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Link to original content: https://pubmed.ncbi.nlm.nih.gov/35169152
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. 2022 Feb 15;13(1):872.
doi: 10.1038/s41467-022-28518-y.

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

Mattia Cenedese  1 , Joar Axås  1 , Bastian Bäuerlein  2   3 , Kerstin Avila  2   3 , George Haller  4
Affiliations

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

Mattia Cenedese et al. Nat Commun. .

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.

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Conflict of interest statement

The authors declare no competing interests.

Figures

Fig. 1
Fig. 1. Examples of non-linearizable systems.
a Snap-through instability of a microelectro-mechanical (MEMS) device with three coexisting equilibria (Sandia National Laboratories). b Wind-tunnel flutter of an airplane prototype, involving a fixed point and coexisting limit cycles (NASA Langley Research Center). c Swirling clouds behind an island in the Pacific ocean, forming a vortex street with coexisting isolated hyperbolic and elliptic trajectories for the dynamical system describing fluid particle motion (USGS/NASA). d Phase portrait of the damped, double-well Duffing oscillator x¨+x˙−x+βx3=0 with β > 0, the most broadly used model for nonlinear systems with coexisting domains of attraction (colored), such as the MEMS device in plot (a). e Nonlinear response amplitude (∣x(t)∣max) in the forced-damped, single-well Duffing oscillator, x¨+x˙+x+βx3=fcosωt with β > 0, under variations of the forcing frequency ω and forcing amplitude f. Coexisting stable and unstable periodic responses show non-linearizable dynamics conclusively for this classic model.
Fig. 2
Fig. 2. Linear vs. nonlinear model reduction.
a Reduction of linear dynamics via Galerkin projection. The slowest spectral subspace, E1 = E1 (green), and the modal subspace, E2 (black), span together the second slowest spectral subspace, E2 = E1 ⊕ E2. The full dynamics (red curve) can be projected onto E1 to yield a reduced slow model without transients. Projection of the full dynamics onto E2 (blue curve) yields a reduced model that also captures the slowest decaying transient. Further, faster-decaying transients can be captured by projections onto slow spectral subspaces, Ek, with k > 1. b Reduction of nonlinearizable dynamics via restriction to spectral submanifolds (SSMs) in the ϵ = 0 limit of nonlinear, non-autonomous systems forced with ℓ frequencies. An SSM, W(E, Ωt; 0), is the unique, smoothest, nonlinear continuation of a nonresonant spectral subspace E. Specifically, the slowest SSM, W(Ek, Ωt; 0) (green), is the unique, smoothest, nonlinear continuation of the slowest spectral subspace, Ek. Nonlinearizability of the full dynamics follows if isolated stationary states coexist on at least one of the SSMs. The time-quasiperiodic SSMs for ϵ > 0, denoted W(E, Ωt; ϵ), are not shown here but they are O(ϵ)Cr-close to the structures shown, as discussed by.
Fig. 3
Fig. 3. Schematics of SSMLearn.
First, he data-driven, SSM-based model reduction algorithm implemented in SSMLearn diagnoses and approximates the dominant SSM from the input data. Next, it constructs a data-driven reduced-order model as an extended normal form on the SSM. Finally, the algorithm uses this model to predict individual unforced trajectories and the response of the system under additional forcing.
Fig. 4
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, M0, 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 M0. (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 SSMTool and with results from numerical integration of the forced-damped beam.
Fig. 5
Fig. 5. Data-driven nonlinear SSM-reduced model on the unstable manifold of the steady solution of the flow past a cylinder.
a Problem setup. b, c Snapshots of the steady solution and the time-periodic vortex-shedding solution (limit cycle, in magenta). d Trajectories projected on the 2-dim. subspace spanned by the two-leading POD modes of the limit cycle. e Model-based reconstruction of the test trajectory (not used in learning the SSM) in terms of velocities and pressures measured at a location q shown in plot a. f The SSM formed by the unstable manifold of the origin, along with some reduced trajectories, plotted over the unstable eigenspace UE ≡ E1; ∥UE⊥∥ denotes the normed projection onto the orthogonal complement UE⊥. g Same but projected over velocity and pressure coordinates.
Fig. 6
Fig. 6. Data-driven nonlinear reduced-order model on the slowest SSM of fluid sloshing in a tank.
a Setup for the sloshing experiment. b Decaying model-testing trajectory and its reconstruction from an unforced, SSM-based model c The geometry of the embedded SSM d Nonlinear damping α(ρ) from the SSM-reduced dynamics e, f Closed form, SSM-based predictions of the FRCs and the response phases ψ0 for three different forcing amplitudes (solid lines), with their experimental confirmation superimposed (dots).
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