Goal‐oriented model reduction of parametrized nonlinear partial differential equations: Application to aerodynamics
Bibliographic record
Abstract
Summary We introduce a goal‐oriented model reduction framework for rapid and reliable solution of parametrized nonlinear partial differential equations with applications in aerodynamics. Our goal is to provide quantitative and automatic control of various sources of errors in model reduction. Our framework builds on the following ingredients: a discontinuous Galerkin finite element (FE) method, which provides stability for convection‐dominated problems; reduced basis (RB) spaces, which provide rapidly convergent approximations; the dual‐weighted residual method, which provides effective output error estimates for both the FE and RB approximations; output‐based adaptive RB snapshots; and the empirical quadrature procedure (EQP), which hyperreduces the primal residual, adjoint residual, and output forms to enable online‐efficient evaluations while providing quantitative control of hyperreduction errors. The framework constructs a reduced model which provides, for parameter values in the training set, output predictions that meet the user‐prescribed tolerance by controlling the FE, RB, and EQP errors; in addition, the reduced model equips, for any parameter value, the output prediction with an effective, online‐efficient error estimate. We demonstrate the framework for parametrized aerodynamics problems modeled by the Reynolds‐averaged Navier‐Stokes equations; reduced models provide over two orders of magnitude online computational reduction and sharp error estimates for three‐dimensional flows.
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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 | 0.001 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.001 | 0.000 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| 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".