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Constrained Sparse Galerkin Regression
Jean-Christophe Loiseau, Steven L. Brunton
TL;DR
Nonlinear low-order fluid models remain difficult to identify accurately from data while retaining physical structure. The paper extends SINDy into constrained Galerkin regression, enforcing physical constraints without requiring a high-fidelity projection solver. Across the studied flow systems, cubic nonlinearities improve low-dimensional predictive models, while a quadratic constrained cavity model can underestimate linear-instability growth.
Problem
Accurately identifying nonlinear reduced-order fluid models is difficult, especially when data-driven regression may lose physical symmetries preserved by Galerkin projection.
Method
The framework combines SINDy-based sparse regression with linear constraints enforcing physical properties such as energy-preserving nonlinearities and symmetries.
Results
Cubic nonlinearities model truncated-mode effects and provide much better predictive capabilities in the studied cylinder and shear-driven cavity flows.
Takeaways & Limitations
Constrained Galerkin regression reproduces key qualitative flow features and offers interpretable data-driven models without requiring a high-fidelity Navier–Stokes projection solver.
Takeaways & Limitations
The quadratic constrained cavity model strongly underestimates the linear-instability growth rate, while constraining the low-dimensional eigenspectrum lies outside convex optimization.
Abstract
from arXiv · showhide
Although major advances have been achieved over the past decades for the reduction and identification of linear systems, deriving nonlinear low-order models still is a chal- lenging task. In this work, we develop a new data-driven framework to identify nonlinear reduced-order models of a fluid by combining dimensionality reductions techniques (e.g. proper orthogonal decomposition) and sparse regression techniques from machine learn- ing. In particular, we extend the sparse identification of nonlinear dynamics (SINDy) algorithm to enforce physical constraints in the regression, namely energy-preserving quadratic nonlinearities. The resulting models, hereafter referred to as Galerkin regression models, incorporate many beneficial aspects of Galerkin projection, but without the need for a full-order or high-fidelity solver to project the Navier-Stokes equations. Instead, the most parsimonious nonlinear model is determined that is consistent with observed mea- surement data and satisfies necessary constraints. Galerkin regression models also readily generalize to include higher-order nonlinear terms that model the effect of truncated modes. The effectiveness of Galerkin regression is demonstrated on two different flow configurations: the two-dimensional flow past a circular cylinder and the shear-driven cavity flow. For both cases, the accuracy of the identified models compare favorably against reduced-order models obtained from a standard Galerkin projection procedure. Present results highlight the importance of cubic nonlinearities in the construction of accurate nonlinear low-dimensional approximations of the flow systems, something which cannot be readily obtained using a standard Galerkin projection of the Navier-Stokes equations. Finally, the entire code base for our constrained sparse Galerkin regression algorithm is freely available online.
1 Introduction
Fluid dynamics seeks reduced-order models because high-dimensional nonlinear flows often evolve on low-dimensional attractors, but data-driven regression can lose physical symmetries. The paper proposes constrained sparse regression to identify nonlinear models while enforcing properties such as energy-preserving nonlinearities.
- Motivation: High-dimensional nonlinear fluid flows often evolve on low-dimensional attractors dominated by coherent structures.These structures can contain significant energy or support control objectives.
- Existing approaches: Traditional Galerkin reduction projects governing equations onto modes using prior Navier–Stokes knowledge and a high-fidelity solver.The modes may be analytical or data-driven, such as POD modes.
- Existing approaches: Data-driven methods identify dynamics from measurements, including DMD, Koopman-based methods, and sparse regression.Sparse regression is presented as a machine-learning route to nonlinear system identification.
- Gap: Regression-based methods may lose governing-equation symmetries that physics-based Galerkin projection preserves.This motivates incorporating physical constraints directly into regression.
- Contribution: The proposed sparse Galerkin regression extends SINDy with constraints such as energy-preserving nonlinearities and symmetry enforcement.It uses constrained least squares and is demonstrated on cylinder and shear-driven cavity flows.
2 Constrained sparse identification
The framework identifies sparse nonlinear dynamical systems from time-series states and derivatives by regressing against candidate-function libraries, then imposing linear physical or sparsity constraints. A Lorenz-system example illustrates how constrained SINDy can improve trajectory agreement with the original system.
- 2.1 Sparse identification of nonlinear dynamics (SINDy): SINDy represents measured dynamics as a sparse combination of candidate functions, avoiding brute-force enumeration of nonlinear model structures.Convex ℓ1-regularized regression promotes parsimonious coefficient vectors.
