Multi-parametric high-order flow sensitivity analysis
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Bibliographic record
Abstract
We present a methodology to automatically generate and solve high order sensitivity equations for multi-dimensional parameter spaces. Given the flow equations of interest (Navier-Stokes, RANS, Burger’s, etc), the methodology uses Newton’s multinomial theorem to automatically derive the set of all terms appearing in the flow sensitivity equations of arbitrary order n with respect to q parameters. We introduce a simple and generic data structure to describe the both the flow and all its sensitivity equations so that one generic solver can solve the differential equations for the flow and its sensitivities. Our approach provides a simple means of extending an existing flow solver to obtain the flow and sensitivity solution fields. A wrapper consisting of a loop over the sensitivity orders calls the main solver for sensitivity orders ranging from 0 to n. The 0 execution of the loop computes the flow while the next iterations compute flow sensitivities up to the requested order n. The k execution of the loop computes all sensitivities of order k for all parameters including all mixed derivatives. The resulting solver is verified by the method of manufactured solutions. Finally, we examine the ability of high-order Taylor series expansions in multi-dimensional parameter spaces to approximate flow solutions over a wide range of parameter values.
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.003 | 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 it