Application of physics-informed neural network on the example of modeling hydrodynamic processes that allow an analytical solution

We consider an actual approach to develop a physically based neural network for solving model problems for the Kovazhny flow, for the geophysical Beltrami flow, and for the flow in a section of the river by the shallow water theory. Physics-informed neural networks (PINN) allow to significantly reduce the computation time compared to conventional computations. There is a different analytical solution for each model flow. The architecture of the DeepXDE software library, its composition by modules, and fragments of program code in the Python programming language are discussed. The PINN model is tested on a test sample. The prediction is evaluated using the MSE metric. The fully connected neural network can contain 4, 7, 10 hidden layers with the number of neurons 50, 50, 100 respectively. The influence of hyperparameters of the neural network on the magnitude of the prediction error is discussed. The calculations performed on a server with Nvidia GeForce RTX 3070 card can significantly reduce the training time for PINN.