- 2.1 Sparse identification of nonlinear dynamics (SINDy): The identification pipeline forms state and derivative data matrices, constructs a library Θ(X), and solves for coefficient matrix Ξ.Derivative estimates can be computed from the data, with total-variation regularization supporting noisy measurements.
- 2.1 Sparse identification of nonlinear dynamics (SINDy): Each coefficient vector determines the active terms in one model equation, while sparsity balances fit accuracy against model complexity.Sequential thresholded least squares can remove small coefficients iteratively to promote parsimonious models and avoid overfitting.
- 2.2 Constrained sparse identification: Constrained sparse identification adds linear equalities Cξ = d that can enforce zero coefficients, known values, or physical relationships.The constrained least-squares problem is solved iteratively using thresholding and KKT-based constrained optimization.
- 2.3 Illustration of constrained sparse identification on the Lorenz system: For the Lorenz example, the quadratic library P2(x) contains all polynomials of degree 2 or less in the state entries.The setup includes up to 30 coefficients and assumes σ = 10 in the x-equation.
- 2.3 Illustration of constrained sparse identification on the Lorenz system: The constrained Lorenz model’s trajectory remains closer to the original system than the trajectory from unconstrained SINDy.Its identified coefficients are close to the original parameters σ = 10, ρ = 28, and β = 8/3.
3 Deriving the constraints
The reduced-order model approximates the perturbation dynamics with a low-dimensional linear operator and nonlinear term while enforcing energy-preserving structure inherited from Navier–Stokes dynamics. Higher-order nonlinearities can model truncated-mode effects, though their physical constraints remain unclear.
- The reduced-order model represents perturbation dynamics using low-dimensional approximations of the linearised operator and quadratic nonlinear term.
- The Navier–Stokes quadratic nonlinearity preserves perturbation energy by redistributing it across lengthscales.
- Because POD amplitudes relate directly to perturbation kinetic energy, the identified quadratic term is constrained to preserve energy.
- Expanding the energy condition in regression coefficients produces coupled linear equality constraints on the reduced equations.
- Standard Galerkin projection may require the first eight POD modes plus a shift mode to reproduce cylinder-flow dynamics faithfully.
- Higher-order polynomial terms can be incorporated automatically to represent truncated-mode effects, but their physical constraints are currently unclear.
4 Flow configurations
The framework is demonstrated on two benchmark flows: cylinder wake dynamics at Re = 100 and shear-driven cavity flow. The cylinder case uses simulations, snapshots, stability information, and POD modes to construct and assess reduced descriptions.
- The study examines incompressible flow past a circular cylinder and shear-driven cavity flow as interpretable benchmark configurations.
- Cylinder flow: The cylinder case uses Re = 100, above vortex-shedding onset and below three-dimensional instability onset.
- Cylinder flow: The cylinder simulation uses a Nek 5000 spectral-element solver in a domain discretised with 1832 seventh-order spectral elements.
- Cylinder flow: The statistically steady cylinder attractor is sampled with M = 1000 velocity snapshots at a frequency about 30 times the vortex-shedding frequency.
- Cylinder flow: The cylinder reduction includes a shift mode measuring base-flow-to-mean-flow distortion and the two most energetic POD modes.
- Shear-driven cavity flow: The cavity flow features a leading two-dimensional instability localized mainly along the shear layer between the external boundary layer and inner cavity flow.
- Shear-driven cavity flow: At Re = 4250, the cavity-flow setup uses an unstable-flow initial condition followed by direct numerical simulation to a statistically steady state.
5 Results and discussion
Quadratic and cubic Galerkin regression models were evaluated against standard Galerkin projection models for cylinder and shear-driven cavity flows. Cubic nonlinearities produced highly accurate cylinder-flow dynamics, while constrained models improved physical fidelity in the cavity-flow identification.
- Model construction: Quadratic models use an energy-preserving quadratic nonlinearity, whereas cubic models add higher-order terms to represent truncated-mode effects.For n = 3 state variables, the cubic library contains polynomials of degree 3 or less and requires up to 57 coefficients.
- Cylinder flow: For cylinder flow, minimal Galerkin projection models overestimate transient duration and develop energy overshoot after nonlinear saturation.Neglected higher-harmonic POD modes cannot absorb excess energy, causing the leading modes and mean-flow distortion to become too large.
