A document by Tengchao Yu . Click on the document to view its contents.

Physics informed neural networks (PINNs) have been a well-known tool in solving forward and inverse partial differential equations (PDEs) problem. In particular, for the forward problem, only the initial/boundary conditions and the equation itself as the physical information are needed to obtain the predicted solution within the definition domain. However, some existing works show that the standard PINN method is insufficient for solving complex problems. For the relatively complex one-dimensional Euler equation problem, due to the limited accuracy of the standard PINN method, the algorithm is improved in two aspects in this study. First, a gradient-related weight function is introduced into the loss function and the framework of gradient-weighted physics-informed neural networks (gwPINNs) is proposed. In particular, the weight function is carefully designed. Second, the sequence-to-sequence (seq2seq) learning is applied in the training process to improve the precision. Then, four cases of Riemann problems are solved using the improved algorithm. The results show that the improved algorithm can obtain high accuracy in solving these problems, and verify the feasibility of solving one-dimensional Euler equation based on the improved PINN method.

It is physically admissible that the numerical solutions of gas dynamics equations are consistent with the second law of thermodynamics, but the other cellcentered Lagrangian (CL) schemes except the hybridized CL scheme introduced by Maire et al. do not satisfy this consistency. For one-dimensional gas dynamics equations, this hybridized CL scheme combines the acoustic approximate (AA) flux and the entropy conservative (EC) flux. By distinguishing expanded zones and compressed zones, the hybridized CL scheme basically satisfies the entropy consistency, but in the part zones of rarefaction waves, the hybridized CL scheme is under-entropic and the EC flux may result in the numerical oscillations in simulating strong rarefaction waves. In this paper, we try to design a new hybridized CL scheme satisfying entropy consistency for one-dimensional complex flows. This new hybridized CL scheme is the combination of the AA flux and the modified entropy conservative (MEC) flux. Just like the EC flux, the MEC flux keeps the entropy conservative in simulating rarefaction waves; while differently, the MEC flux damps effectively numerical oscillations in zones of rarefaction waves. And we find that, by using the third-order TVD-type Runge-Kutta time discretization method, the CL scheme with the improved hybridized flux can completely satisfy entropy consistency. Finally, some numerical examples are tested to prove the characters of our new CL scheme with the improved hybridized flux.

How to simulate shock waves and other discontinuities is a long history topic. As a new and booming method, the physics-informed neural network (PINN) is still weak in calculating shock waves than traditional shock-capturing methods. In this paper, we propose a ‘retreat in order to advance’ way to improve the shock capturing ability of PINN by using a weighted equations (WE) method with PINN. The primary strategy of the method is to weaken the expression of the network in high compressible regions by adding a local positive and compression-dependent weight into governing equations at each interior point. With this strategy, the network will focus on training smooth parts of the solutions. Then automatically affected by the compressible property near shock waves, a sharp discontinuity appears with the wrong inside-shock-points ‘compressed’ into well-trained smooth regions just like passive particles. In this paper, we study one-dimensional and two-dimensional Euler equations. And illustrated by the comparisons with high-order classical WENO-Z method in numerical examples, the proposed method can significantly improve the discontinuity computing ability