Data-Driven Reduced-Order Modeling of Systems with Memory via Spectral Submanifolds

Modeling dynamical systems that exhibit memory effects, such as viscoelasticity or hysteresis, remains a significant challenge, especially in the absence of explicit constitutive laws. This situation frequently arises in soft materials, which display both solid-like and liquid-like behavior. Owing to their compatibility with biological tissues and their adaptability to complex geometries, these materials are increasingly employed in biomedical devices and soft robotics. Their mechanical response depends not only on the current deformation but also on the loading history, giving rise to phenomena such as creep, relaxation, and hysteresis. In this talk, we discuss an extension of data-driven spectral submanifold (SSM) reduction to systems with memory effects. Although memory makes such systems formally infinite-dimensional, we obtain a finite-dimensional reduced-order model by introducing an internal state that encapsulates the memory dynamics and can be inferred directly from data. We validate the approach on experimental data from vibrating hydrogels, showing excellent agreement in predicting backbone and forced response curves using a three-dimensional nonlinear model.