State estimation in minimal turbulent channel ow A comparative study of 4DVar and PINN Yifan Du Mengze Wang Tamer A. Zaki

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State estimation in minimal turbulent channel ﬂow:

A comparative study of 4DVar and PINN

Yifan Du, Mengze Wang, Tamer A. Zaki∗

Department of Mechanical Engineering, Johns Hopkins Univeristy, Baltimore, MD 21218, USA

Abstract

The state of turbulent, minimal-channel ﬂow is estimated from spatio-temporal sparse

observations of the velocity, using both a physics-informed neural network (PINN) and

adjoint-variational data assimilation (4DVar). The performance of PINN is assessed against

the benchmark results from 4DVar. The PINN is eﬃcient to implement, takes advantage

of automatic diﬀerentiation to evaluate the governing equations, and does not require the

development of an adjoint model. In addition, the ﬂow evolution is expressed in terms of

the network parameters which have a far smaller dimension than the predicted trajectory

in state space or even just the initial condition of the ﬂow. Provided adequate observations,

network architecture and training, the PINN can yield satisfactory estimates of the the ﬂow

ﬁeld, both for the missing velocity data and the entirely unobserved pressure ﬁeld. However,

accuracy depends on the network architecture, and the dependence is not known a priori.

In comparison to 4DVar estimation which becomes progressively more accurate over the

observation horizon, the PINN predictions are generally less accurate and maintain the

same level of errors throughout the assimilation time window. Another notable distinction

is the capacity to accurately forecast the ﬂow evolution: while the 4DVar prediction depart

from the true ﬂow state gradually and according to the Lyapunov exponent, the PINN is

entirely inaccurate immediately beyond the training time horizon unless re-trained. Most

importantly, while 4DVar satisﬁes the discrete form of the governing equations point-wise

to machine precision, in PINN the equations are only satisﬁed in an L2sense.

1 Introduction

Simulations and experiments are complementary approaches in the study of wall turbulence.

The respective beneﬁts of both methods can be combined synergistically, when simulations

are infused with experimental measurements. The assimilation of the observations endows the

simulations with a unique level of ﬁdelity, because their predictions are transformed from being

a generic trajectory in state space to the exact one realized in the experiment. In addition, the

simulations compute the ﬂow state at full resolution, far beyond the original measurements, and

can also directly evaluate quantities that are not measured such as pressure. Data assimilation,

or ﬂow estimation from limited observations, can be performed using a variety of techniques

including ﬁltering [1, 2], smoothing using adjoint [3] and ensemble variational algorithms [4, 5]

and recently using physics-informed machine learning [6]. In this study, we consider sparse

observations from an independent simulation of minimal turbulent channel ﬂow, and perform

state estimation using both adjoint-variational data assimilation (4DVar) and physics informed

neural network (PINN). Using the benchmark results from 4DVar, we provide a detailed analysis

on the accuracy of the estimation using PINN.

∗Email address for correspondence: t.zaki@jhu.edu

arXiv:2210.09424v1 [physics.flu-dyn] 17 Oct 2022

Data assimilation methods for ﬂow reconstruction can be classiﬁed based on a number of

criteria, perhaps most important among them in fundamental studies of turbulence is whether

the governing equations are enforced. If the governing equations are not enforced, a statistical

or dynamical model is adopted instead. For example, linear stochastic estimation [7, 8] uses

prior knowledge of two-point correlations to estimate the full velocity ﬁeld from observations;

Extended Kalman ﬁlter provides an optimal update to the state by combining a prediction from

the linearized equations and the observations [9, 10], while ensemble Kalman ﬁlter advances

an ensemble of solutions and optimally weighs them based on the observations to estimate

the new state [11]. Some recent machine-learning algorithms also belong in this class, when

neural networks learn a mapping from limited measurements to full ﬂow ﬁeld, based on training

data from simulations and without direct enforcement of the Navier-Stokes equations. In such

instances, there is no guarantee that predictions, especially outside the training space, satisfy

the governing equations. For example, Fukami et al. [12] trained a convolutional neural network

to map from coarse-grained to full-resolution velocity ﬁelds based on training data from two-

dimensional homogeneous turbulence; In Gundersen et al. [13], a semi-conditional variational

auto-encoder was developed to perform ﬂow reconstruction from sparse measurements in a

probabilistic framework, which predicts the full ﬂow ﬁeld as well as it uncertainty. In these

methods, the predicted ﬁelds do not satisfy the physical governing equations, and some of the

estimated ﬂow structures could be the outcome of generalization errors of the surrogate model

instead of physical ones governed by the Navier-Stokes equations.

