Moving Sampling Physics-informed Neural Networks induced by Moving Mesh PDE

Currently, many researchers have proposed widely used deep learning solvers based on deep neural networks, such as the Deep Ritz method [29], which solve the variational problems arising from PDEs; the Deep BSDE model [4], which is developed from stochastic differential equations and performs well at solving high-dimensional problems, and the DeepONet framework [12], which is used to learn operators accurately and efficiently from a relatively small dataset. In this article, we use physics-informed neural networks (PINN) [17]. In PINN, the governing equations of PDEs, boundary conditions, and related physical constraints are incorporated into the design of the loss function, and an optimization algorithm is used to find the network parameters to minimize the loss function, so that the approximated solution output by the neural networks satisfies the governing equations and constraints.