Data-Driven Modeling and Prediction of Non-Linearizable Dynamics via Spectral Submanifolds

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

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Data-Driven Modeling and Prediction of Non-Linearizable Dynamics via Spectral Submanifolds

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

arXiv:2201.04976v1Jan 13, 2022math.DScs.LGeess.SYnlin.CD

TL;DR

The paper addresses the lack of reliable low-dimensional models for essentially nonlinear dynamical systems using only data. It learns sparse extended normal forms on attracting spectral submanifolds and demonstrates accurate prediction of forced responses from unforced training data across numerical and experimental examples.

Problem
Reliable, robust identification of low-dimensional nonlinear models remains unavailable for essentially nonlinear systems when only data are available.
Method
The method learns sparse extended normal forms for reduced dynamics on low-dimensional, attracting spectral submanifolds using data-driven embeddings and normalization.
Results
The models accurately predict nonlinear forced response despite being trained on unforced data, including beam oscillations, vortex shedding, and water sloshing.
Takeaways & Limitations
SSM reduction provides explicit, deterministic, low-dimensional models that can retain qualitative meaning beyond the training data and under moderate forcing.
Takeaways & Limitations
The approach assumes quasiperiodic external forcing and smooth system dynamics, while higher-dimensional SSMs and multi-harmonic forcing were not considered in this report.

Abstract

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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.

1 Introduction

The paper targets reliable, data-driven low-dimensional models for essentially nonlinear systems, where existing reduction methods are limited. It uses spectral submanifolds and extended normal forms to obtain sparse predictive models that can capture nonlinear and forced dynamics.

Low-dimensional reduced models are needed to reduce computational costs and support interpretation, design optimization, and controllability, but no generally applicable reliable identification procedure has emerged.
POD requires governing equations, while DMD and Koopman-based approaches are mainly justified near fixed points or for globally linearizable systems.
Non-linearizable systems are characterized by coexisting isolated stationary states and arise in phenomena including flutter, buckling, bistable MEMS, traffic jams, and climate tipping points.
The proposed approach learns sparse reduced dynamics as extended normal forms on slow spectral submanifolds, which are nonlinear continuations of non-resonant eigenspaces.
For non-resonant oscillatory systems, two-dimensional SSM models capture asymptotic dynamics and can accurately predict forced response despite training only on unforced data.
The methodology is demonstrated on numerical and experimental data, with implementations available in the open-source SSMLearn package.

2 Results

The paper develops data-driven reduction of nonlinear dynamics by learning sparse extended normal forms on attracting spectral submanifolds, including from observable measurements. These reduced models can capture nonlinear behavior and predict forced response using unforced training data.

Spectral submanifolds: For hyperbolic systems with moderate quasiperiodic forcing, the autonomous dynamics remains relevant to the full forced system.The forcing enters as εf1(x, Ωt; ε), with ε small and Ω containing rationally independent frequencies.
Spectral submanifolds: Spectral submanifolds are the unique smoothest nonlinear continuations of non-resonant spectral subspaces.They provide low-dimensional invariant structures for reducing nonlinear dynamics without increasing the reduced dimension.
Observable embedding: Long-term trajectory data approaches an attracting SSM, enabling reduced modeling even when measurements are available only through observables.Generic observable vectors embed a d-dimensional SSM when p > 2(d + ℓ), while delay embeddings provide an alternative for a single measured quantity.
Data-driven reduced dynamics: The reduced dynamics on an embedded SSM is learned in Poincaré normal form, but extended normal forms are required to retain nonlinear behavior over larger domains.The ordinary normal form near the origin can reduce to the linearized dynamics, whereas the extended form can capture nearby coexisting stationary states.
Forced response prediction: SSM reduction can predict forced response curves and dissipative backbone curves from unforced trajectories.The forced response prediction uses the amplitude-dependent terms α(ρ) and ω(ρ), and the predictions are confirmed by numerical simulation or laboratory experiments.

2.5 Examples

The examples apply data-driven SSM reduction to nonlinear beam, vortex-shedding, and sloshing systems, producing reduced models that reconstruct testing trajectories and predict forced responses.

