Data-driven nonlinear model reduction to spectral submanifolds in mechanical systems

Abstract

While data-driven model reduction techniques are well-established for linearizable mechanical systems, general approaches to reducing nonlinearizable systems with multiple coexisting steady states have been unavailable. In this paper, we review such a data-driven nonlinear model reduction methodology based on spectral submanifolds. As input, this approach takes observations of unforced nonlinear oscillations to construct normal forms of the dynamics reduced to very low-dimensional invariant manifolds. These normal forms capture amplitude-dependent properties and are accurate enough to provide predictions for nonlinearizable system response under the additions of external forcing. We illustrate these results on examples from structural vibrations, featuring both synthetic and experimental data. This article is part of the theme issue ‘Data-driven prediction in dynamical systems’.

Fast data-driven model reduction for nonlinear dynamical systems

“…If the eigenvalues corresponding to V d are nonresonant with the remaining p − d eigenvalues of A, V d admits a unique smoothest, invariant, nonlinear continuation M under addition of the higher-order terms [9]. We refer to M as a spectral submanifold (SSM) [11,22]. In the case of a resonance between V d and the rest of the spectrum of A, we can include the resonant modal subspace in V d and thus obtain a higherdimensional SSM.…”

“…A numerical package, SSMTool, has been developed for the computation of SSMs from arbitrary finite-dimensional nonlinear systems [25,26]. More recently, a data-driven method was developed to compute SSMs purely from observables of the dynamical system [10,11]. In the following, we first review the literature on data-driven reduced-order modeling on SSMs, and then propose a simplified approach that is applicable under further assumptions.…”

“…Data-driven model reduction to SSMs has been successfully applied to both numerical and experimental datasets [10,11,29]. Yet, for very high-dimensional systems, the implicit optimization problems ( 3) and ( 7) may be intractable.…”

“…While data-driven SSM-based model reduction has been successfully applied to both numerical and experimental data for fluid problems, structural dynamics, and fluid-structure interaction [10,11,29], the required implicit optimization can be computationally demanding for high-dimensional systems. This paper introduces an alternative method that explicitly computes an SSM followed by a classic normal form transformation in the reduced coordinates.…”

Data-driven prediction in dynamical systems: recent developments

“…Cenedese et al . [ 186 ] discuss a new data-driven reduced-order modelling approach in the context of mechanical vibrations, which is dynamics-based rather than physics-informed. Built on the recent theory of spectral submanifolds (SSMs), this approach identifies very low-dimensional, sparse models over different time scales by restricting the full system dynamics to a nested family of attractors.…”