Positive-Real Identification of Sparse Mori-Hamiltonians from Partial Observations

Abstract

Discovering the governing equations of a physical system from data is a central goal across the sciences, yet in most experiments only a few states are accessible while the rest stay hidden. Existing approaches treat this partial observability as an obstacle to be removed by first reconstructing the hidden state---a step that is ill-posed under noise and that discards the physical constraints, such as energy conservation, that the true dynamics obey. We show that for conservative (Hamiltonian) systems no reconstruction is needed: projecting the dynamics onto the measured coordinates yields a memory kernel that we prove to be a lossless positive-real rational matrix, whose poles are the hidden natural frequencies and whose positive-semidefinite residues encode the couplings. From this kernel we recover a closed, interpretable governing equation for the observed dynamics---identified from output data alone, passive by construction, and validated by out-of-sample forecasting. Under stronger conditions---an equipartitioned measure with position coupling, or a forced input--output experiment---the bare hidden frequencies and couplings of the underlying Hamiltonian are additionally recoverable. We test the method on linear, nonlinear, and chaotic systems under realistic noise. Because it returns energy-conserving equations of motion from partial measurements, it offers a common tool for problems spanning mechanics, fluid and plasma physics, and beyond.