# Flow Velocity Computation in Solid-Liquid Two-Phase Flow by a Hybrid Network CNN-RKSVM
## 4 Experimental Results and Discussion
### Results and Discussion  
Based on the above sample dataset, the flow velocities of the test dataset obtained by the CNN-RKSVM are compared with the CC, RKSVM, and CNN methods.  
The solid-liquid two-phase flow within dredging engineering has a critical flow velocity (V\({}_{0}\)), and in this experiment, V\({}_{0}\) is nearly 2.3 m/s. When the flow velocity is greater than V\({}_{0}\), the solid phase inside the pipe is in a suspended state, and the flow pattern is under turbulent flow. Conversely, when the flow velocity is less than V\({}_{0}\), the solid phase inside the pipe settles at the bottom of the pipe, and the flow pattern is under laminar flow. Therefore, in order to compare these two flow velocity cases, the results of flow velocity are divided into two groups: low flow velocity and high flow velocity, as shown in Figure 5.  
Figure 5a shows that when the flow velocity is smaller than V\({}_{0}\), the flow velocity obtained by the CC method exhibits a similar trend to the real flow velocity, albeit with a larger error compared to the CNN and CNN-RKSVM methods. In contrast, the RKSVM ha some errors with the real flow velocity. Compared with the CNN, the computed flow velocities from the CNN-RKSVM are very close to the real flow velocities. The CNN-RKSVM uses the RKSVM to map the feature data extracted from the CNN into the high-dimensional space to obtain the flow velocities so that the flow velocities obtained by the CNN-RKSVM are very close to the real flow velocities.  
Figure 5b shows that when the flow velocity exceeds V\({}_{0}\), the CC method fails to accurately capture the flow velocity due to the non-fulfillment of the \"freezing\" model.  
Figure 5: Comparison of flow velocities for four methods. (**a**) V < V\({}_{0}\). (**b**) V > V\({}_{0}\).  
Particularly, when the flow velocity surpasses 4 m/s, the solid phase within the pipeline exhibits a uniform distribution, resulting in a high similarity of ERT measurements. The RKSVM and CNN methods show some deviation from the real flow velocity, whereas the CNN-RKSVM provides a closer approximation.  
To quantitatively evaluate the CNN-RKSVM, we compared it with the CC, RKSCN, and CNN methods. The RMSEs, MAPEs, and runtimes for the four methods of calculating flow velocity are compared with those of real flow velocity, as shown in Table 3.  
Table 3 shows that the RMSEs of the CC method are always greater than 0.8 and that the MAPEs of the CC method are greater than 0.2. The RKSVM method is relatively stable regardless of whether V \(>\) V\({}_{0}\) or V \(<\) V\({}_{0}\), but it may lead to a large error with partial testing samples. Owing to the advantage of deep learning in both the CNN and CNN-RKSVM methods, their RMSEs and MAPEs are much smaller than those of the CC and RKSVM methods. Both the CNN and CNN-RKSVM methods yield a more consistent flow velocity when the flow velocity is less than V\({}_{0}\). However, when the flow velocity is greater than V\({}_{0}\), the RMSE and MAPE of the CNN-RKSVM method are better than those of the CNN method. Since the RKSVM uses the reproducing kernel as the kernel function, it has advantages, including a strong generalization ability and a high computational efficiency. Therefore, the CNN-RKSVM based on both the RKSVM and the CNN can improve the accuracy, generalization ability, and robustness of flow velocity calculations.  
Moreover, the CNN-RKSVM method uses reproducing kernel functions to avoid the computation of inner products in high-dimensional spaces and thus greatly reduces the computational load and shortens the model computation time under the learning process. On the other hand, the bottom line in Table 3 shows the runtimes of the four methods, CC, RKSVM, CNN, and CNN-RKSVM, at any new ERT measurements. It is clear that the CNN-RKSVM has the longest time cost while the CC has the lowest since the CNN-RKSVM must use more blocks and parameters to compute flow velocity. Nevertheless, the runtimes of the CNN-RKSVM are less than 1 s, which is an acceptable response time in dredging engineering.