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-Ampere 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.

翻译：近年来，神经网络被广泛用于物理系统建模中的偏微分方程求解。尽管主流研究聚焦于在预定义的静态网格离散化上学习系统演化，但受限于这些系统的动态特性，部分方法采用强化学习或监督学习技术来构建自适应动态网格。然而，这些方法面临两大挑战：（1）需要昂贵的优化网格数据；（2）网格细化过程中解空间自由度和拓扑结构发生变化。为解决上述问题，本文提出一种配备神经网格适配器的神经PDE求解器。首先，我们设计了一种新颖的无数据神经网格适配器——无数据网格移动器（DMM），其包含两项核心创新：其一，该适配器作为将解映射到自适应网格的算子，通过Monge-Ampère方程进行训练，无需优化网格数据；其二，通过移动现有节点而非增删节点或边的方式动态改变网格。理论分析表明，DMM生成的网格具有最低的插值误差界。在DMM基础上，为高效精确地建模动态系统，我们开发了基于移动网格的神经PDE求解器（MM-PDE），该求解器通过双分支架构嵌入移动网格，并配备可学习插值框架以保留数据中的信息。实验证明，该方法在建模广泛应用的PDE系统时，能够生成适配网格并显著提升求解精度。