Sparse reduced-order modeling : Sensor-based dynamics to full-state estimation

Jean-Christophe Loiseau; Bernd R. Noack; Steven L. Brunton

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Sparse reduced-order modeling : Sensor-based dynamics to full-state estimation

Jean-Christophe Loiseau, Bernd R. Noack, Steven L. Brunton

arXiv:1706.03531v1Jun 12, 2017physics.flu-dyn

TL;DR

Engineering fluid-flow models can be expensive and difficult to interpret, while modal models face challenges from changing conditions and transients. The paper addresses this with sensor-based feature lifting, sparse SINDy dynamics, and optional full-state reconstruction and generalized modes; applied to cylinder flow, the strategy compares favorably with POD and is positioned for estimation, prediction, and control.

Problem
Full-order fluid models are expensive for iterative optimization or in-time control, while POD-Galerkin models are challenged by changing domains, modal deformation, stability, operating conditions, and transients.
Method
The framework lifts sensor signals into feature space, identifies sparse nonlinear dynamics with SINDy, and optionally maps features to velocity fields and generalized modes using full-state snapshots.
Results
The cylinder-flow application accurately captures steady-state and transient behavior, with local linear reconstruction significantly more accurate than a same-order POD-Galerkin model.
Takeaways & Limitations
The resulting gray-box models provide interpretable nonlinear interaction physics and generalized modes, with potential applications in estimation, prediction, and control.
Takeaways & Limitations
The framework still requires demonstrating favorable scaling to systems with higher-dimensional attractors and may require substantial memory for local linear mappings.

Abstract

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We propose a general dynamic reduced-order modeling framework for typical experimental data: time-resolved sensor data and optional non-time-resolved PIV snapshots. This framework contains four steps. First, the sensor signals are lifted to a dynamic feature space. Second, we identify a sparse human-interpretable nonlinear dynamical system for the feature state based on the sparse identification of nonlinear dynamics (SINDy). Third, if PIV snapshots are available, a local linear mapping from the feature state to velocity fields is shown to be orders of magnitudes more accurate than optimal modal expansions of the same order. Fourth, a generalized feature-based modal decomposition identifies coherent structures that are most dynamically correlated with the linear and nonlinear interaction terms in the sparse model, adding interpretability. Steps 1 and 2 define a black-box model. Optional steps 3 and 4 lift the black-box dynamics to a 'gray-box' model of the coherent structures, if non-time-resolved full-state data is available. This gray-box modeling strategy is successfully applied to the transient and post-transient laminar cylinder wake, and compares favorably with a POD model. We foresee numerous applications of this highly flexible modeling strategy, including estimation, prediction and control. Moreover, the feature space may be based on intrinsic coordinates, which are unaffected by a key challenge of modal expansion: the slow change of low-dimensional coherent structures with changing geometry and varying parameters.

1. Introduction

The paper develops an interpretable reduced-order modeling strategy that identifies nonlinear dynamics from sensor measurements and can reconstruct flow fields when full-state snapshots are available.

1. Introduction: Reduced-order models compactly represent fluid behavior, but POD-Galerkin models face challenges with changing domains, boundary conditions, modal deformation, stability, operating conditions, and transients.These limitations motivate a strategy that does not rely primarily on a fixed modal basis.
1.2. Contribution of this work: Sensor measurements are first transformed into a feature vector, after which SINDy identifies a sparse dynamical model in feature space.The feature map may include time-delay coordinates, such as a delayed lift measurement.
1.2. Contribution of this work: The proposed procedure identifies sparse nonlinear models from time-resolved sensor measurements rather than the full fluid state.Sparsity limits overfitting and exposes key nonlinear interaction terms.
1.2. Contribution of this work: When full-state data are available, a local linear KNN mapping reconstructs velocity fields more accurately than a same-order POD-Galerkin model.This optional mapping lifts the sensor-based dynamics toward a gray-box model.
1.2. Contribution of this work: Generalized feature-based modes associate spatial structures with feature variables and specific linear or nonlinear interaction terms, improving physical interpretability.The modes correct linear structures for nonlinear effects and can reveal structures tied to nonlinear interactions.
1.2. Contribution of this work: The framework is designed as a simple, non-invasive procedure whose feature-vector representation is more robust to mode deformation, moving geometry, and varying operating conditions.The authors identify estimation, prediction, and control as potential applications.

2. Flow configuration

The study uses two-dimensional incompressible cylinder flow at Re = 100 as a test case, with simulations spanning transient growth to the developed von Kármán vortex street.

