Asymptotic- and bifurcation-aware scientific machine learning for nonlinear vibration prediction of submerged floating tunnel tethers under hydrodynamic excitation

Abstract

Seabed-anchored submerged floating tunnels transmit most of their transverse stiffness to slender pre-tensioned tethers whose vibration governs the fatigue design of the crossing. A reduced-order model represents the tunnel cross-section as a two-degree-of-freedom rigid tube coupled to small-sag tethers, linearizes the Morison drag into an amplitude-dependent equivalent damping, and yields, through the multiple time scales method, closed-form frequency-response equations for an indirectly forced regime, an indirect parametric regime of Mathieu-Duffing type, and a directly forced regime. These closed forms are exact only as the ordering parameter vanishes, they lose accuracy away from resonance, and their parametric coupling is of order unity, which invalidates small-parameter stability charts. We develop a learning framework that lifts these restrictions while staying close to the analysis. A slow-flow-informed network represents the amplitude and phase modulation, so the fast motion is analytic and the reconstruction error stays uniform over the long horizon rather than growing secularly. A bifurcation-aware operator learns the response in structure-preserving polynomial form, from which steady amplitudes follow as companion-matrix eigenvalues, stability from the modulation Jacobian, folds from the amplitude resultant, cusps from elimination theory, and parametric tongues from a certified Floquet computation. We prove asymptotic-consistency, approximation, and uniform-in-time error theorems. Validated against the closed forms and against direct integration of the unreduced oscillator, the framework reproduces the submerged-tether branch closure, charts the order-unity tongues, and returns a differentiable design manifold with analytic sensitivities, reaching a test coefficient of determination of 0.969 at a query cost several orders of magnitude below direct simulation.