A Dirac-Frenkel-Onsager principle: Instantaneous residual minimization with gauge momentum for nonlinear parametrizations of PDE solutions
For researchers using nonlinear parametrizations (e.g., neural networks) to solve PDEs, this provides a principled regularization method that preserves instantaneous residual minimization.
The authors address ill-conditioning in Dirac-Frenkel instantaneous residual minimization for nonlinear parametrizations of PDE solutions, which causes non-unique parameter dynamics. They introduce a gauge momentum term based on Onsager's principle to select better-conditioned velocities without biasing the residual minimization, leading to increased robustness in singular regimes.
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.