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.

翻译：狄拉克-弗兰克尔瞬时残差最小化方法在时间上演化偏微分方程解的非线性参数化，但病态条件可能导致参数动力学非唯一。我们将这种非唯一性解释为规范自由度：保持时间导数不变的空空间方向可用于选择条件更好的参数速度。基于昂萨格最小耗散原理，我们引入一个可解释为动量的历史变量，并仅沿空空间方向注入该变量。与可能引入偏差的标准正则化方法不同，由此产生的狄拉克-弗兰克尔-昂萨格动力学在保持瞬时残差最小化的同时，促进了参数演化的时间平滑性。实例表明，该方法在奇异和近奇异区域能有效提升鲁棒性。