Globalizing manifold-based reduced models for equations and data

TL;DR

This work introduces globalized spectral submanifold (gSSM) reduction by leveraging Padé approximants to analytically continue local SSM representations, thereby extending reduced-model validity from near fixed points to global dynamical regimes. It develops both equation-driven and data-driven pipelines: equation-driven gSSMs use [N/M] Padé approximants to W(E), R(p), kappa(rho), and omega(rho); data-driven gSSMs employ rational function regression on delay-embedded observations to construct robust reduced dynamics with a common denominator. The authors demonstrate the approach on high-dimensional solid- and fluid-mechanics problems (e.g., Kolmogorov flow, von Kármán beam) and on data-driven inverted-flag experiments, showing accurate reproduction of heteroclinic connections, large-amplitude oscillations, and chaotic attractors, with improved stability and extrapolation compared to Taylor-based SSM reductions. The work highlights practical guidelines for identifying singularities, selecting Padé orders, and ensuring non-singularity, while preserving physical interpretability of reduced-coefficient parameters. Overall, gSSMs offer a principled, scalable pathway to global nonlinear model reduction applicable to complex, data-rich systems with both known and unknown governing equations.

Abstract

One of the very few mathematically rigorous nonlinear model reduction methods is the restriction of a dynamical system to a low-dimensional, sufficiently smooth, attracting invariant manifold. Such manifolds are usually found using local polynomial approximations and, hence, are limited by the unknown domains of convergence of their Taylor expansions. To address this limitation, we extend local expansions for invariant manifolds via Padé approximants, which re-express the Taylor expansions as rational functions for broader utility. This approach significantly expands the range of applicability of manifold-reduced models, enabling reduced modeling of global phenomena, such as large-scale oscillations and chaotic attractors of finite element models. We illustrate the power of globalized manifold-based model reduction on several equation-driven and data-driven examples from solid mechanics and fluid mechanics.

Figures (15)

• Figure 1: Heteroclinic orbits (black) of the Kolmogorov flow connecting $\omega_{1,2}$ and $\omega_{0}$. (a): Projection of the phase space onto three dominant Fourier modes $(1,4)$, ($0,4$) and ($2,4$). We show the slow SSM $\mathcal{W}(E)$ (black), which is tangent to the spectral subspace $E$ (grey), its order-16 Taylor expansion (red) near the fixed point $\omega_{1}$, and the order [5/5] gSSM approximation (orange). The curves are also projected to the horizontal axes. The vorticity fields corresponding to the three fixed points are shown in the insets. (b): A trajectory on the heteroclinic orbit obtained by backward integration and its SSM-reduced and gSSM-reduced counterparts. (c): SSM-reduced and gSSM-reduced dynamics.
• Figure 2: (a): von Kármán beam with clamped-free boundary conditions. (b): Trajectory of the unforced system (black) with its order-16 SSM approximation (red) and gSSM-approximation (orange) with a [5/5] and [5/4] Padé approximant. (c)-(d): Representation of the end point displacement $q_{\text{end}}$ and the full-order trajectory shown in (b) with the SSM and gSSM-reduced trajectories. (e)-(f): Forced response defined as the maximal end point displacement due to a forcing amplitude $\varepsilon = 0.5, 1.7, 2.8$. Panel (e) shows the SSM-prediction and (f) shows the gSSM prediction. A supplementary animation showing the gSSM-prediction is available at https://polybox.ethz.ch/index.php/s/ePwsPfDgHnlzlJ2.
• Figure 3: (a): Sketch of the beam in the buckled configuration, with no external forcing. (b): Trajectories in the unstable manifold of the unstable fixed point (black). Their order-18 SSM-reduced (red) and order [6/6] gSSM-reduced (orange) approximations are also shown. (c)-(d): The SSM and gSSM in the physical space with the direction field of the reduced dynamics indicated on the surface of the manifold. The predicted trajectories connecting the unstable fixed point to the stable fixed points are shown in black.
• Figure 4: (a): Buckled von Kármán beam with periodic external forcing. (b): Time series of the reduced coordinate $\eta_1$ on a chaotic trajectory of the full system (black). Also shown are the SSM-reduced forced model, which diverges immediately (red), and the gSSM-reduced trajectory (orange). (c): Sampling of the Poincaré map of the gSSM-reduced model (orange) and the true system (black). (d)-(e): The autonomous SSM and gSSM with the chaotic trajectory of the full model. A supplementary animation comparing the SSM- and gSSM-predictions is available at https://polybox.ethz.ch/index.php/s/ozX0r0Gx9X2Ryk2
• Figure 5: (a): Experimental setup of the inverted flag and snapshot of the experiment (b), courtesy of Giovanni Berti. The geometric parameters are $H=150$ mm, $L=150$ mm, $U=1\ \text{m}/\text{s}$. (c): Phase portrait of the gSSM-reduced dynamics of the inverted flag obtained from the $[5/5](\boldsymbol{\eta})$ approximation. The unstable fixed points are marked with colored dots. Blue curves denote the stable manifold of the undeflected state which connects to the two coexisting deflected fixed points. The red curve denotes the unstable manifold of the saddle, which wraps onto the stable limit cycle. (d)-(e): Predictions of the reduced model on test trajectories. The black curve is the true trajectory and the dotted orange curve is the gSSM-prediction.