# A Hybrid DSCNN-GRU Based Surrogate Model for Transient Groundwater Flow Prediction
## 3 Results
### Coupled Surrogate Model Results  
To enhance the accuracy and effectiveness of the surrogate model, this study integrates CNN-LSTM, DSCNN-LSTM, and DSCNN-GRU. The coupling method employed for all three models follows the same coupling method as DSCNN-GRU, described in Section 2.2.4. The optimal coupling model is identified by assessing relevant parameters. Given that the training and test sample sets for all three coupled models are identical to c_3, identical evaluation criteria can be used for performance comparison.  
As depicted in Figures 12 and 13, the CNN-LSTM model has exhibited a significant improvement in accuracy compared to traditional CNN models. Specifically, the MSE value decreased from 0.0018 to 0.0012 during training and from 0.009 to 0.0034 during testing. However, this improvement comes at the expense of increased computational load, resulting in a 4.9% increase in training time from 3348 s to 3514 s. Furthermore, it was observed that the MSE value of the CNN-LSTM model exhibited significant fluctuations throughout the entire training cycle, with an absolute difference of up to 0.0008 between consecutive cycles. These fluctuations can primarily be attributed to the nonlinear characteristics of the transient groundwater flow in the time series.  
Figure 11: MSE values, R\({}^{2}\) values, and training time values under the test sample set for each CNN surrogate of Case 1 to Case 3. (**a**) MSE values and training time values. (**b**) R\({}^{2}\) values and SSIM values.  
Figure 10: MSE values achieved using CNN alternative models across various training epochs and different training datasets (c_1: Case 1 model; c_2: Case 2 model; c_3: Case 3 model).  
After integrating DSCNN-LSTM, the training time decreased to 2892 s. Moreover, the limited sampling time of only 540 samples and the small sample size impede LSTM's capability to effectively leverage its memory for acquiring long-term sequence information. Consequently, the MSE value increased to 0.0042 during the training process, and the MSE value rose to 0.059 during the testing process.  
The integration of DSCNN and GRU reduced the training time to 2858 s, which is 490 s lower than that of CNN, 656 s shorter than CNN-LSTM, and 34 s lower than DSCNN-LSTM. This improvement in computational efficiency is significant. The MSE value of the proposed DSCNN-GRU model remained consistently lower than that of traditional CNN models throughout the entire training period. Additionally, the MSE values of the DSCNN-GRU model showed less fluctuations compared to the CNN-LSTM model during the initial 100 to 200 cycles.  
Compared to the CNN-LSTM model, the proposed DSCNN-GRU model has a slightly higher MSE value of 0.0096, compared to the CNN-LSTM's value of 0.0034. However, the DSCNN-GRU model achieves a shorter training time, reduced by 18.6%. The R\({}^{2}\) value of the DSCNN-GRU model is 0.949, which is 0.84% lower than the 0.957 achieved using the CNN-LSTM model. Additionally, the SSIM values for DSCNN-GRU and CNN-LSTM are 0.968 and 0.971, respectively. This indicates that the average prediction error of these two models is within 0.003 m, showing a significant improvement compared to the CNN model's 0.923. Although the proposed DSCNN-GRU model shows only a slight reduction in prediction accuracy compared to CNN-LSTM, it significantly enhances computational  
Figure 12: MSE values obtained from various CNN-RNN coupling surrogate models during different training epochs (c_l: CNN-LSTM; d_l: DSCNN-LSTM; c: CNN in Case 3; d_g: DSCNN-GRU).  
Figure 13: MSE values, R\({}^{2}\) values, and training time values under the test sample set for each surrogate model. (**a**) MSE values and training time values. (**b**) R\({}^{2}\) values and SSIM values.  
efficiency. Considering the trade-off between computational accuracy and efficiency, the proposed DSCNN-GRU model outperforms the other three models in overall performance.