  In this work, we propose an end-to-end adaptive sampling neural network
(MMPDE-Net) based on the moving mesh method, which can adaptively generate new
sampling points by solving the moving mesh PDE. This model focuses on improving
the quality of sampling points generation. Moreover, we develop an iterative
algorithm based on MMPDE-Net, which makes the sampling points more precise and
controllable. Since MMPDE-Net is a framework independent of the deep learning
solver, we combine it with physics-informed neural networks (PINN) to propose
moving sampling PINN (MS-PINN) and demonstrate its effectiveness by error
analysis under some assumptions. Finally, we demonstrate the performance
improvement of MS-PINN compared to PINN through numerical experiments of four
typical examples, which numerically verify the effectiveness of our method.
