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

Abstract

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.