Constrained sparse Galerkin regression

By Jean-Christophe Loiseau, Steven L. Brunton. Journal of Fluid Mechanics, 2016

358 citations recorded by the source corpus.

The sparse identification of nonlinear dynamics (SINDy) is a recently proposed data-driven modelling framework that uses sparse regression techniques to identify nonlinear low-order models. With the goal of low-order models of a fluid flow, we combine this approach with dimensionality reduction techniques (e.g. proper orthogonal decomposition) and extend it to enforce physical constraints in the regression, e.g. 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 high-fidelity solver to project the Navier–Stokes equations. Instead, the most parsimonious nonlinear model is determined that is consistent with observed measurement 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 such an approach is demonstrated on two canonical 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 favourably against reduced-order models obtained from a standard Galerkin projection procedure. Finally, the entire code base for our constrained sparse Galerkin regression algorithm is freely available online.

Subfields: Computational Mechanics, Statistical and Nonlinear Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/3d1b4fedc1b48f92d97a5a820cbb3cc03fcd1f87

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computational Mechanics

In Statistical and Nonlinear Physics

In Statistics and Probability

Citations in the corpus

Enhancing computational fluid dynamics with machine learning (2021, methodology): “In all of these architectures, there is a tremendous opportunity to embed partial knowledge of the physics, such as conservation laws [118], symmetries [120], and invariances [131–133].”
Data-driven discovery of coordinates and governing equations (2019, positive): “SINDy has been widely applied to identify models for fluid flows (27), optical systems (32), chemical reaction dynamics (33), convection in a plasma (34), and structural modeling (35) and for model predictive control (36).”
Integrating Scientific Knowledge with Machine Learning for Engineering and Environmental Systems (2020, positive): “Loiseau et al. [178] used constrained least squares [110] to incorporate energy-preserving nonlinearities or to enforce symmetries in the identified equations for the equation learning process described in Section 2.7.”
SINDy-PI: a robust algorithm for parallel implicit sparse identification of nonlinear dynamics (2020, methodology): “In a short time, the SINDy algorithm has been extended to include inputs and control [48], to identify partial differential equations [29,30], to incorporate physically relevant constraints [34], to include tensor bases [45], and to incorporate integral terms for denoising [49,50].”
Data-Driven Sparse Sensor Placement for Reconstruction: Demonstrating the Benefits of Exploiting Known Patterns (2017, methodology): “Recently, compressed sensing, sparsity-promoting algorithms such as the lasso regression [35], [36], [37], and machine learning have been increasingly applied to characterize and control dynamical systems [38], [39], [40], [41], [42], [43], [44], [45], [46], [47].”
Sparse identification of nonlinear dynamics for model predictive control in the low-data limit (2017, methodology): “Known symmetries, conservation laws, and constraints may be readily included in both the SINDYc and DMDc modeling frameworks [28], as they are both based on least-squares regression, possibly with sequential thresholding.”
Ensemble-SINDy: Robust sparse model discovery in the low-data, high-noise limit, with active learning and control (2021, methodology): “Extensions of the SINDy algorithm include accounting for control inputs [43] and rational functions [44, 45], enforcing known conservation laws and symmetries [27], promoting stability [46], improved noise robustness through the integral formulation [34, 47–51], generalizations for stochastic dynami”
Integrating Physics-Based Modeling with Machine Learning: A Survey (2020, positive): “Loiseau et al. [155] uses constrained least squares [96] to incorporate energy-preserving nonlinearities or to enforce symmetries in the identified equations to the equation learning process described in Section 2.7.”
Deep learning of physical laws from scarce data (2020, background): “…in the form of ﬁrst-order ODEs, alternatively with linear embedding (8, 10), for a wide range of nonlinear dynamical systems, such as ﬂuid ﬂows (13, 14), structural systems (15, 16), biological and chemical systems (17–19), active matter (20), predictive control of nonlinear dynamics (21),…”