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. The governing equation -- and the underlying Hamiltonian -- can therefore be read directly from the autocorrelation of the measured signal, with guarantees of uniqueness and physical passivity, and without neural networks. We validate the approach on linear, nonlinear, and chaotic systems under realistic noise. By recovering interpretable equations of motion that conserve energy by construction from partial measurements, the method offers a common tool for problems spanning mechanics, fluid and plasma physics, and beyond.