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

Abstract

Summary We have recently proposed a data‐driven correction reduced‐order model (DDC‐ROM) framework for the numerical simulation of fluid flows, which can be formally written as follows. The new DDC‐ROM was constructed by using reduced‐order model spatial filtering and data‐driven reduced‐order model closure modeling (for the Correction term) and was successfully tested in the numerical simulation of a two‐dimensional channel flow past a circular cylinder at Reynolds numbers Re = 100,Re = 500, and Re = 1000. In this paper, we propose a physically constrained DDC‐ROM (CDDC‐ROM), which aims at improving the physical accuracy of the DDC‐ROM. The new physical constraints require that the CDDC‐ROM operators satisfy the same type of physical laws (ie, the Correction term's linear component should be dissipative, and the Correction term's nonlinear component should conserve energy) as those satisfied by the fluid flow equations. To implement these physical constraints, in the data‐driven modeling step, we replace the unconstrained least squares problem with a constrained least squares problem. We perform a numerical investigation of the new CDDC‐ROM and standard DDC‐ROM for a two‐dimensional channel flow past a circular cylinder at Reynolds numbers Re = 100,Re = 500, and Re = 1000. To this end, we consider a reproductive regime as well as a predictive (ie, cross‐validation) regime in which we use as little as 50% of the original training data. The numerical investigation clearly shows that the new CDDC‐ROM is significantly more accurate than the DDC‐ROM in both regimes.

A priori analysis on deep learning of subgrid-scale parameterizations for Kraichnan turbulence

“…Several studies have been conducted to model the dynamics of chaotic fluid flow using ML algorithms [25][26][27][28][29][30][31]. Recently there is a growing interest in using the physics of the problem in combination with the data-driven algorithms [28,[32][33][34][35][36][37][38]. The physics can be incorporated into these learning algorithms by adding a regularization term (based on governing equations) in loss function or modifying the neural network architecture to enforce certain physical constraints.In addition to reduced order modeling and chaotic dynamical systems, the turbulence closure problem has also benefited from the application of ML algorithms and has led to reducing uncertainties in .…”

Bayesian identification of a projection-based reduced order model for computational fluid dynamics

“…in which Ψ α (ξ(ω)) are the orthogonal polynomials and ξ := (ξ i ) m i=1 , m = s+N t N r . Following this, Equation 21 rewrites to…”

“…The aim of this work is to extend the previously mentioned approaches in order to deal with the second and third type of inaccuracies and instabilities from a learning perspective. Following the least square idea presented in [21,36], we propose to use a Bayesian strategy to identify correction terms associated with the reduced differential operators derived from a standard POD-Galerkin approach with or without additional physical constrains. A priori modelled as the tensor valued Gaussian random variable, the correction term is learned by exploiting the information contained in the partial observation, e.g.…”

Data-driven correction reduced order models for the quasi-geostrophic equations: a numerical investigation

“…To construct the data-driven correction reduced order model (DDC-ROM) [22,32,53], we use an alternative approach: We start with a new Galerkin truncation, u R = R j=1 a j ϕ j . We emphasize that, since R = O(10 3 ) is the rank of the snapshot matrix, the new Galerkin truncation includes all the information in the available data (snapshots).…”