Reduction of Linear Systems of ODEs with Optimal Replacement Variables
Bibliographic record
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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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 | 0.003 |
| 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 | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.002 | 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".