Data-Driven Nonlinear Model Reduction to Spectral Submanifolds via Oblique Projection

TL;DR

This work addresses the challenge of reducing high-dimensional nonlinear dynamics when the linear part is strongly non-normal, which breaks the effectiveness of normal projection onto the slow subspace. It introduces a data-driven oblique projection that aligns off-SSM trajectories with the primary SSM $\ \mathcal{W}(E)$ by projecting along the fast subspace $F$, and constructs this projection using a compact data workflow that combines DMD/SVD to identify $E$, a B-optimization to minimize backbone oscillations, and a polynomial regression-based SSM parametrization within the SSMLearn framework. The main contributions are (i) a practical oblique-projection algorithm $P=Q(B^TQ)^{-1}B^T$ learned from limited data, (ii) a data-efficient procedure to recover $E$ and $F^ot$ from a single trajectory, and (iii) validated improvements in backbone curves and forced-response predictions across numerical Shaw-Pierre systems and an experimental nonlinear beam. This has significant implications for enabling accurate reduced-order models in highly non-normal settings, including experiments, with reduced data requirements and straightforward integration into existing SSM workflows.

Abstract

The dynamics in a primary Spectral Submanifold (SSM) constructed over the slowest modes of a dynamical system provide an ideal reduced-order model for nearby trajectories. Modeling the dynamics of trajectories further away from the primary SSM, however, is difficult if the linear part of the system exhibits strong non-normal behavior. Such non-normality implies that simply projecting trajectories onto SSMs along directions normal to the slow linear modes will not pair those trajectories correctly with their reduced counterparts on the SSMs. In principle, a well-defined nonlinear projection along a stable invariant foliation exists and would exactly match the full dynamics to the SSM-reduced dynamics. This foliation, however, cannot realistically be constructed from practically feasible amounts and distributions of experimental data. Here we develop an oblique projection technique that is able to approximate this foliation efficiently, even from a single experimental trajectory of a significantly non-normal and nonlinear beam.

Figures (11)

• Figure 1: Subfigures \ref{['1d_ex_orth']} and \ref{['1d_ex_obl']} represent the phase space of a two-dimensional linear system with real eigenvalues, when the slow and fast directions are normal (eq. \ref{['2d_example']}) or non-normal (eq. \ref{['2d_example_nonnormal']}) to each other, respectively. Subfigure \ref{['2d_ex_backones']}: if we consider a linear system with two oscillating modes (eq. \ref{['2d_linear_system']}), the backbone curve of a decaying trajectory lying on a fractional SSM (black) exhibits oscillations around the backbone curve of decaying trajectories on the primary SSM (dashed, red), when the slow and fast subspaces are not normal (non-normal linear part). Otherwise, the backbone curves are identical.
• Figure 2: Aspects of non-normality between the slow spectral subspace $E$ and the fast spectral subspace. \ref{['normal_vs_oblique_backbone']} Two different ways of extracting the backbone curve of system \ref{['2d_linear_system']} with parameter values $\alpha = 0.3$, $\beta = 0.63$, $\omega = 3$, $\nu = 8$, $A = B = C = D = 1$, $\mathbf{x}_0 = \left(1,1,0.8,0.8 \right)^\mathrm{T}$: via normal projection onto $E$ (light blue) and via oblique, $F$-parallel projection onto $E$ (red). \ref{['figure_oblique_projection']} The construction of an $F$-parallel oblique projection $\mathbf{P}$ onto $E$.
• Figure 3: Comparison between normally and obliquely projected reduced coordinates that parametrize the SSM. We note that the $\xi$ coordinate obtained from the oblique projection parametrizes $\mathcal{W}\left(E\right)$ on a much larger domain than the $x_1$ coordinate obtained from normal projection.
• Figure 4: Decaying test trajectory of the linear non-normal system \ref{['2d_linear_system']}, with initial condition on a fractional SSM and its prediction from \ref{['2d_example_SSMLearn_orthogonal']} normal SSM reduction and \ref{['2d_example_SSMLearn_oblique']} from oblique SSM reduction. Both SSMs are computed from data using the SSMLearn algorithm https://github.com/haller-group/SSMLearn.
• Figure 5: Geometry of the two-degree-of-freedom mechanical system studied in shaw_pierre_93, modified by adding a damper between the left mass and the wall.