Constrained sparse Galerkin regression

Abstract

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 challenging 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 learning. 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 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 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. arXiv:1611.03271v2 [physics.flu-dyn]

Physically constrained data‐driven correction for reduced‐order modeling of fluid flows

“…If such a connection exists, we plan to investigate methods that mitigate overfitting. 14,50,71 These alternative methods could eliminate altogether the need for the parameter tol used in the CDDC-ROM's truncated SVD algorithm or yield numerical algorithms with lower parameter sensitivity.…”

“…We note that, in data‐driven modeling, physical constraints have been enforced in, for instance, the works of Kondrashov et al and Loiseau and Brunton . We emphasize, however, that the CDDC‐ROM setting is different from that used in the aforementioned works: indeed, in these works, the authors considered pure DD‐ROMs , which only involve operators and (without A and B ). The new DDC‐ROM (22), on the other hand, is a hybrid data‐driven/projection ROM; hence, it involves both G‐ROM operators ( A and B ) and data‐driven operators ( and ).…”

“…More recently, physical constraints have started to be used in standard (ie, without ROM) LES closure modeling (see, eg, the works of Duraisamy et al and Wang et al). Finally, physical constraints have also been used in standard ROM (ie, without closure modeling) . The CDDC‐ROM proposed in this paper uses physical constraints to improve the physical accuracy of the ROM closure model (ie, the Correction term in the DDC‐ROM).…”

“…Finally, physical constraints have also been used in standard ROM (ie, without closure modeling). [45][46][47][48][49][50][51][52] The CDDC-ROM proposed in this paper uses physical constraints to improve the physical accuracy of the ROM closure model (ie, the Correction term in the DDC-ROM).…”

Methods for data-driven multiscale model discovery for materials

“…have dynamics f(·) with only a few active terms in the space of possible right-hand side functions; for example, the Lorenz equations (see figure 2) only have a few linear and quadratic interaction terms per equation. Here,  Î a r is a low-dimensional state, for example obtained via the singular value decomposition (SVD) [34,39], or constructed from physically realizable measurements.…”