- Cylinder flow: The cubic cylinder-flow regression model fits the original data almost perfectly, with limit-cycle amplitude less than 0.5% higher and mean-flow-distortion saturation differing by less than 0.1%.Its growth rate is slightly over-estimated, but the stabilising cubic terms prevent the overshoot observed in quadratic models.
- Shear-driven cavity flow: For shear-driven cavity flow, Galerkin projection models mispredict transient duration and saturate at excessive mean-flow distortion because higher harmonics are neglected.They also slightly over-predict the growth rate of the linear instability mode.
- Shear-driven cavity flow: Both quadratic cavity-flow regression models reproduce asymptotic dynamics, but the constrained energy-preserving model severely over-predicts transient duration.The unconstrained quadratic model predicts transients better despite a small unphysical overshoot; constraining the linear eigenspectrum is suggested but would require a non-convex constraint.
- Shear-driven cavity flow: Cubic cavity-flow models have similar accuracy, while constraints yield a physically stable mean-flow-distortion direction absent from the unconstrained model.The unconstrained model identifies a spurious linear instability in a∆, whereas coupling equations through energy-preserving constraints produces a more physical system.
6 Conclusion
The framework identifies interpretable nonlinear reduced-order fluid models from data while enforcing physical constraints and allowing higher-order nonlinearities. Across benchmark flows, cubic terms improve predictive capability, while constraints ensure physically meaningful dynamics.
- 6 Conclusion: Galerkin regression combines sparse SINDy-based identification with constraints enforcing symmetries and energy-preserving quadratic nonlinearities.The method retains Galerkin-like interpretability without requiring a high-fidelity Navier–Stokes solver for projection.
- 6 Conclusion: The framework was evaluated on the Lorenz system, two-dimensional cylinder flow, and shear-driven cavity flow, with regression models compared against classical Galerkin models.The identified models qualitatively reproduced linear instability, saturation, and limit-cycle behavior.
- 6 Conclusion: Cubic nonlinearities model truncated-mode effects and substantially improve the predictive capabilities of low-dimensional models.Such terms are not readily obtained through standard Galerkin projection without significant additional post-analysis.
- 6 Conclusion: For the shear-driven cavity, physical constraints corrected the identified mean-flow-distortion dynamics from a spurious unstable manifold to a stable eigenvalue.The constrained model thereby produced a more physical low-dimensional system.
- 6 Conclusion: A constrained quadratic cavity model strongly underestimates the linear-instability growth rate, and constraining the low-dimensional eigenspectrum is outside convex optimization.The authors identify nonlinear optimization as a possible route to address this limitation.
A Coefficients of the different models identified
The appendix reports coefficients for quadratic and cubic Galerkin regression models identified with SINDy and an energy-preserving constraint on the quadratic nonlinear term. Separate models are provided for cylinder and shear-driven cavity flows.
- A Coefficients of the different models identified: The reported models use SINDy extended with an energy-preserving constraint for the quadratic nonlinear term.The tables provide the coefficients of each identified model.
- A Coefficients of the different models identified: Models A1 and B1 are the quadratic and cubic Galerkin regression models for two-dimensional cylinder flow at Re = 100.Their coefficients are listed in tables 1 and 3, respectively.
- A Coefficients of the different models identified: Models A2 and B2 are the quadratic and cubic Galerkin regression models for shear-driven cavity flow at Re = 4250.Their coefficients are listed in tables 2 and 4, respectively.
B Influence of the sparsity knob λ and model selection
The sparsity knob λ controls the trade-off between model complexity and accuracy in Galerkin regression. For the cubic cylinder model, increasing λ can preserve fluctuation accuracy but eventually removes essential mean-flow terms.
- B Influence of the sparsity knob λ and model selection: λ governs sparsity, directly affecting the accuracy and complexity of the identified model.Small λ retains too many terms, whereas large λ may eliminate necessary terms.
- B Influence of the sparsity knob λ and model selection: For the cubic cylinder model, increasing λ to almost 1 has negligible influence on the a priori accuracy of the fluctuation dynamics.The analysis examines this behavior using the r2 score and the number of non-zero coefficients.
- B Influence of the sparsity knob λ and model selection: For λ > 0.1, all coefficients in the mean-flow-distortion equation become zero, producing a sudden drop in its r2 score and non-zero-coefficient count.This behavior indicates over-aggressive sparsity promotion and a kink in the Pareto fronts.
- B Influence of the sparsity knob λ and model selection: The selected model corresponds to the kink in the Pareto fronts and provides the highest a priori accuracy among the examined choices.The passage identifies this criterion for model selection after the abrupt coefficient collapse.