Reduction of Linear Systems of ODEs with Optimal Replacement Variables
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Bibliographic record
Abstract
Abstract In this exploratory study, we present a new method of approximating a large system of ODEs by one with fewer equations, while attempting to preserve the essential dynamics of a reduced set of variables of interest. The method has the following key elements: (i) put a (simple, ad-hoc) probability distribution on the phase space of the ODE; (ii) assert that a small set of replacement variables are to be unknown linear combinations of the not-of-interest variables, and let the variables of the reduced system consist of the variables-of-interest together with the replacement variables; (iii) find the linear combinations that minimize the difference between the dynamics of the original system and the reduced system. We describe this approach in detail for linear systems of ODEs. Numerical techniques and issues for carrying out the required minimization are presented. Examples of systems of linear ODEs and variable-coefficient linear PDEs are used to demonstrate the method. We show that the resulting approximate reduced system of ODEs gives good approximations to the original system. Finally, some directions for further work are outlined.
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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 | 0.001 | 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