Examples: The examples use numerical and experimental trajectory data to build data-driven SSM-based reduced models, with NMTE quantifying reconstruction error.The workflow and error measure are illustrated across beam, cylinder-flow, and sloshing examples.
Finite-element model of a damped-forced beam: A fifth-order delay embedding of beam midpoint displacement embeds the two-dimensional slowest SSM, enabling a seventh-order extended normal form with NMTE = 0.027 on testing data.The model is trained on one decaying trajectory and uses midpoint displacement as its scalar observable.
Finite-element model of a damped-forced beam: The beam model trained on a single decaying trajectory predicts forced-response curves and is compared with analytic SSMTool calculations and numerical simulations.Data-based predictions agree with analytic curves at low forcing but depart at higher amplitudes, where numerical simulations confirm the data-based curve.
Fluid sloshing experiments in a tank: For fluid sloshing, a cubic Stuart–Landau-type model reaches NMTE = 1.88% on testing data, and its closed-form forced-response predictions match experimental FRCs across three forcing amplitudes.The model also predicts the forced-response phase accurately and remains effective for response amplitudes outside the training range.

3 Discussion

The paper presents SSM reduction as a data-driven hierarchy of low-dimensional models for nonlinear systems, demonstrated across three physical examples. The resulting models are sparse, accurate, and can predict forced steady states from unforced decay data, while the report identifies several scope limitations.

Slow SSMs form a nested hierarchy of attracting reduced-order models, with nearby generic trajectories synchronizing exponentially fast to their dynamics.
Two-dimensional extended normal forms on the slowest SSMs provide sparse and accurate models for forced beam oscillations, vortex shedding, and water sloshing.The implementations are available in the open-source SSMLearn MATLAB package.
The report does not consider higher-dimensional SSMs or multi-harmonic forcing, although SSMLearn is equipped to handle them.Higher-dimensional SSMs are needed for internal resonances or more accurate capture of initial transients.
The approach assumes quasiperiodic external forcing and smooth system dynamics for non-autonomous systems.Forcing signals and nonsmooth-system data are discussed as boundaries addressed by approximations or ongoing work.

A.1 Existence of SSMs

SSMs generalize spectral subspaces to nonlinear, possibly quasiperiodically forced systems and provide nested attracting manifolds when suitable non-resonance conditions hold. Resonant subspaces can be enlarged until an SSM exists, linking the mathematical construction to the interacting modes required for reduction.

SSMs are nonlinear invariant-manifold continuations of spectral subspaces, with existence, smoothness, and uniqueness results extended to systems with quasiperiodic time dependence.
For stable SSMs, the absolute spectral quotient is an integer determined by spectral decay rates and governs the relevant non-resonance conditions.
A violating spectral subspace can be enlarged until non-resonance holds; this agrees with including all physically interacting modes in an accurate reduced model.
Slow SSMs form nested local attractors, and nearby trajectories approach one of them exponentially fast; forcing near-resonances can generate non-linearizable dynamics and nontrivial steady states.
For discrete-time systems, the same SSM results apply after replacing continuous-time eigenvalues λ_k with log μ_k and real parts with log|μ_k|.
An invariant spectral foliation near an SSM could provide a nonlinear analogue of linear modal superposition, but constructing it from data remains challenging.

A.2 Embedding the SSM in the observable space

The SSM is embedded and learned in observable space using delay-coordinate results and a graph over its tangent space. Periodic forcing improves the sufficient embedding estimate, while folds can require alternative parametrizations.

For a d-dimensional compact SSM subset under generic assumptions, delay embedding is guaranteed for almost all observables when p > 2(d + l).
For time-periodic forcing, the period-T sampling map has a time-independent SSM, and the sufficient embedding estimate improves to p > 2d.
The embedded autonomous SSM is learned as a graph over its tangent space at the shifted fixed-point origin, which can cover non-linearizable dynamics in suitable observable domains.
Reduced coordinates are defined by orthogonal projection onto the learned tangent space, while the manifold is approximated with linear and higher-order monomial terms fitted by minimizing reconstruction error.
The graph parametrization breaks down when the embedded manifold develops a fold, motivating POD-based tangent directions or additional delayed observations.
Delay-embedded SSMs are nearly flat for small delays and low embedding dimensions, a property described as universal in that regime.
For small forcing, the autonomous SSM captures the bulk nonlinear behavior because forced reduced dynamics can be computed as an additive perturbation of autonomous dynamics.

