  Solving complex fluid-structure interaction (FSI) problems, characterized by
nonlinear partial differential equations, is crucial in various scientific and
engineering applications. Traditional computational fluid dynamics (CFD)
solvers are insufficient to meet the growing requirements for large-scale and
long-period simulations. Fortunately, the rapid advancement in neural networks,
especially neural operator learning mappings between function spaces, has
introduced novel approaches to tackle these challenges via data-driven
modeling. In this paper, we propose a Fourier neural operator-based
fluid-structure interaction solver (FNO-based FSI solver) for efficient
simulation of FSI problems, where the solid solver based on the finite
difference method is seamlessly integrated with the Fourier neural operator to
predict incompressible flow using the immersed boundary method. We analyze the
performance of the FNO-based FSI solver in the following three situations:
training data with or without the steady state, training method with one-step
label or multi-step labels, and prediction in interpolation or extrapolation.
We find that the best performance for interpolation is achieved by training the
operator with multi-step labels using steady-state data. Finally, we train the
FNO-based FSI solver using this optimal training method and apply it to vesicle
dynamics. The results show that the FNO-based FSI solver is capable of
capturing the variations in the fluid and the vesicle.
