Iterative Sizing Field Prediction for Adaptive Mesh Generation From Expert Demonstrations

TL;DR

AMBER tackles adaptive mesh generation by imitation learning, learning to predict per-element sizing fields on intermediate meshes and iteratively refining via a conventional mesh generator to approximate expert meshes. It combines a graph neural network with a replay buffer and an oracle relabeling scheme to align training with inference, propagating refinements for $T$ steps to produce $M^T$ that closely matches the expert mesh $M^*$. Training leverages projections of expert sizing fields onto intermediate meshes and uses stratified replay to mitigate distribution shift, enabling data-efficient learning without handcrafted refinement rewards. Across 2D Poisson problems and 3D automotive-like geometries, AMBER achieves high similarity to expert meshes and outperforms a strong CNN baseline, especially for highly adaptive meshes.

Abstract

Many engineering systems require accurate simulations of complex physical systems. Yet, analytical solutions are only available for simple problems, necessitating numerical approximations such as the Finite Element Method (FEM). The cost and accuracy of the FEM scale with the resolution of the underlying computational mesh. To balance computational speed and accuracy meshes with adaptive resolution are used, allocating more resources to critical parts of the geometry. Currently, practitioners often resort to hand-crafted meshes, which require extensive expert knowledge and are thus costly to obtain. Our approach, Adaptive Meshing By Expert Reconstruction (AMBER), views mesh generation as an imitation learning problem. AMBER combines a graph neural network with an online data acquisition scheme to predict the projected sizing field of an expert mesh on a given intermediate mesh, creating a more accurate subsequent mesh. This iterative process ensures efficient and accurate imitation of expert mesh resolutions on arbitrary new geometries during inference. We experimentally validate AMBER on heuristic 2D meshes and 3D meshes provided by a human expert, closely matching the provided demonstrations and outperforming a single-step CNN baseline.

Figures (13)

• Figure 1: Schematic overview of . During inference, takes an initial mesh $M_0$, predicts a sizing field per element, and combines this with the underlying geometry in a mesh generator which produces an improved mesh. This process is repeated until a final mesh $M_T$ is obtained. When training, is tasked to predict the projected sizing field of an expert mesh for samples from a replay buffer, adding samples to this buffer to maintain a large and accurate distribution of training meshes.
• Figure 2: Intermediate and final meshes on Poisson's equation for an expert mesh with $50$ reference refinements. The colorbar on the left denotes the predicted sizing field per element for each intermediate mesh. This prediction is given to a mesh generator to produce the next mesh. Top: (Mean) converges in a few generation steps at the cost of additional elements in intermediate steps. Bottom: (Max) instead yields more conservative predictions, which take longer to converge but produce less total mesh elements. In both cases, the mesh generator and the architecture favor smooth solutions, generating a mesh that closely matches the expert but has less abrupt variations in local element size.
• Figure 3: Exemplary (Mean) refinements for Poisson's Equation trained on expert meshes with (Left) 25, (Middle) 50, and (Right) 75 refinement steps. We plot the solution of Poisson's Equation as a color plot. The right figure includes a zoom on the concave edge of the domain.
• Figure 4: Zoom-in of a final generated meshes for an unseen test domain on the Console task. (Left) closely matches the expert mesh (Middle), producing finer elements near the hole and coarser elements on the upper border of the mesh. In comparison, the baseline (Right), which acts on a $64\times 64\times 64$$3$D image of the domain, has less variation in the element size and matches the expert less closely.
• Figure 5: Mean and quantiles of the normalized for and the baseline for different input image resolutions. (Left) Poisson Easy; the resulting meshes are comparatively coarse and can be reproduced well with the baseline. (Middle) Poisson Medium; the starts to require a higher input resolution for good refinements. (Right) Poisson Hard; the expert mesh covers elements across multiple scales. Here, the fails to provide good refinements for lower image resolutions and begins to overfit for higher resolutions. In contrast, directly acts on intermediate meshes and can thus dynamically adapt the sampling resolution of the sizing fields it predicts.