# Informative and non-informative decomposition of turbulent flow fields

Gonzalo Arranz  _Department of Aeronautics and Astronautics, Massachusetts
Institute of Technology, Cambridge, Massachusetts 02139, USA_ Adrián Lozano
Durán  _Department of Aeronautics and Astronautics, Massachusetts Institute of
Technology, Cambridge, Massachusetts 02139, USA_

###### Abstract

Not all the information in a turbulent field is relevant for understanding
particular regions or variables in the flow. Here, we present a method for
decomposing a source field into its informative $\bm{\Phi}_{I}(\bm{x},t)$ and
residual $\bm{\Phi}_{R}(\bm{x},t)$ components relative to another target
field. The method is referred to as informative and non-informative
decomposition (IND). All the necessary information for physical understanding,
reduced-order modelling, and control of the target variable is contained in
$\bm{\Phi}_{I}(\bm{x},t)$, whereas $\bm{\Phi}_{R}(\bm{x},t)$ offers no
substantial utility in these contexts. The decomposition is formulated as an
optimisation problem that seeks to maximise the time-lagged mutual information
of the source variable with the target variable while minimising the mutual
information of its residual component. The method is applied to extract the
informative and residual components of the velocity field in a turbulent
channel flow, using the wall-shear stress as the target variable. We
demonstrate the utility of IND in three scenarios: (i) physical insight of the
effect of the velocity fluctuations on the wall-shear stress, (ii) prediction
of the wall-shear stress using velocities far from the wall, and (iii)
development of control strategies for drag reduction in opposition control. In
case (i), IND reveals that the informative velocity related to wall-shear
stress consists of wall-attached high- and low-velocity streaks, collocated
with regions of vertical motions and weak spanwise velocity. This informative
structure is embedded within a larger-scale streak-roll structure of residual
velocity, which bears no information about the wall-shear stress. In case
(ii), the best-performing model for predicting wall shear stress is a
convolutional neural network that uses the informative component of the
velocity as input, while the residual velocity component provides no
predictive capabilities. Finally, in case (iii), we demonstrate that
opposition control based on the informative wall-normal velocity yields
improvements in drag reduction compared to the control based on the total or
residual velocity.

##  1 Introduction

Since the early days of turbulence research, there have been multiple attempts
to decompose the flow into different components to facilitate its physical
understanding, control its behaviour and devise reduced-order models. One of
the earliest examples is the Reynolds decomposition (Reynolds,, 1895), which
divides the velocity field into its mean and fluctuating components. More
sophisticated approaches rapidly emerged aiming at extracting the coherent
structure of the flow through correlations and structure identification
(Robinson,, 1991; Panton,, 2001; Adrian,, 2007; Smits _et al._ ,, 2011;
McKeon,, 2017; Jiménez,, 2018). This interest is justified by the hope that
insights into the dynamics can be gained by analysing a subset of the whole
flow, while the remaining incoherent flow is inconsequential. In this work, we
introduce a method to decompose turbulent flow fields into informative and
non-informative components, referred to as IND, such that the informative
component contains all the useful information for physical understanding,
modelling, and control with respect to a given quantity of interest.

The quest to divide turbulent flows in terms of coherent and incoherent
motions has a long history, tracing back to the work of Theodorsen, (1952),
and has been a subject of active research since the pioneering experimental
visualisations of Kline _et al._ , (1967) and the identification of large-
scale coherent regions in mixing layers by Brown and Roshko, (1974). Despite
this rich history, the field still lacks consensus about the definition of a
coherent structure due to the variety of interpretations proposed by different
researchers. One of the initial approaches to distinguish turbulent regions
was the turbulent/nonturbulent discriminator circuits introduced by Corrsin
and Kistler, (1954). Since then, single- and two-point correlations have
become conventional tools for identifying coherent regions within the flow
(e.g., Sillero _et al._ ,, 2014). The development of more sophisticated
correlation techniques, such as the linear stochastic estimation (Adrian and
Moin,, 1988) and the characteristic-eddy approach (Moin and Moser,, 1989), has
further improved our understanding of the coherent structure of turbulence. An
alternative set of methods focuses on decomposing the flow into localised
regions where certain quantities of interest are particularly intense. The
first attempts, dating back to the 1970s, include the variable-interval time
average method (Blackwelder and Kaplan,, 1976) for obtaining temporal
structures of bursting events and its modified version, the variable-interval
space average method (Kim,, 1985), for characterising spatial rather than
temporal structures. With the advent of larger databases and computational
resources, more refined techniques have emerged to extract three-dimensional,
spatially localised flow structures. These include investigations into regions
of rotating fluid (e.g., vortices Moisy and Jiménez,, 2004; Del Álamo _et al._
,, 2006), motions carrying most of the kinetic energy (e.g., regions of high
and low velocity streaks by Hwang and Sung,, 2018; Bae and Lee,, 2021), and
those responsible for most of the momentum transfer in wall turbulence (e.g.,
quadrant events and uniform momentum zones by Meinhart and Adrian,, 1995;
Adrian _et al._ ,, 2000; Lozano-Durán _et al._ ,, 2012; Lozano-Durán and
Jiménez,, 2014; Wallace,, 2016; de Silva _et al._ ,, 2016).

