A Dirac-Frenkel-Onsager principle: Instantaneous residual minimization with gauge momentum for nonlinear parametrizations of PDE solutions

TL;DR

This paper introduces a gauge-invariant residual minimization method for nonlinear PDE parametrizations, enhancing robustness and smoothness in singular regimes by leveraging a history variable inspired by Onsager's principle.

Contribution

It develops a novel Dirac-Frenkel-Onsager dynamics framework that incorporates gauge freedom and history variables to improve stability and conditioning in PDE solution parametrizations.

Findings

Increased robustness in singular and near-singular regimes.

Preserves instantaneous residual minimization without bias.

Promotes temporally smooth parameter evolutions.

Abstract

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.