Colloque - Benjamin Peherstorfer : Dirac-Frenkel Dynamics with Momentum

Yvon Maday Chaire Informatique et sciences numériques Collège de France Année 2025-2026 Colloque : Aspects mathématiques et appliqués des méthodes de réduction de complexité - Benjamin Peherstorfer : Dirac-Frenkel Dynamics with Momentum Benjamin Peherstorfer Associate professor, Courant Institute of Mathematical Sciences, New York University, USA Résumé Dirac-Frenkel instantaneous residual minimization evolves nonlinear parametrizations of PDE solutions in time, but ill-conditioning can render the parameter dynamics non-unique. We interpret this non-uniqueness as a gauge freedom: nullspace directions that leave the time derivative unchanged can be used to select better-conditioned parameter velocities. Building on Onsager's minimum-dissipation principle, we introduce a history variable (interpretable as momentum) and inject it only along the nullspace directions. The resulting Dirac-Frenkel-Onsager dynamics preserve instantaneous residual minimization, in contrast to standard regularization that can introduce bias, while promoting temporally smooth parameter evolutions. Examples demonstrate that the approach leads to increased robustness in singular and near-singular regimes.