This talk presents a novel method for modeling and estimating the dynamics of a continuous structure based on a limited number of noisy measurements. The goal is reached using a Kalman filter in synergy with the recently developed mathematical framework known as the Theory of Functional Connections (TFC). The TFC allows to derive a functional expression capable of representing the entire space of the functions that satisfy a given set of linear and, in some cases, nonlinear constraints. The proposed approach exploits the possibilities offered by the TFC to derive an approximated dynamical model for the flexible system using the Lagrangian mechanics. The result is a representation of the structural dynamics using a finite number of states, in contrast to the infinite-dimensional model that would be obtained by application of the traditional continuum mechanics models that are based on sets of partial differential equations. The limited number of states enables the application of the well-known Kalman filter framework to improve the estimation of the displacements and displacement velocities. In addition, the continuous displacement field of the structure can be reconstructed with high fidelity. The theoretical development of the method is presented in relation to the case of a Euler-Bernoulli beam. Finally, the obtained model is used to carry out a simulation campaign aimed at assessing the accuracy, efficiency, and robustness of the proposed method.