Data-driven low-dimensional dynamic model of Kolmogorov flow Carlos E. P erez De Jes us1and Michael D. Graham1

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Data-driven low-dimensional dynamic model of Kolmogorov ﬂow

Carlos E. P´erez De Jes´us1and Michael D. Graham1, ∗

1Department of Chemical and Biological Engineering,

University of Wisconsin-Madison, Madison WI 53706, USA

(Dated: August 2, 2023)

Abstract

Reduced order models (ROMs) that capture ﬂow dynamics are of interest for decreasing compu-

tational costs for simulation as well as for model-based control approaches. This work presents a

data-driven framework for minimal-dimensional models that eﬀectively capture the dynamics and

properties of the ﬂow. We apply this to Kolmogorov ﬂow in a regime consisting of chaotic and

intermittent behavior, which is common in many ﬂows processes and is challenging to model. The

trajectory of the ﬂow travels near relative periodic orbits (RPOs), interspersed with sporadic burst-

ing events corresponding to excursions between the regions containing the RPOs. The ﬁrst step

in development of the models is use of an undercomplete autoencoder to map from the full state

data down to a latent space of dramatically lower dimension. Then models of the discrete-time

evolution of the dynamics in the latent space are developed. By analyzing the model performance

as a function of latent space dimension we can estimate the minimum number of dimensions re-

quired to capture the system dynamics. To further reduce the dimension of the dynamical model,

we factor out a phase variable in the direction of translational invariance for the ﬂow, leading to

separate evolution equations for the pattern and phase dynamics. At a model dimension of ﬁve for

the pattern dynamics, as opposed to the full state dimension of 1024 (i.e. a 32 ×32 grid), accurate

predictions are found for individual trajectories out to about two Lyapunov times, as well as for

long-time statistics. Further small improvements in the results occur as dimension is increased to

nine, beyond which the statistics of the model and true system are in very good agreement. The

nearly heteroclinic connections between the diﬀerent RPOs, including the quiescent and bursting

time scales, are well captured. We also capture key features of the phase dynamics. Finally, we use

the low-dimensional representation to predict future bursting events, ﬁnding good success.

arXiv:2210.16708v2 [cs.LG] 1 Aug 2023

I. INTRODUCTION

Development of reduced order dynamical models for complex ﬂows is an issue of long-

standing interest, with applications in improved understanding, as well as control, of ﬂow

phenomena. The classical approach for dimension reduction of these systems consists of

extracting dominant modes from data via principal component analysis (PCA), also known

as proper orthogonal decomposition (POD) and Karhunen-Lo´eve decomposition [1]. PCA

determines a set of basis vectors ordered by their contribution to the total variance (ﬂuc-

tuating kinetic energy) of the ﬂow. Given Nsdata vectors (“snapshots”) xi∈RN, one can

obtain these basis vectors by performing singular value decomposition (SVD) on the data

matrix X= [x1, x2,···]∈RN×Nssuch that X=UΣVT. Projecting the data onto the ﬁrst

dhbasis vectors (columns of U) then gives a low-dimensional representation – a projection

onto a linear subspace of the full state space. To ﬁnd a reduced order model (ROM), a

Galerkin approximation of the Navier-Stokes Equations (NSE) using this basis can be im-

plemented; these have shown some success in capturing the dynamics of coherent structures

[2,3]. Previous research has also used POD as well as a ﬁltered version thereof [4], which

are linear reduction techniques, to reduce dimensions and learn a time evolution map from

data with the use of neural networks (NNs) [5].

Although PCA provides the best linear representation of a data set in dhdimensions, in

general the long-time dynamics of a general nonlinear dynamical systems are not expected

to lie on a linear subspace of the state space. For a primer and more details on data-

driven dimension reduction methods for dynamical systems refer to Linot & Graham [6].

For dissipative systems, such as the NSE, it is expected that the long-time dynamics will lie

on an invariant manifold M, which can be represented locally with Cartesian coordinates,

but may have a complex global topology [7]. In ﬂuid mechanics, this manifold is often

called an inertial manifold [8–10]. Figure 1schematically illustrates a simple example of

this idea. Consider a dynamical system ˙x=F(x) for state variable x∈RN. As time

proceeds, general initial conditions in this space evolve toward an invariant manifold Mof

dimension dM, which in this example can be described by the equation q= Φ(p) where

x=p+q,p∈RdM, q ∈RN−dM. Furthermore, if we write the dynamics in terms of pand q

∗mdgraham@wisc.edu

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Initial Conditions

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q=(p)

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Invariant Manifold

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Trajectories in state space

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FIG. 1: Schematic of state space with initial conditions collapsing onto an invariant manifold

where the long time dynamics occur.

as ˙p=f(p, q),˙q=g(p, q), then trajectories on Mevolve according to ˙p=f(p, Φ(p)): i.e. the

long time dynamics are given by a set of ordinary diﬀerential equations in dMdimensions,

rather than the Ndimensions of the original system. More generally, since Mis invariant

under the dynamics, the vector ﬁeld on Mis always tangent to M, and the dynamics on

Mare determined by this vector ﬁeld. In the present work we do not require that the

manifold be represented in this simple form, but rather a more general form G(x) = 0. In

this example, G(x) = q−Φ(p).

In general one can think of breaking up Minto overlapping regions that cover the domain,

to ﬁnd a local representation. These are called charts and are equipped with a coordinate

domain and a coordinate map [11]. The strong Whitney’s embedding theorem states that

any smooth manifold of dimension dMcan be embedded into a Euclidean space of so-called

embedding dimension 2dM[11,12]. This means that in the worst case we can expect in

principle to be able to ﬁnd a 2dM-dimensional Euclidean space in which the dynamics lie.

