动力系统中的时空信息可互换性：用时间历史替代空间传感器的条件与界限
Space-Time Information Interchangeability in Dynamical Systems: Conditions and Bounds for Replacing Spatial Sensors with Temporal Histories
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中文总结 AI 辅助
本研究提出时空信息交换理论框架，推导线性及非线性动力系统中用时间历史替代空间传感器的条件与界限，通过数值实验验证相关结论，为减少全状态重构的空间传感器覆盖提供依据。
中文摘要 AI 辅助
许多动力状态重构问题旨在仅通过少数位置的测量来推断高维空间状态。由于支配动力学耦合了时空演化，时间测量可包含传感器位置之外的状态分量信息。本研究开发了一个理论框架，用于比较时间测量历史与指定瞬时空间传感器阵列的信息内容。时间历史和空间参考阵列由先验白化信息算子表示，该算子考虑了先验变异性、传感器几何结构和测量噪声，同时保留了不确定性降低的方向分布。一个通用时空信息交换定理给出了在每个状态方向上保留指定比例空间参考信息的充要条件，它回答了三个实际问题：第一，需要多少时间测量以及总信息为多少？第二，指定比例是否可实现，还是存在极限信息上限排除所有有限历史？第三，当可实现时，足够的历史长度是多少？针对收缩、谱分离幺正、同时可对角化耗散线性动力学，以及高斯过程和测量噪声下的随机线性估计，推导了显式的历史长度界限。对于非线性系统，将全局延迟映射单射性与均匀局部费希尔信息条件相结合。数值实验测试了方向交换长度、信息上限和有限记忆界限，并评估了非线性延迟映射可区分性和局部费希尔信息预测。该框架为确定时间测量何时能减少空间传感器覆盖范围以实现全状态重构提供了明确依据。
英文摘要
Many dynamical-state reconstruction problems seek to infer a high-dimensional spatial state from measurements at only a few locations. Because the governing dynamics couple evolution across space and time, temporal measurements can contain information about state components beyond the sensor locations. This work develops a theoretical framework comparing the information content of a temporal measurement history with that of a specified instantaneous spatial sensor array. The temporal history and spatial reference array are represented by prior-whitened information operators that account for prior variability, sensor geometry, and measurement noise while preserving the directional distribution of uncertainty reduction. A general space-time information-exchange theorem gives necessary and sufficient conditions for retaining a prescribed fraction of the spatial-reference information in every state direction. It answers three practical questions. First, how many temporal measurements and how much total information are required? Second, is the prescribed fraction attainable, or does a limiting information ceiling rule out every finite history? Third, when attainable, what history length is sufficient? Explicit history-length bounds are derived for contractive, spectrally separated unitary, and simultaneously diagonalizable dissipative linear dynamics, and for stochastic linear estimation under Gaussian process and measurement noise. For nonlinear systems, global delay-map injectivity is combined with a uniform local Fisher-information condition. Numerical experiments test directional exchange lengths, information ceilings, and finite-memory bounds, and assess nonlinear delay-map distinguishability and local Fisher-information predictions. The framework provides an explicit basis for determining when temporal measurements can reduce spatial sensor coverage for full-state reconstruction.
