Accelerated PDE-constrained Optimization by Adaptive Reduced Order Modelling and Goal-oriented Hyperreduction
Bibliographic record
Abstract
We present a framework to accelerate the solution of optimization problems constrained by nonlinear partial differential equations (PDEs). To reduce the cost of objective function evaluations by several orders of magnitude, we replace the high-fidelity model (HFM) with a reduced order model (ROM). The nonlinearity motivates the use of hyperreduction, for which we use a goal-oriented empirical quadrature procedure. The hyperreduced ROM is trained specifically to preserve zero- and first-order consistency with the HFM, which provides the optimization framework with a convergence guarantee. Several optimization problems are solved using the framework, including a thermal fin problem governed by a nonlinear heat equation, an inverse aerodynamic design problem governed by the Euler equations, and an inverse aerodynamic design problem governed by the Reynolds-averaged Navier-Stokes (RANS) equations. Some speedup is observed for the first two cases, but more research on hyperreduction is necessary before a cost advantage will be seen with RANS.
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How this classification was reachedexpand
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) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.006 | 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 teacher head, 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".