Efficient Simulation, Accurate Sensitivity Analysis and Reliable Parameter Estimation for Delay Differential Equations
Bibliographic record
Abstract
Delay differential equations (DDEs) are a class of differential equations that have received considerable recent attention and \nbeen shown to model many real life problems, traditionally formulated as systems of ordinary differential equations (ODEs), \nmore naturally and more accurately. Ideally a DDE modeling package should provide facilities for approximating the solution, \nperforming a sensitivity analysis and estimating unknown parameters. In this thesis we propose new techniques for efficient simulation, accurate sensitivity analysis and reliable parameter estimation of DDEs. \n \nWe propose a new framework for designing a delay differential equation (DDE) solver which works with any supplied initial value \nproblem (IVP) solver that is based on a general linear method (GLM) and can provide dense output. This is done by treating a \ngeneral DDE as a special example of a discontinuous IVP. We identify a precise process for the numerical techniques used when solving the implicit equations that arise on a time step, such as when the underlying IVP solver is implicit or the delay vanishes. \n \nWe introduce an equation governing the dynamics of sensitivities for the most general system of parametric DDEs. Then, having a similar view as the simulation (DDEs as discontinuous ODEs), we introduce a formula for finding the size of jumps that appear at discontinuity points when the sensitivity equations are integrated. This leads to an algorithm which can compute \nsensitivities for various kind of parameters very accurately. \n \nWe also develop an algorithm for reliable parameter identification of DDEs. We propose a method for adding extra constraints to the \noptimization problem, changing a possibly non-smooth optimization to a smooth problem. These constraints are effectively handled \nusing information from the simulator and the sensitivity analyzer. \n \nFinally, we discuss the structure of our evolving modeling package DDEM. We present a process that has been used for incorporating \nexisting codes to reduce the implementation time. We discuss the object-oriented paradigm as a way of having a manageable design with reusable and customizable components. The package is programmed in C++ and provides a user-friendly calling sequences. The numerical results are very encouraging and show the effectiveness of the techniques.
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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 | 0.002 | 0.009 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| 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".