Solving Euler equations with gradient-weighted multi-input high-dimensional feature neural network

The study found that it is difficult to capture the solutions at the shock wave and discontinuity surfaces when solving Euler equations using physics informed neural network. Thus, this paper proposes an improved neural network based on adaptive weights for multi-input high-dimensional features to solve the Euler equations. First, adaptive weights regarding the velocity are added to the control equation of each residual to train the shock wave region better. Then, more residual training points are used in regions with initial time discontinuities to improve the training efficiency. The problem that the sigmoid activation function is more prone to gradient pathologies than tanh in the training process is also analyzed to show that the Euler equations can be better solved using tanh. Numerical experiments verify that even though the solution process becomes complicated, it outperforms the original physics informed neural network in terms of computational efficiency and computational accuracy and can better portray the physical phenomena of Euler equations.