给变分演化加惯性，让参数更新更稳定可靠。
Dirac-Frenkel dynamics with inertia for nonlinearly parametrized solutions of evolution problems
- 引入惯性项，让参数速度信息在弱约束方向持续保留
- 理论证明参数动力学适定，且给出误差后验界
- 数值实验显示惯性显著提升算法鲁棒性
即使狄拉克-弗伦克尔动力学在函数空间中定义良好，冗余的非线性参数化（如典型神经网络或混合模型）仍可能导致参数动力学非唯一或病态。本文提出在狄拉克-弗伦克尔动力学中加入惯性项，使弱信息方向的参数速度能保留历史信息，而强信息方向仍遵循原动力学。理论上证明了惯性形式下参数动力学适定，并提供后验误差界。时间离散化后，方法需解与标准狄拉克-弗伦克尔相同的正则化线性最小二乘问题，但将前一时刻速度作为锚点。数值实验表明，惯性显著提升了算法的鲁棒性。
原文摘要 · Abstract (English)
Even when Dirac-Frenkel dynamics determine a well-defined evolution in function space, the corresponding parameter dynamics can be non-unique or ill-conditioned for redundant nonlinear parametrizations, such as typical neural networks or mixture models. We propose to add inertia to the Dirac-Frenkel dynamics and show that this allows useful parameter velocity information to persist from the past trajectory in directions that are weakly informed, while well-informed parameter velocity directions continue to follow the Dirac-Frenkel dynamics. We prove that the inertial formulation yields well-posed parameter dynamics and provide a posteriori error bounds. After time discretization, the method requires the solution of the same type of regularized linear least-squares problem as standard Dirac-Frenkel dynamics, but with the previous velocity appearing as an anchor. Numerical experiments demonstrate the increased robustness obtained with inertia.
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