The methods described above offer a local-in-space characterisation of
coherent structures, in contrast to the global-in-space modal decompositions
of turbulent flows (Taira _et al._ ,, 2017, 2020). One of the first
established global-in-space methods is the proper orthogonal decomposition
(POD) (Lumley,, 1967), wherein the flow is decomposed into a series of
eigenmodes that optimally reconstruct the energy of the field. This method has
evolved in different directions, such as space-only POD (Sirovich,, 1987),
spectral POD (Towne _et al._ ,, 2018), and conditional POD (Schmidt and
Schmid,, 2019), to name a few. Another popular approach is dynamic mode
decomposition (DMD) (Schmid,, 2010; Schmid _et al._ ,, 2011), along with
decompositions based on the spectral analysis of the Koopman operator (Rowley
_et al._ ,, 2009; Mezić,, 2013). Similar to POD, various modifications of DMD
have been developed, e.g., the extended DMD (Williams _et al._ ,, 2015), the
multi-resolution DMD (Kutz _et al._ ,, 2016), and the high-order DMD (Le
Clainche and Vega,, 2017) [see (Schmid,, 2022) for a review]. Another
noteworthy modal decomposition approach is empirical mode decomposition, first
proposed by Huang _et al._ , (1998) and recently used in the field of fluid
mechanics (e.g., Cheng _et al._ ,, 2019). While the methods above are purely
data-driven, other modal decompositions, such as resolvent analysis and input-
output analysis, are grounded in the linearised Navier-Stokes equations
(Trefethen _et al._ ,, 1993; Jovanović and Bamieh,, 2005; McKeon and Sharma,,
2010). It has been shown that POD, DMD, and resolvent analysis are equivalent
under certain conditions (Towne _et al._ ,, 2018). Recently, machine learning
has opened new opportunities for nonlinear modal decompositions of turbulent
flows (Brunton _et al._ ,, 2020).

The flow decomposition approaches presented above, either local or global in
space, have greatly contributed to advancing our knowledge about the coherent
structure of turbulence. Nonetheless, there are still open questions,
especially regarding the dynamics of turbulence, that cannot be easily
answered by current methodologies. Part of these limitations stem from the
linearity of most methods, yet turbulence is a nonlinear system. A more
salient issue perhaps lies in the fact that current methods tend to focus on
decomposing source variables without accounting for other target variables of
interest. In general, it is expected that different target variables would
require different decomposition approaches of the source variable. For
example, we might be interested in a decomposition of the velocity that is
useful for understanding the wall-shear stress. Hence, the viewpoint adopted
here aims at answering the question: What part of the flow is relevant to
understanding the dynamics of another variable? In this context, coherent
structures are defined as those containing the useful _information_ needed to
understand the evolution of a target variable.

The concept of information alluded above refers to the Shannon information
(Shannon,, 1948; Cover and Thomas,, 2006), i.e., the average unpredictability
in a random variable. The systematic use of information-theoretic tools for
causality, modelling, and control in fluid mechanics has been recently
discussed by Lozano-Durán and Arranz, (2022). Betchov, (1964) was one of the
first authors to propose an information-theoretic metric to quantify the
complexity of turbulence. Some works have leveraged Shannon information to
analyse different aspects of two-dimensional turbulence and energy cascade
models (Cerbus and Goldburg,, 2013; Materassi _et al._ ,, 2014; Granero-
Belinchon,, 2018; Shavit and Falkovich,, 2020; Lee,, 2021; Tanogami and
Araki,, 2024). Information theory has also been used for causal inference in
turbulent flows (Liang and Lozano-Durán,, 2016; Lozano-Durán _et al._ ,, 2019;
Wang _et al._ ,, 2021; Lozano-Durán and Arranz,, 2022; Martínez-Sánchez _et
al._ ,, 2023), and reduced-order modelling (Lozano-Durán _et al._ ,, 2019).
The reader is referred to Lozano-Durán and Arranz, (2022) for a more detailed
account of the applications of information-theoretic tools in fluid mechanics.

