Data-driven modeling and prediction of non-linearizable dynamics via spectral submanifolds

Abstract

We develop a methodology to construct low-dimensional predictive models from data sets representing essentially nonlinear (or non-linearizable) dynamical systems with a hyperbolic linear part that are subject to external forcing with finitely many frequencies. Our data-driven, sparse, nonlinear models are obtained as extended normal forms of the reduced dynamics on low-dimensional, attracting spectral submanifolds (SSMs) of the dynamical system. We illustrate the power of data-driven SSM reduction on high-dimensional numerical data sets and experimental measurements involving beam oscillations, vortex shedding and sloshing in a water tank. We find that SSM reduction trained on unforced data also predicts nonlinear response accurately under additional external forcing.

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

“…To learn SSMs from data, we use the methodology presented in [63], which is implemented in the open-source MATLAB package, SSMLearn. In what follows, we sketch the main ideas of this method before going into the details of the data-driven reduced-order models that SSMLearn can identify.…”

“…where we assume that v nl : R 2k → R p is a multivariate polynomial from order 2 to M. The matrix V 1 , as well as the coefficients of the polynomial v nl , can be found via constrained maximumlikelihood estimation of (2.4), as discussed in [63].…”

“…In that case, the autonomous SSM will serve as the leading order approximation for a non-autonomous, time-periodic SSM that carries reduced, time-periodic dynamics [52,57,58,60]. With the addition of such forcing, the normal form (2.6) becomes [63]…”

“…These maximal amplitude responses can be used to calibrate the normal form forcing amplitude f with experimentally exerted forcing levels. Equations (2.9) and (2.10) have O(f ρ) accuracy [57,63], but higher-order approximations can improve this accuracy further [59]. We expect, for example, that forced backbone curves depart from those of decaying oscillations at large motion and/or large forcing amplitude values [80,81].…”

“…Our first example is a chain of lumped oscillators, while the other two involve data from laboratory experiments. Additional details and further examples can also be found in [63] and in the MATLAB live-scripts of the SSMLearn repository.…”