🤖 AI Summary
This study addresses the reconstruction of high-dimensional dynamical system states by replacing a subset of spatial sensors with temporal measurements. The authors develop a theoretical framework grounded in prior whitening information operators, integrating Fisher information analysis, delay-map injectivity, and spectral separation in dynamical modeling to establish a general spatiotemporal information exchange theorem. This theorem precisely characterizes the necessary and sufficient conditions for reducing sensor count while preserving a prescribed fraction of spatial information. For both linear and nonlinear systems, the work provides the first explicit upper bounds on the required length of temporal history and demonstrates the existence of an information ceiling. Exact history-length bounds are derived for contractive systems, spectrally separated unitary systems, and simultaneously diagonalizable dissipative systems. Numerical experiments confirm the discriminative power of delay embeddings and the predictive efficacy of local Fisher information.
📝 Abstract
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.