This work is organised as follows: The formulation of the flow decomposition
into informative and non-informative components is introduced in Section 2.
Section 3 demonstrates the application of the method to the decomposition of
the velocity field, using wall-shear stress in a turbulent channel flow as the
target variable. This decomposition is leveraged for physical understanding,
prediction of the wall-shear stress using velocities away from the wall via
convolutional neural networks, and drag reduction through opposition control.
Finally, conclusions are presented in Section 4.

##  2 Methodology

###  2.1 IND of the source variable

Let us denote the _source variable_ by $\bm{\Phi}(\bm{x},t)$ with
$\bm{x}\in\Omega_{\bm{\Phi}}$ and the _target variable_ by
$\bm{\Psi}(\bm{x},t)$ with $\bm{x}\in\Omega_{\bm{\Psi}}$ where $\bm{x}$ and
$t$ represent the spatial and time coordinates, respectively. For example, in
the case of a turbulent channel flow, the source variable could be the
velocity fluctuations defined over the entire domain,
$\bm{\Phi}(\bm{x},t)=\bm{u}(\bm{x},t)$, and the target variable could be the
shear stress vector at every point over one of the walls,
$\bm{\Psi}(\bm{x},t)=\bm{\tau}_{w}(\bm{x},t)$, as shown in figure 1. We seek
to decompose $\bm{\Phi}(\bm{x},t)$ into two contributions: an _informative
contribution_ to the target variable in the future,
$\bm{\Psi}_{+}=\bm{\Psi}(\bm{x},t+\Delta T)$ with $\Delta T\geq 0$, and a
_residual term_ that conveys no information about $\bm{\Psi}_{+}$ (i.e., the
non-informative component):

$\bm{\Phi}(\bm{x},t)=\bm{\Phi}_{I}(\bm{x},t)+\bm{\Phi}_{R}(\bm{x},t),$ (1)

where $\bm{\Phi}_{I}$ and $\bm{\Phi}_{R}$ are the informative and residual
contributions, respectively. The decomposition is referred to as
Informative/Non-informative Decomposition or IND.

Figure 1: Schematic of IND applied to wall-bounded turbulent flow. The source
variable is the velocity fluctuations in the fluid volume
$\bm{x}\in\Omega_{\bm{u}}$ at $t$, and the target variable is the wall shear
stress vector at the wall at $t+\Delta$. For the sake of visualisation, only
the streamwise component of the velocity fluctuations and the wall shear
stress is displayed. The velocity fluctuations at time $t$ are decomposed into
their informative and residual components to the wall shear stress in the
future, $t+\Delta t$.

To find a decomposition of the form shown in Eq. (1), we need to introduce a
definition of _information_. We rely on the concept of Shannon information
(Shannon,, 1948), which quantifies the average information in the variable
$\bm{\Psi}_{+}$ as

$H(\bm{\Psi}_{+})=-\sum_{\bm{S}\in\mathcal{S}}p_{\bm{\Psi}_{+}}(\bm{\Psi}_{+}=\bm{S})\log
p_{\bm{\Psi}_{+}}(\bm{\Psi}_{+}=\bm{S})\geq 0,$ (2)

where $H(\bm{\Psi}_{+})$ is referred to as the Shannon entropy or information
of $\bm{\Psi}_{+}$, $p_{\bm{\Psi}_{+}}(\bm{\Psi}_{+}=\bm{S})$ denotes the
probability of $\bm{\Psi}_{+}$ being in the state $\bm{S}$, and $\mathcal{S}$
represents the set of all possible states of $\bm{\Psi}_{+}$. The remaining
information in $\bm{\Psi}_{+}$, after discounting for the information in
$\bm{\Phi}$, is measured by the conditional Shannon information:

$H(\bm{\Psi}_{+}|\bm{\Phi})=-\sum_{\bm{S}\in\mathcal{S}}\sum_{\bm{R}\in\mathcal{R}}p_{\bm{\Psi}_{+},\bm{\Phi}}(\bm{S},\bm{R})\log\frac{p_{\bm{\Psi}_{+},\bm{\Phi}}(\bm{S},\bm{R})}{p_{\bm{\Phi}}(\bm{R})}\geq
0,$ (3)

where $p_{\bm{\Psi}_{+},\bm{\Phi}}$ is the joint probability distribution of
$\bm{\Psi}_{+}$ and $\bm{\Phi}$, $\bm{R}$ is a particular state of
$\bm{\Phi}$, and $\mathcal{R}$ is the set of all possible states of
$\bm{\Phi}$. The difference between Eq. (2) and Eq. (3) quantifies the amount
of shared information between the variables

