Fig. 4. Construction of a data-driven nonlinear reduced-order model on the slowest SSM of a von Kármán beam.

(a) System setup and the initial condition for the decaying training trajectory shown in (b) in terms of the midpoint displacement. (c) The SSM, ${\mathcal{M}}_0$, in the delay embedding space, shown along with the reconstructed test trajectory in extended normal form coordinates. (d) Zoom of the prediction of the reduced order model for the test trajectory not used in learning ${\mathcal{M}}_0$. (e) Closed-form backbone curve and forced response curve (FRC) predictions (ϵ > 0, ℓ = 1) by SSMLearn are compared with analytic FRC calculations performed by SSMTool³⁶ and with results from numerical integration of the forced-damped beam.