Computation of empirical eigenfunctions of parabolic PDEs with time-varying domain
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Bibliographic record
Abstract
In this work, we explore a methodology to compute the empirical eigenfunctions for the order-reduction of nonlinear parabolic partial differential equations (PDEs) system with time-varying domain. The idea behind this method is to obtain the mapping functional, which relates the time-evolution scalar physical property solution ensemble of the nonlinear parabolic PDE with the time-varying domain to a fixed reference domain, while preserving space invariant properties of the raw solution ensemble. Subsequently, the Karhunen-Lo'eve decomposition is applied to the solution ensemble with fixed spatial domain resulting in a set of optimal eigenfunctions that capture the most energy of data. Further, the low dimensional set of empirical eigenfunctions is mapped (“pushed-back”) on the time-varying domain by an appropriate mapping resulting in the basis for the construction of the reduced-order model of the parabolic PDEs with time-varying domain. Finally, this methodology is applied in the representative cases of calculation of empirical eigenfunctions in the case of one and two dimensional model of nonlinear reaction-diffusion parabolic PDE systems with analytically defined domain evolutions. In particular, the design of both mappings which relate the raw data and function spaces transformations from the time-varying to time-invariant domain are designed to preserve dynamic features of the scalar physical property and we provide comparisons among reduced and high order fidelity models.
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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 | 0.000 | 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 | 0.000 |
| 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.
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