A.3 SSM dynamics via extended normal forms

The reduced dynamics are transformed into extended normal forms that retain near-resonant nonlinear terms, allowing data-driven models to represent behavior beyond the locally linearizable regime. Coefficients are learned from Jacobian estimates and conjugacy errors over observed data.

Classical normal-form transformations linearize the hyperbolic reduced dynamics only on a sufficiently small domain where the system is linearizable.
Extended normal forms retain polynomial terms whose removal would create small denominators and shrink the transformation’s convergence domain.
The extended normal form uses n(z; N) = Λz + Nz2:N together with polynomial transformations h(z; H) and h^-1(η; H⋆).
Near-resonances are identified from integer combinations of eigenvalues, using a threshold of 10^-8 by default in SSMLearn.
Near-resonant transformation coefficients are set to zero while their corresponding monomials remain in the normal-form dynamics.
For two-dimensional complex-conjugate dynamics, the cubic polar form follows from z = (ρe^iθ, ρe^-iθ) and h21 = β + iγ.
The data-driven construction estimates the reduced Jacobian, determines admissible monomials, and fits nonzero transformation and normal-form coefficients by minimizing conjugacy error.
Finite differences are suitable when sampling is fast relative to the SSM’s fastest timescale; otherwise, the discrete-time formulation uses one-step prediction error.

A.4 Prediction of forced response from unforced training data

The forced SSM-reduced normal form incorporates resonant interactions between SSM eigenfrequencies and external forcing, enabling forced-response prediction from unforced data after calibrating observable-space forcing amplitude.

The quasiperiodic SSM-reduced normal form includes forcing amplitudes and phases for resonant forcing harmonics affecting each SSM mode.The resonant harmonic sets identify which forcing frequencies interact with each mode.
Resonant interactions between SSM eigenfrequencies and external forcing frequencies can produce non-linearizable dynamics in the reduced model.Numerical continuation can locate coexisting steady states under varying forcing amplitudes and frequencies.
The forced response curve is predicted from the autonomous SSM normal form, whose terms are constructed using unforced trajectories.The observable-space forcing amplitude must first be related to the physical forcing amplitude through measured periodic responses.
Calibration relates physical forcing amplitude and frequency to normalized reduced response amplitude before applying the forced reduced model.The procedure uses one forcing amplitude-frequency pair, measures the periodic observable response, and computes the corresponding normalized reduced response amplitude.

A.5 Summary of the algorithm

SSMLearn takes measured unforced trajectories and user-specified reduction settings, constructs an embedded SSM and extended normal form, and supports forced-response analysis and continuation.

SSMLearn accepts measured trajectories of the autonomous system, SSM dimension, polynomial approximation orders, and discrete- or continuous-time dynamics.The package automatically augments observables when their number is insufficient for manifold embedding.
SSMLearn provides analysis tools for reduced dynamics and forced-response prediction from unforced training data.Its package includes the COCO numerical-continuation core for computing steady states and supporting nonlinear-control design.
Qualitative or partial knowledge of linearized modes and frequencies can help select suitable initial trajectories for SSMLearn.The resonance decay method empirically isolates a resonant periodic motion on a targeted two-dimensional stable SSM in experiments.
The algorithm embeds data in an observable space, identifies manifold coordinates, and estimates normalized reduced dynamics through automated identification.The prescribed embedding dimension satisfies p > 2d.

Competing interests

The paper reports no competing interests.

The authors declare no competing interests.

1 Details and additional analysis for the examples

Additional analyses assess model-order selection, noisy-data robustness, strong SSM deformation, and comparisons with SINDy and DMD across beam, vortex-shedding, and sloshing examples.

