Geometric Deep Learning for Electrostatics and Magnetostatics Problems
Bibliographic record
Abstract
Conventional numerical methods such as the Finite Element Method (FEM) and Finite Difference Method (FDM) have been employed to solve electrostatics and magnetostatics problems involving Partial Differential Equations (PDEs).With the recent boom in deep learning, alternative ways to solve such PDEs subject to boundary value constraints have been proposed and implemented, in contrast to classical numerical solvers.One such entity is a Graph Neural Network (GNN), which precisely depicts the internodal connectivity in a graph structure comprising nodes and edges.At the structural level, a GNN is analogous to the tessellations (here, triangular simplices) of a finite element mesh.This thesis investigates the prediction quality of a GNN model with spectral graph convolution, to learn and approximate the solution for a time-independent, boundary value problem in the domains of electrostatics and magnetostatics.This network is trained in a supervised manner on datasets generated by a FEM solver with diverse shapes and charge/current non-uniformities.For the given datasets and the GNN framework, experimental results demonstrate that introducing mesh augmentations i.e., structural variations in the finite element meshes, enhance GNN predictions over unseen geometries and inhomogeneities, in comparison to commonly used regularization techniques.
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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 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.002 | 0.001 |
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".