A kinematic smoothing method for tightening convex relaxations of ordinary differential equations
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Bibliographic record
Abstract
This article presents a new approach for constructing convex enclosures of reachable sets of parametric ordinary differential equations (ODEs), for use in deterministic methods for global dynamic optimization. In our new approach, we modify an established ODE relaxation framework by Scott and Barton (2013), using kinematic intuition to replace certain discontinuous transitions between discrete modes with tighter, smoother transitions, and ultimately producing tighter, smoother relaxations of the original ODE solution that are more amenable to integration by off-the-shelf numerical ODE solvers. We refer to our new relaxation approach as “kinematic smoothing”. Our new ODE relaxations are straightforward to construct automatically based on established tools, and we present several numerical examples based on a proof-of-concept implementation in Julia.
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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 | 0.006 |
| 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)
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Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
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