Characteristic Values in the Scaling of Differential Equations in Engineering
Bibliographic record
Abstract
This work introduces, for the first time, a formal approach to the estimation of characteristic values of differential and other related expressions in the scaling of engineering problems. The methodology introduced aims at overcoming the inability of the traditional approach to match the exact solution of asymptotic cases. This limitation of the traditional approach often leaves in doubt whether the scaling laws obtained actually represent the desired phenomena. The formal approach presented yields estimates with smaller error than traditional approaches; these improved estimates converge to the exact solution in simple asymptotic cases and do not diverge from the exact solution in cases in which the error of traditional approaches is unbounded. The significance of this contribution is that it extends the range of applicability of scaling estimates to problems for which traditional approaches were deemed unreliable, for example, cases in which the curvature of functions is large, or complex cases in which the accumulation of estimation errors exceeds reasonable limits. This research is part of a larger effort towards a computational implementation of scaling, and it is especially valuable for approximating multicoupled, multiphysics problems in continuum mechanics (e.g., coupled heat transfer, fluid flow, and electromagnetics) that are often difficult to analyze numerically or empirically.
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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 | 0.005 | 0.022 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.003 | 0.001 |
| Science and technology studies | 0.001 | 0.006 |
| Scholarly communication | 0.002 | 0.005 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| 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".