$I(\bm{\Psi}_{+};\bm{\Phi})=H(\bm{\Psi}_{+})-H(\bm{\Psi}_{+}|\bm{\Phi}),$ (4)

and is referred to as the mutual information between $\bm{\Psi}_{+}$ and
$\bm{\Phi}$. The condition $H(\bm{\Psi}_{+})\geq H(\bm{\Psi}_{+}|\bm{\Phi})$
–known as _information can’t hurt_ (Cover and Thomas,, 2006)– guarantees that
the $I(\bm{\Psi}_{+};\bm{\Phi})$ is always non-negative. The mutual
information is equal to 0 only when the variables are independent, i.e.,
$p_{\bm{\Psi}_{+},\bm{\Phi}}(\bm{S},\bm{R})=p_{\bm{\Psi}_{+}}(\bm{S})p_{\bm{\Phi}}(\bm{R})$
for all possible states $\bm{S}\in\mathcal{S}$ and $\bm{R}\in\mathcal{R}$.

We are now in a position to define the conditions that $\bm{\Phi}_{I}$ and
$\bm{\Phi}_{R}$ must satisfy. First, for the residual term, $\bm{\Phi}_{R}$,
not to have any information about $\bm{\Psi}_{+}$, we need

$I(\bm{\Psi}_{+};\bm{\Phi}_{R})=0.$ (5)

Secondly, the informative contribution should maximise
$I(\bm{\Psi}_{+};\bm{\Phi}_{I})$ from Eq. (4), which is achieved when

$H(\bm{\Psi}_{+}|\bm{\Phi}_{I})=0.$ (6)

The condition $H(\bm{\Psi}_{+}|\bm{\Phi}_{I})=0$ is mathematically equivalent
to expressing $\bm{\Psi}_{+}$ as a function of $\bm{\Phi}_{I}$, namely,
$\bm{\Psi}_{+}=\bm{\mathcal{F}}(\bm{\Phi}_{I})$. Since the Shannon information
is based on the joint probability distribution of the variables, rather than
their specific values, there are infinitely many functions that satisfy Eqs.
(5) and (6). To identify a unique solution, we impose the condition that the
informative contribution $\bm{\Phi}_{I}$ must reconstruct as much of the
energy of the source variable $\bm{\Phi}$ as possible. With that, the final
optimisation problem for IND is posed as

$\arg\min_{\bm{\Phi}_{I}}\quad\|\bm{\Phi}-\bm{\Phi}_{I}\|^{2}\quad\mathrm{s.t.}\quad
H(\bm{\Psi}_{+}|\bm{\Phi}_{I})=0,\quad I(\bm{\Phi}_{R};\bm{\Phi}_{I})=0,$ (7)

where we have replaced $I(\bm{\Phi}_{R};\bm{\Psi}_{+})=0$ by the more
restrictive condition $I(\bm{\Phi}_{R};\bm{\Phi}_{I})=0$ (Cover and Thomas,,
2006) to satisfy the property:

$\|\bm{\Phi}\|^{2}=\|\bm{\Phi}_{I}\|^{2}+\|\bm{\Phi}_{R}\|^{2}.$ (8)

If $\bm{\Phi}$ contains no information about $\bm{\Psi}_{+}$, then
$\|\bm{\Phi}_{I}\|^{2}/\|\bm{\Phi}\|^{2}\simeq 0$ and
$\|\bm{\Phi}_{R}\|^{2}/\|\bm{\Phi}\|^{2}\simeq 1$. Conversely, if $\bm{\Phi}$
exclusively contains all the information necessary to understand
$\bm{\Psi}_{+}$, then $\|\bm{\Phi}_{I}\|^{2}/\|\bm{\Phi}\|^{2}=1$. Note that,
in general, $\bm{\Phi}_{I}$, $\bm{\Phi}_{R}$ and $\bm{\mathcal{F}}$ are
functions of $\Delta T$, which has been omitted here for the sake of
simplicity in the notation.

###  2.2 IND of the target variable

Alternatively, we can seek to decompose the target variable as
$\bm{\Psi}=\bm{\Psi}_{I}+\bm{\Psi}_{R}$, where $\bm{\Psi}_{I}$ and
$\bm{\Psi}_{R}$ are, respectively, the informative and residual components of
$\bm{\Psi}$ with respect to $\bm{\Phi}_{-}=\bm{\Phi}(\bm{x},t-\Delta T)$, with
$\Delta T>0$. The problem is formulated as

$\arg\min_{\bm{\Psi}_{I}}\quad\|\bm{\Psi}-\bm{\Psi}_{I}\|^{2}\quad\mathrm{s.t.}\quad
H(\bm{\Phi}_{-}|\bm{\Psi}_{I})=0,\quad I(\bm{\Psi}_{R};\bm{\Psi}_{I})=0.$ (9)

In this case, $\bm{\Psi}_{I}$ corresponds to the part of $\bm{\Psi}$ that can
explain the source variable $\bm{\Phi}$ in the past, while $\bm{\Psi}_{R}$ is
the remaining term, which is agnostic to the information in the source
variable.

