Mohammad A. Ayoubi | Searcharxiv

arXiv2026

SPIRAL-PO: Symbolic Identification of Partially Observed Nonlinear Dynamics with Application to Rotating Machinery

Many engineering systems have states that cannot be directly measured-tilt angles in rotating machines, internal flow variables, aeroelastic modes-yet these hidden states couple into the measured outputs. This paper addresses identifying the governing equations from such partial observations. We present SPIRA-PO (Symbolic Physics-Informed Residual Augmentation Loop-Partially Observed), a framework that treats hidden-state effects not as nuisances to be eliminated but as structured, physically interpretable signatures in the observable equations of motion. From a minimal physics seed in measured coordinates, the method fits a multi-output residual network to the projection residual, projects the learned structure onto a physics-constrained candidate library, and admits terms through a sequential statistical gating protocol. We give sufficient conditions for unique recovery of the hidden-coupling coefficients, a matching impossibility result showing that insufficiently rich excitation makes recovery impossible for any estimator, and a closed-form sample-complexity bound. These guarantees concern the projected hidden-coupling coefficients, not the hidden trajectory itself, which is reconstructed separately by a state estimator. The framework is demonstrated on a vertical flexible rotor with Duffing supports, where only the lateral displacements are measured while the tilt angles remain hidden. A constant-speed run is provably unidentifiable, whereas a speed sweep restores identifiability; from noisy coast-down data, SPIRAL-PO recovers the gyroscopic coupling, the Duffing nonlinearity, and the translation-tilt cross-coupling, each with a standard error and t-statistic. An Extended Kalman Filter built on the validated model then reconstructs the hidden tilt trajectory from observed displacements alone.

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arXiv2026

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

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.

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