This talk explores the structure of kernel functions and Lagrange multipliers to develop a dual method for data-driven discrepancy modeling in finite strain elastodynamics. The proposed method leverages the structure of Lagrange multipliers to derive loss functions that are composed of residuals from the first-principles theory and measurement terms containing the sparse data. It facilitates the derivation of discrepancies-informed closure terms that are operated upon by the kernel function. The structure of the kernel function is analyzed, and its role in data embedding through least-squares regression analysis is examined. The effect of the variationally embedded loss function on the system’s time-dependent response is evaluated under various types of discrepancies. Test cases are presented to demonstrate that when reference data is provided, the Data-Driven Variational (DDV) method can accurately reconstruct the displacement and velocity fields in the presence of model and parameter discrepancies. DDV method also successfully reconstructs the time histories of strain and kinetic energy and restores the eigenvalues and eigenvectors of deficient dynamical system to closely match those of the target. The DDV reconstructed displacement time history provides the synthetic data that is employed in an inverse problem to extract the material properties of the target system, and error bounds on the extracted material parameters are discussed.