School of Mathematics, Sichuan University, 610065, Chengdu, China.
Neural Netw. 2024 Dec;180:106706. doi: 10.1016/j.neunet.2024.106706. Epub 2024 Sep 10.
In this work, we propose an end-to-end adaptive sampling framework based on deep neural networks and the moving mesh method (MMPDE-Net), 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 sampling points distribute more precisely and controllably. Since MMPDE-Net is independent of the deep learning solver, we combine it with physics-informed neural networks (PINN) to propose moving sampling PINN (MS-PINN) and show the error estimate of our method 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.
在这项工作中，我们提出了一种基于深度神经网络和移动网格方法（MMPDE-Net）的端到端自适应采样框架，该框架可以通过求解移动网格 PDE 自适应地生成新的采样点。该模型专注于提高采样点生成的质量。此外，我们还开发了一种基于 MMPDE-Net 的迭代算法，使采样点的分布更加精确和可控。由于 MMPDE-Net 独立于深度学习求解器，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），并在一些假设下给出了我们方法的误差估计。最后，我们通过四个典型示例的数值实验证明了 MS-PINN 相对于 PINN 的性能提升，数值实验验证了我们方法的有效性。
