标题： 神经ODE中无遗忘调优的几何基础 显示英文标题

标题： Geometric Foundations of Tuning without Forgetting in Neural ODEs

摘要： 在我们之前的工作中，我们为神经ODE的顺序训练引入了无需遗忘的调谐原则（TwF），其中训练样本是迭代添加的，并且参数在保持输出标签流形上先前学习样本端点映射的第一阶近似意义下的控制函数子空间内进行更新。 在本信中，我们证明了在非奇异控制下，这个参数子空间形成一个有限余维的巴拿赫子流形，并且我们对其切空间进行了表征。 这表明TwF对应于沿该巴拿赫子流形切空间的控制函数的延续/变形，从而在顺序训练中精确地提供了其映射保持（不遗忘）的理论基础，超越了一阶近似。 显示英文摘要

摘要： In our earlier work, we introduced the principle of Tuning without Forgetting (TwF) for sequential training of neural ODEs, where training samples are added iteratively and parameters are updated within the subspace of control functions that preserves the end-point mapping at previously learned samples on the manifold of output labels in the first-order approximation sense. In this letter, we prove that this parameter subspace forms a Banach submanifold of finite codimension under nonsingular controls, and we characterize its tangent space. This reveals that TwF corresponds to a continuation/deformation of the control function along the tangent space of this Banach submanifold, providing a theoretical foundation for its mapping-preserving (not forgetting) during the sequential training exactly, beyond first-order approximation.