Control of parabolic PDEs with time-varying spatial domain: Czochralski crystal growth process
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Bibliographic record
Abstract
This paper considers the optimal control problem for a class of convection-diffusion-reaction systems modelled by partial differential equations (PDEs) defined on time-varying spatial domains. The class of PDEs is characterised by the presence of a time-dependent convective-transport term which is associated with the time evolution of the spatial domain boundary. The functional analytic description of the PDE yields the representation of the initial and boundary value problem as a nonautonomous parabolic evolution equation on an appropriately defined infinite-dimensional function space. The properties of the time-varying evolution operator to guarantee existence and well posedness of the initial and boundary value problem are demonstrated which serves as the basis for the optimal control problem synthesis. An industrial application of the crystal temperature regulation problem for the Czochralski crystal growth process is considered and numerical simulation results are provided.
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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 | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.001 | 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.
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