Data-driven Koopman linearization of nonlinear elastodynamics problems

This work investigates Koopman-based data-driven approximations for the reconstruction of smooth nonlinear elastodynamic wave responses. A one-dimensional longitudinal motion problem for an elastic bar governed by a convex strain-energy density is considered as a benchmark. Reference snapshots are generated through a finite-difference spatial discretization combined with Runge--Kutta time integration, and are subsequently used to construct standard Dynamic Mode Decomposition (DMD), polynomial Extended Dynamic Mode Decomposition (EDMD), and Gaussian Kernel Dynamic Mode Decomposition (KDMD) models. The objective is to assess whether finite-dimensional linear representations, obtained directly from time-resolved data, can reproduce the nonlinear elastodynamic response with a reduced number of modes. The numerical comparisons show that DMD accurately reconstructs the reference solution when a sufficient number of modes is retained, while EDMD improves the low-rank approximation by lifting the dynamics to a polynomial observable space. Among the tested approaches, the Gaussian KDMD formulation provides the most accurate reconstruction and yields a stable discrete-time eigenvalue distribution over the simulated interval. The results indicate that Koopman-based decompositions can serve as non-intrusive reduced-order reconstruction tools for smooth nonlinear elastodynamic problems, provided that sufficiently resolved snapshot data are available. 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.