 3 is selected for further modeling because its hydraulic conductivity complexity at each time point closely resembles the actual conditions in the study area. To make Case 3 more close to the real situation in the study area, two new variables representing pumping rates of two water wells are introduced. The locations of these wells are obtained from a report published by local authorities [23]. The values of these two new variables are randomly assigned and integrated with image data during training, similar to the treatment of hydraulic conductivity. A fully connected layer is then applied to learn the relationships between the data.