###  2.3 Approximate IND

The optimisation problem in Eq. (7) (similarly for Eq. (9)) requires
calculating high-dimensional joint probability distributions, which might be
impractical due to limited data and computational resources. The curse of
high-dimensionality comes from both the high dimensionality of $\bm{\Phi}$ and
$\bm{\Psi}$ and the large number of points in $\bm{x}$. To make the problem
tractable, we introduce the approximate IND or aIND for short. First, the
source and target variables are restricted to be scalars, $\Phi$ and $\Psi$,
respectively. Second, we consider only two points in space: $\Phi(\bm{x},t)$
and $\Psi_{+}(\bm{x}-\Delta\bm{x},t+\Delta T)$, where $\bm{x}$ and
$\Delta\bm{x}$ are fixed. This reduces the problem to the computation of two-
dimensional joint probability distributions, which is trivially affordable in
most cases.

Another difficulty arises from the constraint in Eq. (5), which depends on the
probability distribution of the unknown variable $\Phi_{R}$. To alleviate this
issue, $I(\Phi_{R};\Phi_{I})$ is incorporated as a penalisation term in the
cost function, rather than as a hard constraint. Thus, the formulation of the
aIND is posed as

$\arg\min_{\Phi_{I},\mathcal{F}}\quad\|\Phi-\Phi_{I}\|^{2}+\gamma
I(\Phi_{R};\Phi_{I})\quad\mathrm{s.t.}\quad\Psi_{+}=\mathcal{F}(\Phi_{I}),$
(10)

where $\gamma\geq 0$ is a regularisation constant. Equation (10) yields the
informative and residual components for a given $\bm{x}$, $\Delta\bm{x}$, and
$t$, denoted as $\Phi_{I,\Delta}(\bm{x},t;\Delta\bm{x})$ and
$\Phi_{R,\Delta}(\bm{x},t;\Delta\bm{x})$, together with the mapping
$\mathcal{F}$. More details about the solution of Eq. (10) are provided in
Appendix A.1. We can find the best approximation to IND by selecting the value
of $\Delta\bm{x}$ that maximises the informative component. To that end, we
introduce the relative energy of $\Phi_{I,\Delta}$ as

$E_{I}(\Delta\bm{x};\bm{x},\Delta
T)=\frac{\|\Phi_{I,\Delta}\|^{2}}{\|\Phi\|^{2}}.$ (11)

High values of $E_{I}$ define the _informative region_ of $\Phi_{I,\Delta}$
over $\Psi_{+}$ and constitute the information-theoretic generalisation of the
two-point linear correlation (see Appendix B). We define $\Delta\bm{x}^{\max}$
as the shift $\Delta\bm{x}$ that maximises $E_{I}$ for a given $\bm{x}$ and
$\Delta T$. Hence, we use $\Delta\bm{x}=\Delta\bm{x}^{\max}$ for aIND and
simply refer to the variables in this case as $\Phi_{I}$ and $\Phi_{R}$.
Finally, given that
$I(\bm{\Phi}_{R};\bm{\Psi}_{+})\leq\sum_{\bm{x}}I(\Phi_{R};\Psi_{+})$, the
validity of the aIND can be assessed _a posteriori_ by ensuring that
$I(\Phi_{R};\Psi_{+})$ remains small for all $\bm{x}$.

###  2.4 Validation

The methodology presented in §2.1 and its numerical implementation (Appendix
A.1) have been validated with several analytical examples. In this section, we
discuss one of these examples that also illustrates the use and interpretation
of the IND.

Consider the source and target fields:

source: $\displaystyle\quad\Phi(\bm{x},t)=f(\bm{x},t)+g(\bm{x},t),$ (12)
target:
$\displaystyle\quad\Psi_{+}(\bm{x},t)=\Psi(\bm{x},t+1)=0.5f(\bm{x},t)^{2}-0.2f(\bm{x},t)+\epsilon(\bm{x},t),$
(13)

where

$\displaystyle f(\bm{x},t)$ $\displaystyle=2\sin\left(2\pi
x-2t\right)\sin\left(2\pi y\right),$ $\displaystyle g(\bm{x},t)$
$\displaystyle=\frac{1}{5}\sin({7\sqrt{2}\pi x-0.1t})\sin({8\sqrt{3}\pi
y-0.5t}).$

The source field is a combination of the streamwise travelling wave, $f$, and
the lower amplitude, higher wavenumber travelling wave, $g$. The target is a
function of $f$ and $\epsilon$, where the latter is a random variable that
follows the pointwise normal distribution with zero mean and standard
deviation ($\sigma$) equal to 0.1: $\epsilon(\bm{x},t)\sim\mathcal{N}(0,0.1)$.
Snapshots of $\Phi$ and $\Psi$ are shown in figures 2 and 2, respectively.

