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Better Neural PDE Solvers Through Data-Free Mesh Movers

ICLR 2024 · Probabilistic and scientific modeling › Differential equations and PDEs · approach: Neural PDE solvers · 1 citations (OpenAlex)

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Proposes Data-free Mesh Mover (DMM), a neural operator trained with a Monge-Ampère physics loss (no optimal-mesh data) to produce r-adaptive moving meshes, and MM-PDE, a two-branch GNN solver that embeds these moving meshes with a learnable interpolation framework.

Problem

Neural PDE solvers on static meshes waste resources on non-uniform, multi-scale dynamics; existing adaptive approaches need expensive optimal-mesh supervision or change node count/topology during h-refinement.

Approach

Train a DeepONet-style mesh mover to output a moving-mesh map by solving the Monge-Ampère equation with a physics loss and monitor-function-proportional sampling, then embed the moving mesh in a two-branch GNN with learnable interpolation weights and a residual cut network.

Key results

• On Burgers' equation, MM-PDE achieves MSE 1.04e-05 versus 1.19e-05 for bigger GNN, 1.18e-05 for FNO, and 3.10e-05 for GNN.
• DMM meshes reduce cell-volume std from 0.1027 to 0.0469 and range from 0.7524 to 0.2061 on Burgers' equation.
• On flow around a cylinder, MM-PDE error is 0.0846 versus 0.1548 for bigger GNN, 0.2892 for GNN, 0.4040 for LAMP, and 0.2844 for Geo-FNO.

Measured effects reported in the paper

Surprising findings

• Physics-loss-trained DMM (data-free) used as initialization with end-to-end fine-tuning (mm+end2end) gives lower error than training DMM only end-to-end, suggesting the physics-loss pretraining is a good initial state.
• Uniform meshes with the two-branch architecture did not improve over the base model, while the moving-mesh variant did.

Limitations

• Monitor function is based on the current state at time t and may not capture future state variations, as the authors note.
• Mesh tangling may occur when the Jacobian determinant is negative; not addressed beyond the convexity loss.
• Memory consumption of DMM training is high, requiring batching and optimizing only the last decoder layer with BFGS.

Experimental setup

Datasets: 2-D Burgers' equation simulations flow around a circular cylinder (FEniCS Navier-Stokes) 3-D Gray-Scott reaction-diffusion

Benchmarks: 2-D Burgers' equation flow around a circular cylinder 3-D Gray-Scott equation

Metrics: MSE std and range of cell volumes under monitor function inference time RMSE physics loss

Baselines: GNN bigger GNN CNN FNO LAMP Geo-FNO MP-PDE MeshGraphNets

Builds on: DeepONet MP-PDE LAMP Geo-FNO FNO

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