This study builds on previous work developing Finite Element-based Physics-Informed Neural Networks (FE-PINNs), which combine the Finite Element (FE) method with Physics-Informed Neural Networks (PINNs) to solve inverse problems. In earlier research, we introduced the FE-PINN framework and demonstrated its effectiveness in performing parameter regression on simple structural models. In this work, we propose a temporally discrete variant, which extends the original method by discretizing the governing equations not only in space but also in time. We compare the performance of both frameworks --Continuous FE-PINN and Discrete FE-PINN -- under varying levels of noise, initialization errors, and available system measurements. Continuous FE-PINN achieved reliable convergence with up to 30\% initialization error and across all tested noise levels. In contrast, Discrete FE-PINN, while more computationally efficient, was sensitive to noise and performed reliably only in homogeneous media. Both methods required at least two measured degrees of freedom (DOFs) for accurate parameter estimation. Continuous FE-PINN is better suited for noisy environments, while Discrete FE-PINN is advantageous in noise-free settings with limited computational resources. These results provide a foundation for applying FE-PINNs to structural analysis and lay the groundwork for future extensions to more complex systems.