RPINNs: Rectified-physics informed neural networks for solving stationary partial differential equations.

Due to the development of high performance computing, deep learning algorithm has made a significant progress in many fields such as computational mathematics. The physics-informed neural networks have put forward a innovative idea for tackling a broad range of forward and inverse problems of partial differential equations. Motivated by the philosophy of physics-informed neural network and the multigrid method, we introduce the gradient information of numerical solution of physics-informed neural network into the new neural networks and propose the rectified-physics informed neural network for solving stationary partial differential equations. And for solving multi-objective optimization of neural networks, the dynamic weight strategy is adopted to balance numerical difference among terms in the loss function, and effectively alleviate the gradient ill-conditioned phenomenon. Finally, we perform a series of numerical experiments to demonstrate effectiveness of the RPINNs method which is combined with the dynamic weight strategy to improve calculation accuracy. • RPINNs based on multi-grid are proposed to solve the stationary PDEs. • RPINNs are improved by the dynamic weight strategy. • Incorporate the dynamic weight strategy based on NTK to RPINNs. • Study efficiency of calculating gradients by automatic differentiation and finite difference. [ABSTRACT FROM AUTHOR]