Beam analysis: Normal-form order 7 is selected for the beam despite the conjugacy-error minimum at order 9 because it yields a simpler model with only slightly higher error.
Beam analysis: SSMLearn produces more accurate beam frequency-response results at O(7) than SSMTool, whose predictions remain mismatched with numerical integration even at O(15).The SSMTool prediction error decreases with order but converges slowly near the Taylor-expansion convergence boundary.
Noisy beam data: 7.6% NMTE: the beam model trained on noisy data reconstructs trajectories accurately, and its predicted FRCs align with noise-free direct simulations.The noisy-data workflow uses a 200-dimensional delay embedding and an O(7) reduced dynamics.
Vortex shedding: The vortex-shedding SSM folds over the unstable eigenspace while showing no comparable fold over the leading POD-mode coordinates.Trajectories initially follow the unstable eigenspace before converging to the limit cycle.
Vortex shedding: 108% NMTE: the POD-coordinate model has a sizable reconstruction error, whereas sparse SINDy reaches 10.2% and the sparse SSM model reaches 3.86%.SINDy’s fitted model is dense because all polynomial coefficients through degree 5 are nonzero.
Vortex shedding: 500-mode DMD fits the training trajectory accurately but cannot capture the limit cycle, with long-term predictions diverging to infinity.Low-rank DMD fits fail to predict from the initial condition, while the 500-mode fit becomes accurate over the training interval.
Fluid sloshing: Higher-amplitude sloshing forcing may reveal an apparent 1:3 resonance, but available measurements cannot identify the higher modes needed for an accurate four-dimensional resonant SSM model.Focused resonance-decay experiments at higher amplitudes are identified as necessary future work.
Fluid sloshing: Forced DMD is adequate at very small sloshing amplitudes but misses the softening nonlinearity at moderate and large forcing.The experimental overhangs indicate coexisting isolated stationary states and thus non-linearizable dynamics.

2 Geometry of SSMs in delay-embedding spaces

Delay embeddings of scalar measurements make the SSM approximately flat when sampling delays are sufficiently small and moderate, while the reduced dynamics on that flat manifold can remain nonlinear. This geometry supports data-driven SSM modeling but does not make forced dynamics linear.

2 Geometry of SSMs in delay-embedding spaces: Scalar-observable SSMs can be constructed in delay-embedding spaces when only delayed measurements are available.In the cited examples, the embedded SSM is close to a plane.
2 Geometry of SSMs in delay-embedding spaces: For any smooth scalar time series, delay-embedding vectors can be analyzed through the signal and its time derivatives.
2 Geometry of SSMs in delay-embedding spaces: A DMD model fitted to unforced data can match the decaying unforced center-of-mass signal when its delay-embedding dimension is high enough.
2 Geometry of SSMs in delay-embedding spaces: Forced DMD necessarily produces linear response and therefore fails at higher forcing amplitudes in the non-linearizable regime.It is approximately correct only for very low forcing amplitudes.
2 Geometry of SSMs in delay-embedding spaces: With sufficiently small sampling intervals and few delays, the embedding lies approximately on a plane spanned by two constant vectors.This approximation requires that higher-order derivatives are not too large relative to the sampling time scale.
2 Geometry of SSMs in delay-embedding spaces: The flatness of the embedded SSM does not imply that its reduced dynamics are linear.

3 The quasiperiodically forced, extended SSM normal form

The paper formulates data-driven reduction on low-dimensional SSMs for systems with quasiperiodic forcing, deriving extended normal forms through recursive invariance equations. The construction separates autonomous nonlinear dynamics from forcing effects and accommodates resonances through selected reduced terms.

3 The quasiperiodically forced, extended SSM normal form: The reduction focuses on a slow SSM of dimension 2m associated with the first m complex-conjugate eigenvalue pairs.
3 The quasiperiodically forced, extended SSM normal form: The SSM is parameterized by w(z, ϕ; ϵ), while the reduced dynamics n(z, ϕ; ϵ) are determined together through the SSM invariance equation.
3.1 Solving the autonomous invariance equation: Taylor expansions and cohomological equations are solved recursively by polynomial order to obtain the SSM parametrization and autonomous reduced dynamics.
3.1 Solving the autonomous invariance equation: The extended normal-form parametrization avoids small denominators in the SSM parametrization and enhances its domain of convergence.
3.1 Solving the autonomous invariance equation: Near-resonance tolerance δ determines which nonlinear terms are retained in the extended normal form.
3.2 Solving the non-autonomous invariance equation: Forcing corrections are obtained from the O(ϵ) invariance equation, with resonant forcing frequencies influencing the reduced dynamics.
3.3 SSM-reduced dynamics in polar coordinates: The polar SSM normal form preserves complex-conjugate modal structure, recovers the periodically forced two-dimensional case, and has leading-order accuracy O(ϵ∥ρ∥).Forcing coefficients can be calibrated to experiments; resonances involving omitted modes may require increasing the SSM dimension and collecting additional modal data.