A Continuous Sensitivity Equation of Arbitrary High Order
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Bibliographic record
Abstract
We present an approach to automatically generate and solve the flow sensitivities with respect to a given single parameter up to an arbitrary order n. We use the Newton multinomial theorem to automatically derive the set of terms constituting the sensitivity equations of any order. Hence, given the flow equations at hand (Navier-Stokes, RANS, Burgers, etc), our methodology automatically produces the corresponding equations for the flow sensitivities of an arbitrary high order n. In our approach, the flow and sensitivity equations are not calculated by different solvers resulting from different source codes. Rather, we extend an existing flow solver by adding an extra loop over the sensitivity order (i.e. from 0 to n, the 0 order flow sensitivity being the flow itself) on top of the main solution procedure. Thus, during the execution of the loop the first iteration computes the flow as before and the next iterations compute the flow sensitivities up to the requested order n. We present the necessary generic data structure to do so. The verification of the flow-and-sensitivity solver is performed by the method of the manufactured solution. The computed sensitivities are validated by comparison to sensitivities obtained by second-order finite-differences. Finally, we examine the ability of high-order Taylor series expansions in parameter space 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 | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 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) | 0.000 | 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