对称性下逆问题的可重构性：区分结构、有效与物理上界
Reconstructability of Inverse Problems under Symmetry: Separating Structural, Effective, and Physical Upper Bounds
- Graduate School of Science and Engineering, Kansai University（关西大学理工学研究科）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文提出重构维度的三层结构（结构、有效、物理），区分对称性决定的上界与物理过程实现后的上界，并通过片状颗粒和二体问题示例说明，为物理逆问题提供重构框架。
AI中文摘要：
在我们先前的工作中，我们引入了重构维度，这是一个仅由对称性决定的可重构自由度上界。然而，在实际物理系统中，状态量是通过由原因$O$引发的物理过程$P_o$产生的。因此，状态空间通常是简并的，由表示论结构决定的上界无法达到。这要求我们区分仅由对称性决定的上界与物理过程实现后的上界。为此，我们引入了上界的三层结构：结构重构维度（SRD）、有效重构维度（ERD）和物理重构维度（PRD），它们分别由中间空间的表示论结构、重构映射的具体设计以及物理过程$P_o$下状态空间的简并性决定。对于每个不可约分量，它们构成层级关系$\mathrm{SRD}\ge\mathrm{ERD}\ge\mathrm{PRD}$，因此两个间隙标识了可重构性在哪个层级上丧失。该框架通过两个对比系统加以说明：片状颗粒的取向动力学（其中原因是速度梯度）和二体问题（其中原因本身通过对称性定义）。本文为物理逆问题中的重构提供了一个具体框架。
英文摘要：
In our previous work we introduced the Reconstruction Dimension, an upper bound on the reconstructable degrees of freedom determined solely by the symmetry. In actual physical systems, however, the state quantities are generated through physical processes $P_o$ induced by the cause $O$. The state space is therefore often degenerate, and the upper bound determined by the representation-theoretic structure is not reached. This calls for a distinction between the upper bound determined solely by symmetry and the one realized after the physical process. We therefore introduce a three-level structure of upper bounds: the Structural, Effective, and Physical Reconstruction Dimensions (SRD, ERD, PRD), determined respectively by the representation-theoretic structure of the intermediate space, by the concrete design of the reconstruction map, and by the degeneracy of the state space under the physical process $P_o$. For each irreducible component they form the hierarchy $\mathrm{SRD}\ge\mathrm{ERD}\ge\mathrm{PRD}$, so that the two gaps identify at which level reconstructability is lost. The framework is illustrated through two contrasting systems: the orientation dynamics of flake-like particles, where the cause is a velocity gradinet, and the two-body problem, where the cause itself is defined through the symmetry. This paper provides a concrete framework for reconstruction in physical inverse problems.
