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AMBER: Adaptive Mesh Generation by Iterative Mesh Resolution Prediction
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Proposes AMBER, a supervised iterative adaptive mesh generation method where a message-passing GNN predicts sizing fields on intermediate meshes, trained with a self-generated replay buffer whose meshes are relabeled from expert meshes.
Problem
Adaptive mesh generation for FEM requires costly task-specific heuristics or manual expert design; learned methods struggle with extreme local size variation and distribution shift between coarse training inputs and inference-time intermediate meshes.
Approach
Iteratively predict a scalar, piecewise-linear sizing field on each intermediate mesh with a hierarchical message-passing GNN, generate the next mesh with an off-the-shelf mesher (gmsh), and train with a DAgger-like replay buffer where generated meshes are automatically labeled by projecting expert mesh sizes.
Key results
- On Poisson (hard), AMBER reduces DCD relative to supervised baselines such as Image and GraphMesh and the RL baseline ASMR, with consistent gains across all six datasets.
- AMBER (Mixed), trained jointly on Poisson (hard), Laplace and Airfoil, matches single-task AMBER (0.226 vs 0.224 on Poisson hard; 0.222 vs 0.222 on Laplace; 0.102 vs 0.103 on Airfoil).
- AMBER generalizes to over 100,000 elements via a single scalar scaling of the sizing field, while the largest expert mesh has 31,510 elements; it outperforms the expert heuristic runtime by over an order of magnitude on meshes above 30,000 elements.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Train jointly on Poisson hard, Laplace, and Airfoil instead of training a separate model per dataset | AMBER · Poisson hard | DCD: 0.224 ± 0.004 -> 0.226 ± 0.011 | No clear effect |
| Train jointly on Poisson hard, Laplace, and Airfoil instead of training a separate model per dataset | AMBER · Laplace | DCD: 0.222 ± 0.003 -> 0.222 ± 0.005 | No clear effect |
| Train jointly on Poisson hard, Laplace, and Airfoil instead of training a separate model per dataset | AMBER · Airfoil | DCD: 0.103 ± 0.002 -> 0.102 ± 0.002 | No clear effect |
| Use iterative mesh generation instead of a single generation step (ablation) | AMBER · Six evaluation datasets | DCD | Helped |
| Use a hierarchical graph architecture (ablation) | AMBER · Evaluation datasets | Mesh quality | Helped |
| Use the proposed loss (ablation) | AMBER · Evaluation datasets | Mesh quality | Helped |
| Normalize inputs and outputs (ablation) | AMBER · Evaluation datasets | Mesh quality | Helped |
| Use piecewise-linear rather than piecewise-constant sizing fields (ablation) | AMBER · Evaluation datasets | Mesh quality | Helped |
| Increase the number of mesh-generation steps (ablation) | AMBER · Evaluation datasets | Mesh quality | Helped |
| Replace ASMR's reference-mesh reward with the error-indicator reward (ablation) | ASMR · Poisson | Mesh quality | Hurt |
Surprising findings
- Baselines (Image, GraphMesh) diverge without the softplus transformation, producing overly fine meshes; adding it is needed for fair comparison.
- ASMR degrades on fine meshes beyond about 30,000 elements due to its fixed-depth reference mesh.
Limitations
- Predicts scalar, piecewise-linear sizing fields, limiting expressiveness for extreme density variation and anisotropic refinement.
- Evaluation relies on DCD and error-indicator proxies rather than direct simulation error on real-world scenarios.
Experimental setup
Datasets: Poisson Laplace Airfoil Console Mold Beam
Benchmarks: Poisson Laplace Airfoil Console Mold Beam
Metrics: directed Chamfer distance (DCD) symmetric relative projected L2 error of sizing fields norm of FEM error indicator runtime
Baselines: GraphMesh Image (CNN) Image (Variant) ASMR AMBER (1-Step)
Builds on: DAgger message passing neural networks GraphMesh ASMR
Closest papers by meaning
- G-Adaptivity: optimised graph-based mesh relocation for finite element methods (ICML 2025 · similarity 0.87)
- EvoMesh: Adaptive Physical Simulation with Hierarchical Graph Evolutions (ICML 2025 · similarity 0.85)
- 3DID: Direct 3D Inverse Design for Aerodynamics with Physics-Aware Optimization (NeurIPS 2025 · similarity 0.84)
- ARMesh: Autoregressive Mesh Generation via Next-Level-of-Detail Prediction (NeurIPS 2025 · similarity 0.84)
- UGM2N: An Unsupervised and Generalizable Mesh Movement Network via M-Uniform Loss (NeurIPS 2025 · similarity 0.84)
- Mesh-RFT: Enhancing Mesh Generation via Fine-grained Reinforcement Fine-Tuning (NeurIPS 2025 · similarity 0.83)
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