Recently, neural networks have been extensively employed to solve partial differential equations (PDEs) in physical system modeling. While major studies focus on learning system evolution on predefined static mesh discretizations, some methods utilize reinforcement learning or supervised learning techniques to create adaptive and dynamic meshes, due to the dynamic nature of these systems. However, these approaches face two primary challenges: (1) the need for expensive optimal mesh data, and (2) the change of the solution space's degree of freedom and topology during mesh refinement. To address these challenges, this paper proposes a neural PDE solver with a neural mesh adapter. To begin with, we introduce a novel data-free neural mesh adaptor, called Data-free Mesh Mover (DMM), with two main innovations. Firstly, it is an operator that maps the solution to adaptive meshes and is trained using the Monge-Amp\`ere equation without optimal mesh data. Secondly, it dynamically changes the mesh by moving existing nodes rather than adding or deleting nodes and edges. Theoretical analysis shows that meshes generated by DMM have the lowest interpolation error bound. Based on DMM, to efficiently and accurately model dynamic systems, we develop a moving mesh based neural PDE solver (MM-PDE) that embeds the moving mesh with a two-branch architecture and a learnable interpolation framework to preserve information within the data. Empirical experiments demonstrate that our method generates suitable meshes and considerably enhances accuracy when modeling widely considered PDE systems. The code can be found at: https://github.com/Peiyannn/MM-PDE.git.

翻译：近年来，神经网络被广泛应用于物理系统建模中的偏微分方程求解。现有研究主要关注在预定义的静态网格离散化上学习系统演化，但部分方法考虑到这些系统的动态特性，采用强化学习或监督学习技术来创建自适应和动态网格。然而，这些方法面临两大挑战：（1）需要昂贵的优化网格数据，以及（2）网格细化过程中解空间的自由度和拓扑结构发生变化。为解决这些问题，本文提出了一种配备神经网格适配器的神经偏微分方程求解器。首先，我们引入了一种新型的无数据神经网格适配器——无数据网格移动器（Data-free Mesh Mover, DMM），其创新之处体现在两个方面：其一，它是一个将解映射到自适应网格的算子，并在无优化网格数据的条件下通过Monge-Ampère方程进行训练；其二，它通过移动现有节点而非增删节点和边来动态改变网格。理论分析表明，DMM生成的网格具有最低的插值误差界。基于DMM，为高效准确地建模动态系统，我们开发了基于移动网格的神经偏微分方程求解器（MM-PDE），该求解器通过双分支架构嵌入移动网格，并采用可学习插值框架以保留数据中的信息。实验结果表明，该方法能够生成合适的网格，并在广泛考虑的偏微分方程系统建模中显著提升精度。代码可访问：https://github.com/Peiyannn/MM-PDE.git。