This study introduces a novel model- and data-driven methodology for addressing soil-structure interaction (SSI) problems in elastodynamics. The proposed approach combines the finite element method (FEM) for modelling the structure with a physics-informed neural networks (PINN) to simulate elastic wave propagation in the soil, all formulated in the time domain. PINNs, as a promising class of NN-based partial differential equations (PDE) solvers, utilise standard feedforward networks but redefine the loss function to incorporate the PDE for elastic wave propagation as well as initial/boundary conditions. In this work, FEM and PINN are coupled by enforcing displacement/traction continuity at the soil-structure interface, with these conditions included in the PINN loss function. Traditional numerical approaches like FEM-BEM or FEM-SBM for SSI problems can be computationally expensive, particularly when solving Green's functions for large numbers of evaluation points in the soil domain. By contrast, the proposed framework offers improved computational efficiency due to the PINN's capacity to handle complex wave propagation scenarios without explicitly solving Green’s functions. The integration of benefits provided by FEM, such as the ability to model complex geometries and materials, along with the computational efficiency and accuracy presented by PINN, results in a robust, user-friendly, and precise framework for solving SSI problems in elastodynamics. The accuracy of this method is validated by comparing it with traditional 2D FEM and 2D FEM-BEM approaches, applied to a case study involving a thin cylindrical shell embedded in a homogeneous full-space medium. The results are assessed in both the time and frequency domains, where the proposed 2D FE-PINN approach shows excellent agreement with established reference solutions. The results highlight the advantages of the proposed method in terms of modelling simplicity, numerical efficiency, and robustness compared to previous methodologies. Its computational efficiency makes this approach particularly suitable for real-time predictions.