For $\Delta T=1$ and values of $\sigma\to 0$, the analytical solution of the
IND is

$\Phi_{I}^{\text{exact}}=f,\quad\Phi_{R}^{\text{exact}}=g,$ (14)

where the mapping to comply with $H(\Psi_{+}|\Phi_{I}^{\text{exact}})=0$ is
$\mathcal{F}^{\text{exact}}(\Phi_{I})=0.5\Phi_{I}^{2}-0.2\Phi_{I}$, and the
residual term satisfies the condition
$I(\Phi_{I}^{\text{exact}};\Phi_{R}^{\text{exact}})=0$, since the variables
are independent.

The results of solving the optimisation problem using aIND, denoted by
$\Phi_{I}$, $\Phi_{R}$, and $\mathcal{F}$ are displayed in figures 2, 2 and 2.
It can be observed that $\Phi_{I}$ approximates well the travelling wave
represented by $\Phi_{I}^{\text{exact}}=f$. The small differences between
$\Phi_{I}$ and $\Phi_{I}^{\text{exact}}$, also appreciable in $\Phi_{R}$, are
localised at values of $f\approx 0.2$ and can be explained by the small
discrepancies between $\mathcal{F}$ and $\mathcal{F}^{\text{exact}}$ at the
inflection point as seen in figure 2. These are mostly a consequence of
$\epsilon$ and the numerical implementation (see Appendix A.1), and they
diminish as $\sigma\rightarrow 0$.

(a)(b)(c)(d)$x$$x$$y$$y$

(e)$\Phi$$\Psi_{+}$

Figure 2: Validation of aIND for the system in Eq. (12). (a,b,c,d) Snapshots
of $\Phi$, $\Psi_{+}$, $\Phi_{I}$ and $\Phi_{R}$, respectively. The contours
range from the minimum value (dark blue) to the maximum value (dark red) for
each quantity; these correspond to $\Phi,\Phi_{I}\approx[-2,2]$,
$\Psi_{+}\approx[-1.1,1.4]$, and $\Phi_{R}\approx[-0.35,0.35]$. (e) Contours
of the joint probability $(\Phi,\Psi_{+})$ from (white) lower to (blue) higher
probability. Analytical solution
$\mathcal{F}^{\text{exact}}(\Phi_{I})=0.5\Phi_{I}^{2}-0.2\Phi_{I}$ (dashed
black) and numerical solution $\mathcal{F}(\Phi_{I})$ (orange).

##  3 Results

We study the aIND of the streamwise ($u$), wall-normal ($v$) and spanwise
($w$) velocity fluctuations in a turbulent channel flow using as target the
streamwise component of the shear stress at the wall,
$\tau_{x}(x,z,t)=\nu\partial U(x,0,z,t)/\partial y$, where $\nu$ is the
kinematic viscosity, $U$ is the instantaneous streamwise velocity and $x$, $y$
and $z$ are the streamwise, wall-normal, and spanwise directions,
respectively. The wall is located at $y=0$. The data are obtained from direct
numerical simulation in a computational domain of size $8\pi h\times 2h\times
4\pi h$ in the streamwise, wall-normal, and spanwise directions, respectively,
where $h$ represents the channel half-height. The flow is driven by a constant
mass flux imposed in the streamwise direction. The Reynolds number, based on
the friction velocity $u_{\tau}$, is $\mathit{Re}_{\tau}=u_{\tau}h/\nu\approx
180$. Viscous units, defined in terms of $\nu$ and $u_{\tau}$, are denoted by
superscript $*$. The time step is fixed at $\Delta t^{*}=5\cdot 10^{-3}$, and
snapshots are stored every $\Delta t_{s}^{*}=0.5$. A description of the
numerical solver and computational details can be found in Lozano-Durán _et
al._ , (2020).

The source and target variables for aIND are

$\displaystyle\text{source}:$ $\displaystyle\ u(\bm{x},t),\ v(\bm{x},t)\
\text{or}\ w(\bm{x},t),$ (15) $\displaystyle\text{target}:$ $\displaystyle\
\tau_{x,+}=\tau_{x}(x-\Delta x^{\max}_{\square},z-\Delta
z^{\max}_{\square},t+\Delta T),$ (16)

where $\square=u$, $v$ or $w$. The aIND gives

$\displaystyle u(\bm{x},t)$ $\displaystyle=u_{I}(\bm{x},t)+u_{R}(\bm{x},t),$
(17) $\displaystyle v(\bm{x},t)$
$\displaystyle=v_{I}(\bm{x},t)+v_{R}(\bm{x},t),$ (18) $\displaystyle
w(\bm{x},t)$ $\displaystyle=w_{I}(\bm{x},t)+w_{R}(\bm{x},t),$ (19)