2. Flow configuration: The test case is two-dimensional incompressible viscous flow past a circular cylinder at Re = 100.This Reynolds number lies above vortex-shedding onset and below three-dimensional instability onset.
2. Flow configuration: The simulations use the linearly unstable steady solution plus a small random white-noise velocity perturbation as the initial condition.The perturbation has zero mean and unit variance.
2. Flow configuration: Each simulation runs for 150 convective time units and provides M = 1200 uniformly sampled velocity snapshots with associated lift and drag measurements.The data cover the unforced transient from the steady solution to the fully developed von Kármán vortex street.
2. Flow configuration: The lift coefficient CL is used to show temporal evolution and the system trajectory in the phase plane (CL, dCL/dt).Highlighted times correspond to the vorticity snapshots in figure 3.
2. Flow configuration: The modeling objective is to reproduce the observed dynamics and flow structures without requiring the full flow field or modal basis.The cylinder wake provides a canonical setting for evaluating this objective.

3. Sparse sensor-based modeling

The framework builds sparse nonlinear reduced-order models from limited sensor measurements by lifting signals into feature space and identifying governing dynamics with SINDy. When full-state snapshots are available, local mapping and feature-based modes enable interpretable full-state estimation.

3.2. Sparse Identification of Nonlinear Dynamics: The identified model advances the feature state rather than the full flow field, providing a computationally efficient reduced-order representation.The procedure uses time-series data, derivative data, and a library of candidate nonlinear functions to fit the dynamics.
3.1. From sensor signals to feature space: The workflow uses sensor measurements to construct a low-dimensional feature vector before identifying reduced-order dynamics.Feature lifting can include functions of sensor measurements, such as time delays or derivatives.
3.1. From sensor signals to feature space: For the cylinder wake, lift measurements are augmented with a properly scaled time derivative, while drag is modeled algebraically from the feature vector.The chosen features represent shedding amplitude, shedding phase, and base-flow deformation.
3.2. Sparse Identification of Nonlinear Dynamics: SINDy identifies parsimonious nonlinear dynamics by selecting active terms from a candidate function library through sparse regression.The library contains polynomial and other symbolic functions, while sparse regression penalizes the number of active terms.
3.3.1. Local linear mapping: A local linear mapping estimates high-dimensional velocity fields from low-dimensional feature states using stored snapshots from transient evolutions.The mapping interpolates within a Delaunay triangulation of the feature-state plane and can require substantial snapshot storage.
3.3.2. Feature-based modal expansion: Feature-based modes provide physical interpretability by identifying structures dynamically correlated with terms in the sparse model, although they are typically less accurate than local mapping.The modal representation generalizes POD or DMD by accounting for nonlinear interaction terms.

4. Results

The cylinder-wake results show that a sparse low-dimensional dynamical system captures transient and saturated dynamics, while algebraic and local-linear mappings extend it to drag estimation and accurate flow reconstruction.

4.1.1. Dynamical system: The model is identified from feature vectors computed from transient sensor trajectories using SINDy and a polynomial candidate-function library.The polynomial library is chosen as a natural representation for a nonlinear oscillator.
4.1.1. Dynamical system: The identified two-degree-of-freedom system predicts the sensor evolution in close agreement with direct numerical simulation and captures the cylinder flow’s key physics.The comparison includes trajectories initialized near the linearly unstable baseflow.
4.1.1. Dynamical system: The system has one linearly unstable fixed point and an attracting limit cycle, with saturated frequency ω◦ = 1.119, less than 1.5% below the DNS value ω• = 1.132.Its instantaneous growth rate has the quadratic dependence 2σ(a) = 0.28(1 − a1^2 − a2^2).
4.1.2. Descriptor system: Adding a nonlinear algebraic measurement equation produces a descriptor system governing both instantaneous lift and drag coefficients.This addresses the limitation that lift-derived dynamics alone cannot directly estimate drag.
4.1.2. Descriptor system: The descriptor system correctly infers drag evolution, despite a small amplitude misprediction for 45 ⩽ t ⩽ 65, and is reported as a simple, accurate, interpretable model at Re = 100.The measurement equation has a distorted-cone structure that differs from the mean-field paraboloid of a three-dimensional POD-Galerkin model.
4.2. Flow field estimation: The local linear mapping largely outperforms 5-mode POD and generalized modal estimators, averaging two to three orders of magnitude higher accuracy while requiring more storage.A rank-50 snapshot approximation has almost no effect on estimation error and reduces storage requirements by nearly 24-fold.

