Data-driven variational method for discrepancy modeling: Dynamics with small-strain nonlinear elasticity and viscoelasticity

The effective inclusion of a priori knowledge when embedding known data in physics-based models of dynamical systems can ensure that the reconstructed model respects physical principles, while simultaneously improving the accuracy of the solution in the previously unseen regions of state space. This paper presents a physics-constrained data-driven discrepancy modeling method that variationally embeds known data in the modeling framework. The hierarchical structure of the method yields fine scale variational equations that facilitate the derivation of residuals which are comprised of the first-principles theory and sensor-based data from the dynamical system. The embedding of the sensor data via residual terms leads to discrepancy-informed closure models that yield a method which is driven not only by boundary and initial conditions, but also by measurements that are taken at only a few observation points in the target system. Specifically, the data-embedding term serves as residual-based least-squares loss function, thus retaining variational consistency. Another important relation arises from the interpretation of the stabilization tensor as a kernel function, thereby incorporating a priori knowledge of the problem and adding computational intelligence to the modeling framework. Numerical test cases show that when known data is taken into account, the data driven variational (DDV) method can correctly predict the system response in the presence of several types of discrepancies. Specifically, the damped solution and correct energy time histories are recovered by including known data in the undamped situation. Morlet wavelet analyses reveal that the surrogate problem with embedded data recovers the fundamental frequency band of the target system. The enhanced stability and accuracy of the DDV method is manifested via reconstructed displacement and velocity fields that yield time histories of strain and kinetic energies which match the target systems. The proposed DDV method also serves as a procedure for restoring eigenvalues and eigenvectors of a deficient dynamical system when known data is taken into account, as shown in the numerical test cases presented here.