The second class of methods aims to predict a trajectory of the ﬂow in state space, that

both satisﬁes the governing equations and optimally reproduces the measurements. In this class,

four-dimensional adjoint variational data assimilation [14, 15], or 4DVar, casts the problem as an

optimization constrained by the governing equations. Starting from an estimate of the unknown

ﬂow state, a forward simulation produces the full ﬂow trajectory over the time horizon where

measurements are available. The disparity between the measurements and their estimates from

the simulation deﬁnes the loss function, and also features in the adjoint equations which are

solved backward in time. A complete forward-adjoint loop yields the gradient of the loss with

respect to the unknown ﬂow state, which can be adopted to improve the estimate of the state.

Since the governing equations are nonlinear, the procedure is repeated till convergence, whereby

the optimal ﬂow state is identiﬁed and accurately reproduces the entire history of observations.

The method has been adopted in a wide range of applications, including prediction of scalar

sources from remote measurements [16], estimation of transitional and turbulent Taylor-Couette

ﬂows from limited observations [17], and estimation of turbulence in channel ﬂow [18, 19, 20, 21].

An important property of 4DVar is that the computational cost of evaluating the gradient of

the loss function is one forward-adjoint loop, independent of the size of the control vector being

optimized. This eﬃciency presents an advantage compared to other optimization algorithms

that only adopt the forward model. For example, in ensemble-variational (EnVar) approaches

[22, 23] the gradient of the loss function is evaluated from an ensemble of forward solutions

whose size, and hence the associated computational cost, are proportional to the dimension of

the control vector. Mons et al. [24] performed a comparison of 4DVar, ensemble-variational

(EnVar) method and ensemble Kalman ﬁlter, and concluded that 4DVar is the most accurate.

In this study, the adjoint reconstruction of ﬂow ﬁeld will be adopted as the benchmark for

evaluating the performance of the physics-informed, machine-learning approach.

Recent innovation in machine learning has presented new opportunities for data assimilation

and ﬂow estimation [25, 26, 27, 28, 29]. Our primary focus will be on physics-informed neural

networks (PINNs). Similar to the adjoint-variational approach, ﬂow estimation using PINNs

is formulated as a minimization problem. The network inputs are the spatial and temporal

coordinates and its outputs are the ﬂow variables at the corresponding coordinates. The loss

function for training the PINN is comprised of diﬀerent parts: (a) The ﬁrst part is due to

the mismatch between the network predictions and the available ﬂow measurements; (b) The

Figure 1: Schematic of the reference simulation of turbulent channel ﬂow and sample observa-

tions. Right two panels show a zoomed-in region of the ﬂow, and contrast the ﬁne and coarse

observations (color rectangle) to the background line contours of the full, unknown ﬂow ﬁeld.

second part is in terms of the residuals of the governing equations and other constraints, e.g. the

boundary conditions. The minimization of such loss function leads to a neural network which

predicts correct observation values and satisﬁes the governing equations at the training spatio-

temporal locations, in a weak sense. Raissi et al. [6] introduced the notion of PINNs for

solving partial diﬀerential equations (PDEs). The ease of implementation led to a number

of applications with diﬀerent types of PDEs and ﬂow conditions, for example Mao et al. [25]

considered forward and inverse problems in high-speed ﬂows; Lou et al. [26] examined rariﬁed

ﬂow with Bolzmann-BGK formulation; Mowlavi and Nabi [30] adopted the PINN framework for

optimal control and validated the approach against adjoint-based nonlinear control; and Yang

et al. [31] developed the Bayesian PINN methodology for inverse PDE problems with noisy

data.

In contrast to adjoint methods which enforce the governing equations exactly and preserve

causality in the predicted state-space trajectory, PINNs methods use the residual of the govern-

ing equation as a penalty in the optimization problem. As such, the PINN-estimated ﬂow only

satisfy the governing equations in L2sense. The advantage of PINN, however, is the ease of

implementation for any set of forward evolution equations, and without the need for an adjoint

model. We will provide a practical guide for application of PINN in estimation of turbulent

ﬂows from limited observations, and discuss various properties of these networks.