5. Conclusions

The paper presents a modular, sensor-based reduced-order modeling framework that identifies sparse nonlinear feature dynamics, estimates full states from optional synchronized snapshots, and connects model terms to coherent structures. Applied to cylinder flow, the strategy supports interpretable modeling and may extend to estimation, prediction, and control.

5. Conclusions: The four-step framework lifts sensor measurements into features, identifies sparse dynamics with SINDy, estimates full states, and extracts dynamically correlated coherent structures.The first two steps form a black-box model; full-state estimation and generalized modal decomposition add a gray-box interpretation when synchronized snapshots are available.
5. Conclusions: Local linear mapping reconstructs full states from non-time-resolved synchronized snapshots, while generalized modal expansion prioritizes structures correlated with SINDy interaction terms.The mapping interpolates between historically similar flow fields based on feature dynamics; the modal expansion favors interpretability rather than maximum reconstruction accuracy.
5. Conclusions: The methodology is demonstrated on two-dimensional cylinder flow at Re = 100 using lift and drag measurements as physically relevant sensor inputs.Models range from a single lift-based dynamical system to a three-degree-of-freedom representation using lift, its derivative, and drag.
5. Conclusions: The identified sparse models avoid the transient-duration overestimation and energy overshoots associated with Galerkin projection, while offering simple explanations for nonlinear saturation.The study uses direct numerical simulation data, but the authors state that the overall strategy is generally applicable to real experiments with minor modifications.
5. Conclusions: Sensor-based intrinsic coordinates may reduce sensitivity to coherent-structure deformation caused by changing geometry and varying parameters.The framework is proposed for broader applications in simulations and experiments, including interpretable reduced-order models and future flow-control studies.
5. Conclusions: Future extensions target higher-dimensional attractors, lower-memory local modal libraries, compressed or sparse sampling, and reconstruction-aware feature selection.These extensions are presented as possible improvements to framework performance and scalability.

Appendix A. Model Selection

The appendix uses information criteria and physical constraints to select sparse SINDy models that balance accuracy and complexity. For cylinder-flow identification, adding a constraint improves the selected cubic model relative to unconstrained alternatives.

Appendix A. Model Selection: Model selection combines SINDy’s sparsity sweep with AICc to choose a parsimonious candidate balancing prediction accuracy and model complexity.AIC penalizes models with more free parameters when accuracy is equal, and relative scores rank candidates across the Pareto front.
Appendix A. Model Selection: Relative AICc values classify models with Δ ≤ 2 as strongly supported, 4 ≤ Δ ≤ 7 as weakly supported, and Δ ≥ 10 as unsupported.The model with Δ = 0 is only the best among the tested candidates, not necessarily the globally best possible model.
Appendix A. Model Selection: The unconstrained Pareto-front model with four terms provides the best accuracy-complexity balance among the initial candidates.The candidate library contains polynomials in a1 and a2 up to seventh degree.
Appendix A. Model Selection: The unconstrained model predicts a limit-cycle amplitude different from unity, violating the normalized periodic-orbit behavior.This amplitude error motivates adding equality constraints to the identification problem.
Appendix A. Model Selection: The constrained cubic model outperforms all other identified models, including the unconstrained cubic model, according to its AIC ranking.The constraint enforces a physically motivated parameter relation involving the leading instability rate.

Appendix B. Identifying a discrete-time dynamical system

The appendix extends SINDy to identify a discrete-time nonlinear model from time-shifted feature data. In cylinder flow, the resulting first-order sparse vector autoregressive model agrees well with direct numerical simulation, while noisy experimental data require preprocessing or robust differentiation.

Appendix B. Identifying a discrete-time dynamical system: SINDy can identify discrete-time dynamics by replacing the derivative data matrix with a time-shifted copy of the feature-data matrix.For cylinder flow at Re = 100, the example uses a time lag τ = 0.125.
Appendix B. Identifying a discrete-time dynamical system: The resulting discrete-time model is a sparse first-order nonlinear vector autoregressive system.Its predicted feature trajectory is compared with the corresponding trajectory from direct numerical simulation.
Appendix B. Identifying a discrete-time dynamical system: The predicted a1 trajectory agrees well with direct numerical simulation for an initial condition near the linearly unstable base flow.This supports the use of discrete-time SINDy for the illustrated low-dimensional cylinder-flow dynamics.
Appendix B. Identifying a discrete-time dynamical system: Experimental application must address noisy or corrupted measurements through preprocessing, total-variation derivative estimation, or parameter-robustness assessment.The appendix notes that the presented simulations provide ideal noise-free training data, unlike many real experiments.