# Application of Physics-Informed Neural Networks to River Sifting Simulation
## 3 Results
### Numerical Result from ANSYS
The former provides smoother solutions and reduces the computational complexity but may not highlight some physical effects, while the latter allows for more accurate modeling of physical processes but may require higher computational resources. Figure 9 shows comparisons of PINN solutions with exact solutions from Ansys at various x locations. We can observe that while we can achieve accurate results for density, our ability to make precise predictions for velocity and pressure is limited. This limitation arises from the fact that the flow has not yet reached the area between the shock and the right boundary. Thus, attempting to incorporate pressure information from this region into the avail-able data and equations would not uncover the actual pressure and velocity fields within the desired domain. As the results show, numerical simulation more clearly shows the shock effect, where the boundaries of pressure, velocity, and density changes are. And the obtained results of the PINN method are smoother, since using a neural network, training was carried out considering the physical constraints and conditions of the problem. However, the calculation time (including training time) using the PINN method is much less than when using numerical simulation. The analysis of the conducted research leads to the conclusion that comparing the results obtained using numerical modeling and the PINN method is significant for solving problems related to river dynamics and pollution processes. Both methods have their strengths and are applicable depending on the objectives and characteristics of the task. Numerical modeling provides precise solutions and allows for capturing explicit physical effects, such as shock waves. This is achieved via the careful discretization of space into a grid and the numerical solution of the equations on each grid element. This approach is better suited to tasks where it is necessary to describe physical processes and effects in detail. Figure 9: A comparison of density, pressure, and velocity with the exact solutions. (**a**) Density. (**b**) Pressure. (**c**) Velocity. On the other hand, the PINN method offers smoother and more continuous solutions, as neural networks are trained to integrate physical laws and conditions. This method reduces the computational complexity and can be effective in cases where obtaining fast results is crucial and when explicit physical effects are not the primary concern. Therefore, the choice between numerical modeling and the PINN method depends on the specific objectives and requirements of the task. The former allows for a more precise description of physical phenomena but may require substantial computational resources. The latter provides smoother solutions and faster computations over a short period but may not highlight certain physical effects. Comparing results, as shown in Figure 9, can assist researchers and engineers in selecting the approach most suitable for their specific task. This enables the attainment of more reliable and efficient solutions when modeling river dynamics and water pollution processes.