  In this paper, we present a deep learning-based reduced-order model (DL-ROM)
for the stability prediction of unsteady 3D fluid-structure interaction
systems. The proposed DL-ROM has the format of a nonlinear state-space model
and employs a recurrent neural network with long short-term memory (LSTM). We
consider a canonical fluid-structure system of an elastically-mounted sphere
coupled with incompressible fluid flow in a state-space format. We develop a
nonlinear data-driven coupling for predicting unsteady forces and
vortex-induced vibration (VIV) lock-in of the freely vibrating sphere in a
transverse direction. We design an input-output relationship as a temporal
sequence of force and displacement datasets for a low-dimensional approximation
of the fluid-structure system. Based on the prior knowledge of the VIV lock-in
process, the input function contains a range of frequencies and amplitudes,
which enables an efficient DL-ROM without the need for a massive training
dataset for the low-dimensional modeling. Once trained, the network provides a
nonlinear mapping of input-output dynamics that can predict the coupled
fluid-structure dynamics for a longer horizon via the feedback process. By
integrating the LSTM network with the eigensystem realization algorithm (ERA),
we construct a data-driven state-space model for the reduced-order stability
analysis. We investigate the underlying mechanism and stability characteristics
of VIV via an eigenvalue selection process. To understand the frequency lock-in
mechanism, we study the eigenvalue trajectories for a range of the reduced
oscillation frequencies and the mass ratios. Consistent with the full-order
simulations, the frequency lock-in branches are accurately captured by the
combined LSTM-ERA procedure. The proposed DL-ROM aligns with the development of
physics-based digital twin of engineering systems involving fluid-structure
interactions.
