Better Neural PDE Solvers Through Data-Free Mesh Movers

TL;DR

This paper introduces a data-free neural mesh adapter called DMM that dynamically moves mesh nodes for PDE solving, improving accuracy without requiring optimal mesh data and handling topology changes.

Contribution

It proposes a novel data-free neural mesh adaptor that moves existing nodes based on the Monge-Ampère equation, enabling adaptive mesh refinement without expensive data.

Findings

Meshes generated by DMM have minimal interpolation error.

The MM-PDE solver with DMM improves accuracy in modeling PDE systems.

The method effectively adapts meshes for dynamic PDE solutions.

Abstract

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.