Better Neural PDE Solvers Through Data-Free Mesh Movers

TL;DR

The paper tackles the challenge of neural PDE solvers on dynamic, nonuniform meshes by introducing a data-free mesh adaptor (DMM) grounded in Monge-Ampère theory to produce moving meshes with fixed topology. Building on DMM, it presents MM-PDE, a two-branch graph neural solver that transfers information between the original and moving meshes via learnable interpolation and a residual-cut network, preserving data fidelity. The authors prove that the derived monitor function minimizes interpolation error and demonstrate, across 2-D Burgers', flow around a cylinder, and 3-D Gray-Scott equations, that moving meshes improve accuracy over baselines, with ablations highlighting the importance of the moving mesh and interpolation design. This approach eliminates the need for costly optimal-mesh data, preserves data structure, and scales to 3D, offering a practical framework for data-free, adaptive neural PDE solvers.

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-Ampère 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.

Figures (9)

• Figure 1: Model Architecture Summary: (a) Encodes state $u$ and coordinates $(t, x)$ using two networks, followed by a third network that processes their combined outputs. (b) MM-PDE has two branches: the bottom branch is a graph neural network for original meshes, and the top branch uses DMM-generated moving meshes for evolution learning. Interpolation of the discrete state onto moving meshes is done with $Itp_1$, processed by a graph neural network, then re-interpolated to original meshes using $Itp_2$. Both interpolations involve weights from $Itp_1$ and $Itp_2$. A residual cut network ensures the preservation of previous state information during interpolation.
• Figure 2: Meshes generated by previously trained DMM with different resolution data. Four lines respectively correspond to $24\times24$ resolution, $96\times96$ resolution, 1938 resolution and 3286 resolution.
• Figure 3: $l_{convex}$ of the Burgers' equation during training. The horizontal axis represents the number of training epochs, while the vertical axis depicts the value of the loss.
• Figure 4: Moving meshes of DMM with reduced weight of $l_{bound}$ (top) and $l_{equation}$ (bottom).
• Figure 5: Moving meshes of the 2-D Burgers' equation generated by DMM. The left three figures and the right three ones present moving meshes of $20\times20$ resolution and monitor functions of two trajectories at $t=0,12,24s$ respectively.