where the informative and residual components are also a function of $\Delta
T$. We focus our analysis on $\Delta T^{*}\approx 25$ unless otherwise
specified. This value corresponds to the time shift at which
$H(\tau_{x,+}|\tau_{x})/H(\tau_{x,+})\lesssim 0.03$, meaning that $\tau_{x,+}$
shares no significant information with its past. For $\Delta T^{*}>25$, the
value of $H(\tau_{x,+}(\Delta T)|\tau_{x})$ gradually diminishes towards 0
asymptotically. This value is similar to the one reported by Zaki and Wang,
(2021), who found using adjoint methods that wall observations at $\Delta
T^{*}\approx 20$ are the most sensitive to upstream and near-wall velocity
perturbations. The shift $\Delta\bm{x}_{\square}^{\max}=[\Delta
x^{\max}_{\square},\Delta z^{\max}_{\square}]$ for $\square=u$, $v$ or $w$ is
computed by a parametric sweep performed in Appendix C. Their values are a
function of $y$, but can be roughly approximated by
$\Delta\bm{x}_{u}^{\max}/h\approx[-1,0]$,
$\Delta\bm{x}_{v}^{\max}/h\approx[-1.2,0]$ and
$\Delta\bm{x}_{w}^{\max}/h\approx[-0.8,\pm 0.15]$. Due to the homogeneity and
statistical stationarity of the flow, the mapping $\mathcal{F}$ is only a
function of $y$ and $\Delta T$. The validity of the approximations made in the
aIND is discussed in Appendix D, where it is shown that the residual component
of $u$ contains almost no information about the future wall-shear stress.

###  3.1 Coherent structure of the informative and residual components of
${\bf u}$ to $\tau_{x}$

We start by visualising the instantaneous informative and residual components
of the flow. We focus on the streamwise component, as it turns out to be the
most informative to $\tau_{x}$, as detailed below. Figure 3 displays iso-
surfaces of $u(\bm{x},t)$, revealing the alternating high- and low-velocity
streaks attached to the wall along with smaller detached regions. The
informative and residual components, $u_{I}(\bm{x},t)$ and $u_{R}(\bm{x},t)$,
are shown in figures 3 and 3, respectively. The structures in $u_{I}$ exhibit
a similar alternating pattern as in the original field, with the high- and
low-velocity streaks located roughly in the same positions as $u(\bm{x},t)$.
These structures are also attached to the wall but do not extend as far as the
streaks in the original field, especially for $u_{I}(\bm{x},t)>0$. In
contrast, the residual field $u_{R}(\bm{x},t)$ lacks most of the elongated
streaks close to the wall but resembles $u(\bm{x},t)$ far away, once the flow
bears barely no information about $\tau_{x,+}$.

(a)

(b)$\partial\mathcal{F}^{*}/\partial u_{I}^{*}$

(c)

Figure 3: Instantaneous flow structures. (a) Iso-contours of the streamwise
velocity fluctuations, $u$; (b) iso-contours of the informative streamwise
velocity fluctuations, $u_{I}$; and (c) iso-contours of the residual
streamwise velocity fluctuations, $u_{R}$. In (a) and (c) the iso-contours
correspond to: (blue) $u^{*}\approx-2.7$, (red) $u^{*}\approx 2.7$; and in
(b): (blue) $u_{I}^{*}\approx-1.8$ and (orange) $u_{I}^{*}\approx 1.8$. The
wall is coloured by the instantaneous wall shear stress at $\Delta T$, from
(white) $\tau_{x}^{*}\approx 0.5$ to (black) $\tau_{x}^{*}\approx 2$.

Figure 4 displays the root-mean-squared turbulence intensities as a function
of the wall distance. Note that, by construction, $\langle
u^{2}\rangle(y)=\langle u_{I}^{2}\rangle(y)+\langle u_{R}^{2}\rangle(y)$
(similarly for the other components). From figure 4, we observe that $\langle
u_{I}^{2}\rangle^{1/2}$ is predominantly located within the region $y^{*}\leq
50$. This finding aligns with our earlier visual assessments from figure 3.
The residual component $\langle u_{R}^{2}\rangle^{1/2}$ also has a strong
presence close to the wall, although it is shifted towards larger values of
$y$. Interestingly, about half of the streamwise kinetic energy in the near-
wall region originates from $\langle u_{R}^{2}\rangle$, despite its lack of
information about $\tau_{x,+}$. This phenomenon is akin the inactive motions
in wall turbulence (e.g. Townsend,, 1961; Jiménez and Hoyas,, 2008; Deshpande
_et al._ ,, 2021) with the difference that here inactive structures are
interpreted as those that do not react to time variations of the wall-shear
stress. Another interesting observation is that $\langle
u_{I}^{2}\rangle^{1/2}$ peaks at $y^{*}\approx 10$, which is slightly below
the well-known peak for $\langle u^{2}\rangle^{1/2}$, whereas $\langle
u_{R}^{2}\rangle^{1/2}$ peaks at $y^{*}\approx 30$. This suggests that the
near-wall peak of $\langle u^{2}\rangle^{1/2}$ is controlled by a combination
of active and inactive motions as defined above.

