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

Mattia Cenedese,Kerstin Avila,George Haller,Bastian Bäuerlein,Joar Axås

doi:10.1038/s41467-022-28518-y

Mattia Cenedese, Kerstin Avila + Show 3 more

Open Access

https://doi.org/10.1038/s41467-022-28518-y

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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.

Highlights

ObjectivesOur objective is to learn from numerically generated trajectory data the reduced dynamics on the slowest, two-dimensional spectral submanifolds (SSMs), W(E1), of the system, defined over the slowest two-dimensional (d = 2) eigenspace E1 of the linear part
A recent result in dynamical systems is that all eigenspaces of linearized systems admit unique nonlinear continuations under well-defined mathematical conditions
forced response curve (FRC) are hallmarks of non-linearizable dynamics and cannot be captured by the model reduction techniques we reviewed in the Introduction for linearizable systems

Summary

Objectives

Our objective is to learn from numerically generated trajectory data the reduced dynamics on the slowest, two-dimensional SSM, W(E1), of the system, defined over the slowest two-dimensional (d = 2) eigenspace E1 of the linear part

Methods

Results

Discussion

Conclusion