Data-driven Koopman linearization of nonlinear elastodynamics problems

Abstract A data-driven Koopman procedure is utilized to linearize the one-dimensional nonlinear constitutive equations of elastodynamics. Initially, we transform the nonlinear elastodynamics problem into a nonlinear system of equations, which is integrated through an established numerical scheme. The integration produces high-dimensional data (displacements), which constitutes the input of our approach. Then, the Koopman theory and the Dynamic Mode Decomposition (DMD) algorithm for system identification are applied, which produces a numerical linearization setup of the initial problem. The results demonstrate that, although the Koopman operator is an infinite-dimensional linear operator, it admits accurate low-dimensional approximations. The significance of this approach lies in its ability to leverage machine learning and system identification techniques to address nonlinear elastodynamics problems. Moreover, the method is well-suited for elastodynamic experimental applications, as it requires only displacement measurements as input and is independent of the underlying numerical scheme.