In this study, we perform estimation of turbulent ﬂow in a minimal channel conﬁguration,

from sparse velocity measurements using 4DVar and PINNs. The results from the adjoint-

variational approach provide the benchmark for evaluating the accuracy of the PINN predic-

tions. In §2, we formulate the data assimilation problem and present the details of the 4DVar

and PINNs algorithms. The results are presented and discussed in §3, followed by a summary

of the main conclusions in §4.

2 Methodology

2.1 The state-estimation problem

In this section, we formulate the data-assimilation problem, where we attempt to estimate the

initial state of channel-ﬂow turbulence from sparse measurements. The channel half height h∗

and bulk ﬂow velocity U∗are adopted as the reference scales, where the star denotes dimensional

quantities. The ﬂow domain, shown in ﬁgure 1, is a minimal rectangular ﬂow unit, Ω =

{(x, y, z)∈[0, Lx]×[0,2] ×[0, Lz]}. The ﬂuid motion is bounded by two ﬁxed walls at y= 0

and y= 2, is periodic on both the horizontal xand zdirections, and is governed by the

Domain Size Grid points Grid resolution Reynolds number

L+

xL+

yL+

314 360 157

NxNyNz

65 385 65

∆x+∆y+

min ∆y+

max ∆z+

4.90 0.40 1.40 2.45

Re Reτ

2800 180

Table 1: Domain sizes and grid resolutions of reference DNS.

non-dimensional incompressible Navier-Stokes equations,

∇ · u= 0 (1)

∂u

∂t +u· ∇u+∇p−1

Re∇2u= 0 (2)

where u= (u, v, w) is the velocity, and pis the pressure. The bulk Reynolds number is

Re ≡U∗h∗/ν∗, where ν∗is the ﬂuid kinematic viscosity. The velocity satisﬁes no slip at the

walls, and is periodic on xand zboundaries. The ﬂow is statistically stationary, and is sustained

by a known constant mean pressure gradient in the xstreamwise direction. The total pressure

is decomposed into p=P+p0, where Pis the mean pressure and p0is the ﬂuctuating pressure

that is periodic in xand zdirections. The velocity and pressure boundary conditions can be

expressed in operator form as,

B[u, p0] = 0.(3)

The friction velocity is deﬁned as u∗

τ≡pτ∗

w/ρ∗, where τ∗

wis the mean shear stress on the

walls, and the friction Reynolds number is given by Reτ≡u∗

τh∗/ν∗. The discussion in the main

text will be focused on Reτ= 180, and the eﬀect of a higher Reynolds number (Reτ= 392)

will be discussed in Appendix B. When viscous scaling is adopted for non-dimensionalization,

quantities will be marked by superscript ‘+’. For example, the domain sizes in viscous units,

L+=L∗u∗

τ/ν∗, are reported in table 1. The present values are motivated by previous studies of

the minimal ﬂow unit, which determined the domain sizes required to sustain wall turbulence

[32, 33, 34]. Speciﬁcally, the statistics of minimal channel in the near-wall region agree with

those in large channels when the spanwise and streamwise sizes of the former are larger then

100 wall units [32].

The ﬂow estimation problem relies on the availability of some measurements, or observations.

In the present study, the measurements are spatio-temporal sparse velocity data, without any

knowledge of the pressure ﬁeld. Our goal is to reconstruct the full spatio-temporal velocity and

pressure ﬁelds. The objective is therefore to identify the initial ﬂow state that, when evolved

using the governing equations (1), reproduces all the available measurements. The estimation

of the initial condition can therefore be cast as the constrained optimization problem:

min

u,p Jm=1

2kM (u)−mk2,

subject to N[u, p]=0,B[u, p]=0,

(4)

where Jmis the measurement loss, mrepresents sparse measurement velocity data from exper-

iments or reference numerical simulations, and Mis the measurement operator that maps the

continuous spatio-temporal solution uto its values at the same discrete measurement locations.

