The optimal transport problem and its application to dissipative partial differential equations
Bibliographic record
Abstract
The optimal transport problem has found many applications in mathematics and physical sciences, in part due to the importance of the Wasserstein gradient flow.To appreciate this importance, we first introduce the optimal transport problem in the formulations of Monge and Kantorovich and present a numerical approach to the discrete equivalent problem.This numerical procedure is used to visualize optimal transport plans.We then prove the result of Gangbo andMcCann that, under standard assumptions, there exists a unique optimal transport plan to problems involving strictly convex cost functions.This background allows us to build the Wasserstein gradient flow from its discretization, the Jordan-Kinderlehrer-Otto scheme.We use this procedure to justify that the Fokker-Planck equation is the Wasserstein gradient flow of a physical energy functional and conclude by briefly presenting similar applications to other dissipative equations. iv R SUM La thorie du transport optimal est aujourd'hui applique dans plusieurs domaines des sciences physiques et mathmatiques.Cette omniprsence s'explique en partie par la puissance de la descente de gradient par la mtrique de Wasserstein.Pour apprcier l'importance de cette technique, on introduit le problme de Monge et de Kantorovich ainsi qu'une approche numrique et visuelle au problme discret.On montre le rsultat de Gangbo et McCann qu'il n'existe qu'une unique solution aux problmes avec un cot strictement convexe.On construit ensuite la descente de gradient de Wasserstein partir de sa discrtisation, la mthode de Jordan, Kinderlehrer et Otto.On tablit ainsi que l'quation de Fokker-Planck est la descente de gradient de Wasserstein d'une fonctionnelle avec une interprtation manifestement physique.On conclut par un bref sommaire des applications de cette descente de gradient aux quations de dissipation.
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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 | 0.001 | 0.001 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.002 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.000 | 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".