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.

翻译：本文提出了一种基于移动网格方法的端到端自适应采样神经网络（MMPDE-Net），该方法通过求解移动网格偏微分方程自适应生成新的采样点。该模型主要关注提升采样点生成质量。此外，我们开发了基于MMPDE-Net的迭代算法，使采样点更加精确可控。由于MMPDE-Net是一个独立于深度学习求解器的框架，我们将其与物理信息神经网络（PINN）相结合，提出了移动采样PINN（MS-PINN），并通过在若干假设下的误差分析验证了其有效性。最后，我们通过四个典型例子的数值实验展示了MS-PINN相比PINN的性能提升，从数值上验证了所提出方法的有效性。