通过引入规范动量，提升非线性偏微分方程求解的参数演化稳定性。
A Dirac-Frenkel-Onsager principle: Instantaneous residual minimization with gauge momentum for nonlinear parametrizations of PDE solutions
- 在残差最小化框架中引入沿零空间的动量变量，缓解参数动力学病态问题。
- 相比传统正则化，避免引入偏差，保持瞬时残差最小化特性。
- 适用于奇异或近奇异条件下求解，提升数值鲁棒性，适合高精度模拟场景。
Dirac-Frenkel瞬时残差最小化用于演化非线性参数化的偏微分方程解，但病态性可能导致参数动力学不唯一。我们将其不唯一性解释为规范自由度：不改变时间导数的零空间方向可用于选择更良态的参数速度。基于Onsager最小耗散原理，引入一个历史变量（可解释为动量），仅在零空间方向注入。所得的Dirac-Frenkel-Onsager动力学保持瞬时残差最小化，不同于标准正则化可能引入的偏差，同时促进参数演化的时序平滑。实例表明该方法在奇异和近奇异区域显著提升鲁棒性。
原文摘要 · Abstract (English)
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.
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