AMBER: Adaptive Mesh Generation by Iterative Mesh Resolution Prediction

TL;DR

This work proposes Adaptive Meshing By Expert Reconstruction (AMBER), a supervised learning approach to mesh adaptation that generalizes to unseen geometries and consistently outperforms multiple recent baselines, including ones using Graph and Convolutional Neural Networks, and Reinforcement Learning-based approaches.

Abstract

The cost and accuracy of simulating complex physical systems using the Finite Element Method (FEM) scales with the resolution of the underlying mesh. Adaptive meshes improve computational efficiency by refining resolution in critical regions, but typically require task-specific heuristics or cumbersome manual design by a human expert. We propose Adaptive Meshing By Expert Reconstruction (AMBER), a supervised learning approach to mesh adaptation. Starting from a coarse mesh, AMBER iteratively predicts the sizing field, i.e., a function mapping from the geometry to the local element size of the target mesh, and uses this prediction to produce a new intermediate mesh using an out-of-the-box mesh generator. This process is enabled through a hierarchical graph neural network, and relies on data augmentation by automatically projecting expert labels onto AMBER-generated data during training. We evaluate AMBER on 2D and 3D datasets, including classical physics problems, mechanical components, and real-world industrial designs with human expert meshes. AMBER generalizes to unseen geometries and consistently outperforms multiple recent baselines, including ones using Graph and Convolutional Neural Networks, and Reinforcement Learning-based approaches.

Figures (17)

• Figure 1: learns adaptive mesh generation on complex geometries for simulation applications from an expert dataset. Left: During training, predicts a sizing field, as indicated by the mesh's color, from labels projected from an expert mesh $M^*$. continuously updates a replay buffer with newly generated meshes to preserve a diverse and accurate training data distribution. Right: During inference, starts from an initial mesh $M_0$, predicts a sizing field per element, and feeds it into a mesh generator that refines the mesh using the underlying geometry $\Omega$. This process is repeated until a final mesh $M_T$ is produced. On the car seat crossmember shown above, learns that the expert assigns more mesh elements to holes and sharp bends, which are particularly interesting for strength and durability analyses.
• Figure 2: Exemplary meshes for each dataset. The color represents the local element size, with smaller elements being red. We propose six novel and challenging datasets for mesh generation. (a)Poisson uses an L-shaped domain with a multimodal load function. (b)Laplace features parameterized $2$D lattices with complex Dirichlet boundaries. (c)Airfoil includes geometries representative of aerodynamic flow setups. (d)Console consists of $3$D car seat crossmembers. (e)Mold includes complex $3$D plates used in injection molding contexts. (f)Beam covers elongated, perforated beams inducing long-range mesh dependencies.
• Figure 3: Mean and two times standard error of expert mesh similarity evaluated by (lower is better). achieves the best results across all datasets, demonstrating its ability to generate highly accurate meshes on diverse and challenging domains. All methods perform well on Poisson (easy). As task complexity increases, the baselines and eventually variants become less reliable. (1-Step) remains strong across tasks, while the full model achieves further improvements through iterative refinement.
• Figure 4: Log-log plot of error indicator norm versus number of mesh elements (lower left is better) for , and the expert across Poisson (easy, medium, hard). Each marker shows the mean over the test set for a given seed. and evaluations are obtained by scaling the final predicted sizing field and tuning the element penalty, respectively. closely matches or even exceeds expert performance in terms of indicator error, and generalizes to meshes that are more than $3{\times}$ finer, maintaining the expected error-element trend beyond $100\,000$ elements.
• Figure 5: Full views and close-ups of generated Mold test meshes. The element size is denoted by color, with red indicating small elements. closely matches the expert mesh, producing finer elements near the hole and coarser elements near the mesh's border. In comparison, the Image baselines have less variation in the element size, matching the expert less closely.

Theorems & Definitions (2)

• Theorem 1
• proof