Despite the significant progress in recent years in numerical simulations using physics-informed neural networks (PINNs), it remains challenging to apply PINNs to solve coupled fluid–structure interaction problems. This work employs PINNs to approximate two-dimensional fluid–structure interaction in both forward and inverse problem settings. For the forward problem, data are randomly sampled from the fluid domain and combined with governing physical laws, including the Robin boundary condition and the Euler–Bernoulli equation, to infer the beam displacement. For the inverse problem, in which a parameter in the Euler–Bernoulli equation is unknown, the problem is solved using both data and physical constraints to infer the fluid pressure and pressure gradient at the interface, from which the beam pressure and acceleration are computed via the Robin boundary condition. Finally, using the data obtained in the previous step, both the beam displacement and the unknown parameter are identified. The results demonstrate that PINNs can accurately solve fluid–structure coupled problems by enforcing conservation laws at randomly distributed collocation points, achieving relative L2 errors below 1%, particularly in inverse problems, for which conventional techniques are often ineffective.