# Application of Physics-Informed Neural Networks to River Sifting Simulation
## 3 Results  
### Numerical Result from PINN  
The problem of hydrodynamics with a shock tube is a standard hydrodynamic test problem. The problem is used as a test problem to test the ability of numerical methods to capture characteristics unique to solving conservation laws. The solution for each physical quantity gives a contact discontinuity and a shock wave. The numerical calculation of the shock wave and the contact discontinuity is difficult because the numerical scheme creates artificial dispersion or scattering near the discontinuity points.  
To explain the predictive versatility of physics-informed neural network (PINN) modeling, the case of the Euler problem was selected for analysis in this research. For a general universal PINN model, it is proposed to analyze several different cases representing diverse conditions and carefully justify the choice and representativeness of each of them. The PINN model presented can predict the clogging relationships in different scenes accurately.  
In our study, we considered the following architecture of PINN. The input to this architecture is two vectors of values t and tix, and the result is three vectors of values u, p, and \(\rho\) (velocity, pressure, and density). The main hyperparameters of PINN are presented in Table 1. The hyperparameter of the neural network is four hidden layers with 30 neurons in each layer, and the optimizing procedure is the Adam optimizer with a learning rate of 0.001. For each layer, the function of activation is the tanh function. The loss function is the MSE for the given data and initial condition, as presented in Equations (4) and (5).  
Figures 4-6 show the prediction results using PINN for density, pressure, and velocity over time = 0.2, 0.6, 0.8, respectively. In these figures, we can notice that when the shock moved, the density changed from x = 0.5 and t = 0.2 to x = 0.7 and t = 0.8. It showed that the consistency of fluid was changing, i.e., the flow became saturated with rock particles; in our case, we observed that sedimentation occurred. The shock wave has a smooth trend, although it should be a sharp feature. In this paper, we will compare the PINN solution with the Ansys solution.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **PINN** \\ \hline Number of layers & 6 \\ \hline Activation function & Tanh \\ \hline Optimizer & Adam \\ \hline Number of epochs & 7500 \\ \hline Learning rate & 0.001 \\ \hline Loss function & MSE \\ \hline \hline \end{tabular}
\end{table}
Table 1: The main hyperparameters of PINN architecture.  
The use of the PINN approach opens up possibilities for solving a general class of discontinuous solutions of compressible Euler equations. This is achieved due to the ability to capture physical phenomena such as impacts, contact breaks, and rarefaction.  
Figure 4: Solving the problem using PINN for a period of time = 0.2.  
Figure 5: Solving the problem using PINN for a period of time = 0.6.  
Figure 6: Solving the problem using PINN for a period of time = 0.8.  
In Figures 5-7, the velocity behavior shows scattering near the impact angles. This allows for computational errors, and one of the goals of PINN is to find solutions that are as accurate as possible.