Polytopal mesh agglomeration via geometrical deep learning for three-dimensional heterogeneous domains
Bibliographic record
Abstract
Agglomeration techniques can be successfully employed to reduce the computational costs of numerical simulations and stand at the basis of multilevel algebraic solvers. To automatically perform mesh agglomeration, we propose a novel Geometrical Deep Learning-based algorithm that can exploit the geometrical and physical information of the underlying computational domain to construct the agglomerated grid and -- simultaneously -- guarantee the agglomerated grid's quality. In particular, we propose a bisection model based on Graph Neural Networks (GNNs) to partition a suitable connectivity graph of computational three-dimensional meshes. The new approach has a high online inference speed. It can simultaneously process the graph structure of the mesh, the geometrical information of the mesh (e.g., elements' volumes, centers' coordinates), and the physical information of the domain (e.g., physical parameters). Taking advantage of this new approach, our algorithm can agglomerate meshes of a domain composed of heterogeneous media, automatically respecting the underlying heterogeneities. The proposed GNN approach is compared with the k-means algorithm and METIS, which are widely employed approaches for graph partitioning and are meant to process only the connectivity information on the mesh. We demonstrate that the performance of our algorithms outperforms the k-means and METIS algorithms in terms of quality metrics and runtimes. Moreover, we demonstrate that our algorithm also shows a good level of generalization when applied to complex geometries, such as three-dimensional geometries reconstructed from medical images. Finally, the model's capability to perform agglomeration in heterogeneous domains is evaluated when integrated into a polytopal discontinuous Galerkin finite element solver.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.002 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.002 | 0.002 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.002 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".