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.

本文提出了一种利用神经网格适配器的神经 PDE 求解器和基于移动网格的神经 PDE 求解器，以解决传统方法中昂贵优化网格数据和自由度拓扑变化的挑战，并通过实验证明了方法的有效性。