To ﬁnd a dM-dimensional Euclidean space one would in general need to develop overlapping

local representations and evolution equations – this avenue is not pursued in the present

work but has been done elsewhere [13]. In this work we aim to ﬁnd a high-ﬁdelity low-

dimensional dynamical model using data from simulations of two-dimensional Kolmogorov

ﬂow. In this work, the governing Navier-Stokes Equations will only be used to generate

the data – the models will only use this data, not the equations that generated it. Neural

networks (NNs) will be used to map between the full state space and the manifold, as well

as for the dynamical system model on the manifold.

A number of previous studies have focused on ﬁnding data-driven models for ﬂuid ﬂow

problems with the use of NNs. Srinivasan et al. [14] developed NN models to attempt

to predict the time evolution of the Moehlis-Faisst-Eckhardt (MFE) model [15], which is

a nine-dimensional model for turbulent shear ﬂows. They used two approaches to ﬁnding

discrete-time dynamical systems. The ﬁrst is to simply use a neural network as a discrete-

time map, yielding a Markovian representation of the time evolution. The second is to

use a long short-term memory (LSTM) network, which yields a non-Markovian evolution

equation. Despite the fact that the dynamics are in fact Markovian, the LSTM approach

worked better, yielding reasonable agreement with the Reynolds stress proﬁles. Page et al.

used deep convolutional autoencoders (CAEs) to learn low-dimensional representations for

two-dimensional (in physical space) Kolmogorov ﬂow, showing that these networks retain

a wide spectrum of lengthscales and capture meaningful patterns related to the embedded

invariant solutions [16]. They considered the case where bursting dynamics is obtained at

a Reynolds number of Re = 40 and n= 4 wavelengths in the periodic domain. Nakamura

et al. used CAEs for dimension reduction combined with LSTMs and applied it to minimal

turbulent channel ﬂow for Reτ= 110 where they showed to capture velocity and Reynolds

stress statistics [17]. They studied various degrees of dimension reduction, showing good

performance in terms of capturing the statistics; however for drastic dimension reduction

they showed how only large vortical structures were captured. Hence, the selection of the

minimal dimension to accurately represent the state becomes a challenging task. Reservoir

networks have also shown great potential in learning nonlinear models for time evolution.

For example, Doan et al. trained what they call an Auto-Encoded Reservoir-Computing

(AE-RC) framework where the latent space is fed into an Echo State Network (ESN) to

model evolution in discrete time [18]. By considering the two-dimensional Kolmogorov ﬂow

for Re = 30 and n= 4 good performance was obtained when comparing the kinetic energy

and dissipation evolution in time. They also showed how the model captures the velocity

statistics. However, the nature of the reservoir in the ESN stores past history, making the

model non-Markovian.

Although previous research has found data-driven ROMs for ﬂuid ﬂow problems, the

focus on these has not been to ﬁnd the minimal dimension required to capture the data

manifold and dynamics. Linot & Graham have addressed this issue for the Kuramoto-

Sivashinsky equation (KSE) [6,19]. They showed that the mean squared error (MSE) of

the reconstruction of the snapshots using an AE for the domain size of L= 22 exhibited an

orders-of-magnitude drop when the dimension of the inertial manifold is reached. Further-

more, modeling the dynamics with a dense NN at this dimension either with a discrete time

map [19] or a system of ordinary diﬀerential equations (ODE) [6] yields excellent trajectory

predictions and long-time statistics. Increasing domain size to L= 44 and L= 66, which

makes the system more chaotic, aﬀects the drops of MSE signiﬁcantly. However a drop is

still seen, and when obtaining the dynamics and calculating long time statistics, good agree-

ment with the true data is obtained. This work, denoted “Data-driven manifold dynamics”

(DManD) has been extended to incorporate reinforcement learning control for reduction of

dissipation in the KSE, yielding a very eﬀective control policy [20].

We aim to extend this approach to the NSE, speciﬁcally to the two-dimensional Kol-

mogorov ﬂow, where an external forcing drives the dynamics. As Re increases, the trivial

state becomes unstable, giving rise to periodic orbits (POs), relative periodic orbits (RPOs)

and eventually chaos. Relative periodic orbits correspond to periodic orbits in in a moving

reference frame, such that in a ﬁxed frame, the pattern at time t+Tis a phase-shifted

replica of the pattern at time t. The nature of the weakly turbulent dynamics at a Reynolds

number of Re = 14.4, and connections with RPO solutions are the focus of this study.

Due to the symmetries of the system the chaotic dynamics travels between unstable RPOs

[21] through bursting events [22] that shadow heteroclinic orbits connecting the RPOs. A

past study [23] shows that low-dimensional representations can be found with PCA for two-

dimensional Kolmogorov ﬂow where in the case of weakly turbulent data, the ﬁrst two PCA

basis in the streamfunction formulation capture most of the energetic content when ﬁltering

out the bursting events before the analysis, and including a third basis function captures

the bursting information. This point hints at the low-dimensional nature of this system,

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摘要：

Data-drivenlow-dimensionaldynamicmodelofKolmogorovflowCarlosE.P´erezDeJes´us1andMichaelD.Graham1,∗1DepartmentofChemicalandBiologicalEngineering,UniversityofWisconsin-Madison,MadisonWI53706,USA(Dated:August2,2023)AbstractReducedordermodels(ROMs)thatcaptureflowdynamicsareofinterestfordecreasingcompu-t...

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Data-driven low-dimensional dynamic model of Kolmogorov flow Carlos E. P erez De Jes us1and Michael D. Graham1

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