The operator Nmaps functions uand pto the residuals of equations (1). When the equations

are strongly enforced, the residuals are exactly zero, and the instantaneous forward velocity

ﬁelds are fully determined by the initial condition, u=u(u0). As such, the loss function is

fully determined by the initial condition, Jm=Jm(u(u0)), and its derivative is denoted as

DJm/Du0.

In order to ensure accuracy and have full control of the resolution of measurements, we

acquired them from an independent reference direct numerical simulation (DNS). Once the

Case ∆Mx∆My∆Mz∆Mt∆x+

m∆y+

m∆z+

m∆t+

mT+

F 6 24 4 32 32 [9.6, 33.2] 10 2.27 50.0

C 12 48 8 64 64 [19.2, 66.4] 20 4.54 50.0

Table 2: Parameters of observations extracted from the reference simulation. The sampling rate

relative to the DNS is ∆M, with a subscript that denotes the spatial or temporal dimension.

The spatio-temporal resolution of the measurements (subscript ‘m’) and the time horizon Tare

also provided, in viscous units (superscript ‘+’).

measurements are collected, the true ﬂow is hidden throughout the data assimilation procedure,

and is only revisited to quantify the accuracy of the estimated initial conditions. The algorithm

for the reference DNS was used in a number of earlier studies of wall turbulence, and has

been extensively validated [35, 36]. It adopts a fractional-step approach with a local volume

ﬂux formulation on a staggered grid [37]. The advection terms are discretized by the Adams-

Bashforth scheme, and the Crank-Nicolson scheme is adopted for the diﬀusion terms. The

pressure Poisson equation is solved using Fourier transforms in the periodic directions and

tri-diagonal inversion in the wall-normal direction. The reference solution uris then sub-

sampled in both space and time. The sampling gaps are expressed in terms of the number of

discretization points (∆Mx,∆My,∆Mz,∆Mt) in table (2). Previous studies have demonstrated

that the maximum observation gap that allows accurate reconstruction of the full velocity ﬁeld

corresponds to the correlation lengthscale of the turbulence [20, 38], which is measured by the

Taylor microscale:

Λξ

r=−1

d2Rξ(r)

dr2−1/2

(5)

where Rξ(r) is the two-point correlation function of ξvelocity component in the rcoordinate

direction.

Two resolutions of the measurements will be considered, the ﬁner of which is denoted case

‘F’ and the coarse as case ‘C’ (see table 2 and also ﬁgure 1). The gap between observations in the

former case is one Taylor microscale, and twice that scale in case ‘C’. In all cases, observations

are available within a time horizon 0 ≤t≤T, where in viscous units T+= 50. This duration

is approximately one Lyapunov timescale (τ+

σ≈48 for Reτ= 180), which has weak Reynolds-

number dependence in the range considered [21, 39]. In all cases, the spatio-temporal resolution

of the observations is not suﬃcient to achieve synchronization of the estimated state to the true

ﬂow [38]. As a result, the 4DVar and PINN are not expected to perfectly reproduce the true

ﬂow, and we can compare their prediction accuracy.

2.2 Adjoint-variational state estimation

The optimization problem (4) can be reformulated in terms of the Lagrangian,

L=Jm+Dq†,N[u, p]E+Dβ†,B[u, p]E(6)

where the forward state is q≡(u, v, w, p) and its adjoint is q†= (p†, u†, v†, w†). For convenience,

we also introduce forward and adjoint variables, βand β†, for enforcing the boundary conditions.

The ﬁrst order optimality condition yields the following equalities:

Dq†= 0,DL

Dβ†= 0,

Dq= 0,DL

Dβ= 0,

(7)

where Ddenotes the functional derivative. The ﬁrst two conditions are gradients of the La-

grangian with respect to the adjoint variables, which lead to the Navier-Stokes equations (1) and

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Stateestimationinminimalturbulentchannelow:Acomparativestudyof4DVarandPINNYifanDu,MengzeWang,TamerA.Zaki*DepartmentofMechanicalEngineering,JohnsHopkinsUniveristy,Baltimore,MD21218,USAAbstractThestateofturbulent,minimal-channelowisestimatedfromspatio-temporalsparseobservationsofthevelocity,usingbotha...

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State estimation in minimal turbulent channel ow A comparative study of 4DVar and PINN Yifan Du Mengze Wang Tamer A. Zaki

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