AMBER: Adaptive Mesh Generation by Iterative Mesh Resolution Prediction

Niklas Freymuth¹, Tobias Würth², Nicolas Schreiber¹, Balazs Gyenes¹, Andreas Boltres^1,3, Johannes Mitsch², Aleksandar Taranovic¹, Tai Hoang¹, Philipp Dahlinger¹, Philipp Becker¹, Luise Kärger², Gerhard Neumann¹

¹Autonomous Learning Robots, Karlsruhe Institute of Technology, Karlsruhe
²Institute of Vehicle System Technology, Karlsruhe Institute of Technology, Karlsruhe
³SAP SE

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.

Method

We approach adaptive mesh generation as a supervised learning problem, factorizing the process into two key stages. First, a learnable model predicts a spatially-varying, scalar-valued sizing field. Second, a non-parametric mesh generator consumes this field to produce the final mesh. AMBER's central idea is to perform this prediction iteratively, allowing the model to gradually refine its own sampling resolution. Starting with a coarse, uniform mesh $M^0$, AMBER processes the corresponding graph to predict a vertex-level discrete sizing field. This collection of point-based predictions is then transformed into a smooth, continuous field using a linear interpolant, similar to a linear basis function in the FEM. The resulting continuous field then guides the generation of a new, more refined mesh, $M^{t+1}$.

A single AMBER refinement step can be expressed as

$$ M^{t+1} = g_\texttt{msh}\left(\Omega, \mathcal{I}_{M^t}\left(\sigma\left(\text{MPN}_{\theta}(G(M^t))\right)\right)\right)\text{.}$$

Here, the current mesh $M^t$ is first converted into a graph representation $G(M^t)$. The message passing network $\text{MPN}_{\theta}$ processes this graph to predict vertex-level scalars, which are transformed into a discrete sizing field using some transformation $\sigma$. This discrete field is then transformed into a smooth, continuous field by a linear interpolant, $\mathcal{I}_{M^t}$. The mesh generator, $g_\texttt{msh}$, uses this continuous field to produce the refined mesh for the next iteration, $M^{t+1}$. This iterative loop and the flexibility of the underlying message passing network allow AMBER to progressively focus its resolution on the most critical regions of a provided arbitrary geometry.

Qualitative Results

AMBER-generated meshes on six novel datasets for adaptive mesh generation.
The datasets include classical physics problems, mechanical components, and real-world industrial designs.

AMBER-generated meshes on the Poisson and Laplace test domains.
Both tasks feature a FEM solution in the loop.

Results on Airfoil, Console, Mold and Beam

AMBER-generated meshes on the Airfoil, Console, Mold and Beam test domains.
Meshes are generated from the domain geometry to match an expert strategy.

Quantitative Results

Main results comparing AMBER to baselines

AMBER generates high-quality meshes on all datasets, outperforming recent image- and graph-based baselines.

Different Poisson mesh resolutions for AMBER, ASMR and the expert

Estimated simulation error for different mesh resolutions on Poisson.
AMBER's meshes are on par with the expert that it learns from, while recent RL methods show worse performance and more variability.
AMBER can generalize to larger meshes during inference by scaling the predicted sizing field, maintaining mesh quality for over 100k elements.

Runtime comparison between AMBER and an expert heuristic

Runtime comparison between AMBER and the expert heuristic used to generate its training meshes.
AMBER achieves comparable mesh quality at a fraction of the runtime, especially for finer meshes.

Multi-task performance of AMBER trained simultaneously on Poisson (hard), Laplace and Airfoil, compared to training on each task individually.
AMBER straightforwardly adapts to the multi-task setting, and is able to provide accurate meshes for different scenarios without sacrificing performance.

BibTeX

@article{freymuth2025amber, title={AMBER: Adaptive Mesh Generation by Iterative Mesh Resolution Prediction}, author={Freymuth, Niklas and W{\"u}rth, Tobias and Schreiber, Nicolas and Gyenes, Balazs and Boltres, Andreas and Mitsch, Johannes and Taranovic, Aleksandar and Hoang, Tai and Dahlinger, Philipp and Becker, Philipp and others}, journal={arXiv preprint arXiv:2505.23663}, year={2025} }

AMBER: Adaptive Mesh Generation by Iterative Mesh Resolution Prediction

Abstract

Method

Qualitative Results

AMBER-generated meshes on six novel datasets for adaptive mesh generation. The datasets include classical physics problems, mechanical components, and real-world industrial designs.

AMBER-generated meshes on the Poisson and Laplace test domains. Both tasks feature a FEM solution in the loop.

AMBER-generated meshes on the Airfoil, Console, Mold and Beam test domains. Meshes are generated from the domain geometry to match an expert strategy.