# Application of Physics-Informed Neural Networks to River Sifting Simulation
## 2 Materials and Methods  
### Neural Networks Based on Physics  
The physics-informed neural networks (PINN) method is extremely popular in fluid flow simulation, and numerous studies demonstrate the success of using this method ([PERSON] et al., 2020, [30]; [PERSON] et al., 2021, [31]; [PERSON] et al., 2023, [32]). However, free surface problems are a major problem for the PINN method in its original formulation. So far, studies on solving free surface flow problems using the PINN method are limited, and free surface problems have been solved only in the shallow water approximation ([PERSON], 2011, [33]), where all calculations are performed on the free surface ([PERSON] and [PERSON], 2010, [34]) and where the basic equations are simpler. Therefore, a PINN structure for solving the general problem of a free surface in rough water conditions is not available and, therefore, is very necessary. The application of the PINN method to solving the Euler equations and detecting shock wave evolution is a complex task that requires a deep understanding of the physical processes, the creation of suitable training data, and network configuration. Detecting shock wave evolution and solving the Euler equations using a physics-informed neural network (PINN) involves specific steps. The general procedure for implementing the PINN method is presented below:  
* Definition of the physical problem: Begin by clearly defining the physical problem you want to solve. In your case, it involves the Euler equations describing gas dynamics and shock wave evolution. * Formulation of the mathematical model: Translate the physical problem into a mathematical model. The Euler equations for gas dynamics constitute a system of hyperbolic equations. Solving this system of equations is essential for modeling shock wave evolution. * Preparation of training data: Collect the data to be used for network training. This may include both simulated data and experimental data, if available. * Definition of PINN: Create a neural network that will be trained to solve the Euler equations. This network will be \"physically informed\" (PINN), meaning it integrates the physical equations into the training process. * Definition of the loss function: Specify the loss function to be minimized during PINN training. The loss function should encompass both the Euler equations' conditions and the training data. This ensures that the network provides physically accurate solutions. * Network training: Utilize the training data and loss function to train the network. The training process should be capable of capturing the shock wave evolution and solving the Euler equations. * Results validation: After training, verify the network's results with test data or real experiments. Ensure that the solutions align with physical reality and shock wave evolution. * Refinement and optimization: If the results do not meet your requirements, make adjustments to the model, loss function, and training data, then repeat the process. This translation captures the essence of the original text and conveys the steps involved in implementing the PINN method for solving the Euler equations and detecting shock wave evolution. The PINN implementation approach is shown in Figure 1, which shows the concept scheme for solving the objective function optimization problem. The PINN (physics-informed neural network) method is designed to solve partial differential equations or inverse problems by incorporating fundamental physics into the neural network architecture. By adding regularization related to partial differential equations to the loss function, the model is structured in such a way that it considers physical laws during the training process. This approach reduces the data requirements and accelerates the training process. The structure of PINN consists of three main components: an input layer, hidden layers, and an output layer. The hidden layers in PINN are responsible for learning the relationships between the input variables and the desired outcome.