Order‐reduction of parabolic PDEs with time‐varying domain using empirical eigenfunctions
Bibliographic record
Abstract
A novel methodology for the order‐reduction of parabolic partial differential equation (PDE) systems with time‐varying domain is explored. In this method, a mapping functional is obtained, which relates the time‐evolution of the solution of a parabolic PDE with time‐varying domain to a fixed reference domain, while preserving space invariant properties of the initial solution ensemble. Subsequently, the Karhunen–Loève decomposition is applied to the solution ensemble on fixed spatial domain resulting in a set of optimal eigenfunctions. Further, the low dimensional set of empirical eigenfunctions is mapped on the original time‐varying domain by an appropriate mapping, resulting in the basis for the construction of the reduced‐order model of the parabolic PDE system with time‐varying domain. This methodology is used in three representative cases, one‐ and two‐dimensional (1‐D and 2‐D) models of nonlinear reaction‐diffusion systems with analytically defined domain evolutions, and the 2‐D model of the Czochralski crystal growth process with nontrivial geometry. © 2013 American Institute of Chemical Engineers AIChE J, 59: 4142–4150, 2013
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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 | 0.001 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.001 | 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".