The root-mean-squared velocities for the cross flow are shown in figures 4 and
4. The informative component of the wall-normal velocity $\langle
v_{I}^{2}\rangle^{1/2}$ is predominantly confined within the region $y^{*}\leq
70$, although its magnitude is small. The residual component, $\langle
v_{R}^{2}\rangle^{1/2}$, is the major contributor to the wall-normal
fluctuations across the channel height. The dominance of $\langle
v_{R}^{2}\rangle^{1/2}$ has important implications for control strategies in
drag reduction, which are investigated in §3.3. A similar observation is made
for $\langle w^{2}\rangle^{1/2}$, with $\langle w_{I}^{2}\rangle^{1/2}$ being
negligible except close to the wall for $y^{*}<40$.

$y^{*}$$\langle{u_{\square}^{*}}^{2}\rangle^{1/2}$(a)

$y^{*}$$\langle{v_{\square}^{*}}^{2}\rangle^{1/2}$(b)

$y^{*}$$\langle{w_{\square}^{*}}^{2}\rangle^{1/2}$(c)

Figure 4: Root-mean-squared turbulence intensities of the (a) streamwise, (b)
wall-normal, and (c) spanwise velocity components. (solid) original flow
field; (dashed) informative flow field; and (dash-dot) residual field.

The statistical coherence of the informative and residual velocity in the
wall-parallel plane is quantified with the two-point auto-correlation

$C_{\phi\phi}(\Delta x,\Delta
z;y_{\text{ref}})=\frac{\langle\phi(x,y_{\text{ref}},z,t)\phi(x+\Delta
x,y_{\text{ref}},z+\Delta
z,t)\rangle}{\langle\phi(x,y_{\text{ref}},z,t)^{2}\rangle},$ (20)

where $\phi$ is any component of the velocity field, and
$y_{\text{ref}}^{*}=15$. The auto-correlations are shown in figure 5 for the
total, informative, and residual components of the three velocities. The shape
of informative structure is elongated along the streamwise direction for the
three correlations $C_{u_{I}u_{I}}$, $C_{v_{I}v_{I}}$, and $C_{w_{I}w_{I}}$.
The results for $u$, shown in figure 5, reveal that $u_{I}$ closely resembles
the streaky structures of $u$ in terms of streamwise and spanwise lengths. On
the other hand, $u_{R}$ consists of more compact and isotropic eddies in the
$(x,z)$-plane. Figure 5 shows that $v_{I}$ captures the elongated motions in
$v$, which represents a small fraction of its total energy, whereas the
shorter motions in $v$ are contained in $v_{R}$. A similar conclusion is drawn
for $w$, as shown in figure 5, where both $w$ and $w_{R}$ share a similar
structure, differing from the elongated motions of $w_{I}$. The emerging
picture from the correlations is that informative velocities tend to comprise
streamwise elongated motions, whereas the remaining residual components are
shorter and more isotropic. The differences between the structure of $v$ and
$w$ with their informative counterparts are consistent with the lower
intensities of $v_{I}$ and $w_{I}$ discussed in figure 4.

$\Delta x^{*}$$\Delta z^{*}$(a)$\Delta x^{*}$$\Delta z^{*}$(b)$\Delta
x^{*}$$\Delta z^{*}$(c) Figure 5: Auto-correlation coefficient of the velocity
fluctuations in the $y^{*}=15$ plane. (a) Streamwise component, (b) wall
normal component, and (c) spanwise component. (Grey) Original field, (blue)
informative field, and (orange) residual field. The contours correspond to
$C_{\square\square}=[0.05,0.1]$.

We now analyse the average coherent structure of the flow in the
$(y,z)$-plane. It is widely recognised in the literature that the most
dynamically relevant energy-containing structure in wall turbulence comprises
a low-velocity streak accompanied by a collocated roll (e.g. Kline _et al._ ,,
1967; Kim _et al._ ,, 1987; Lozano-Durán _et al._ ,, 2012; Farrell and
Ioannou,, 2012). A statistical description of this structure can be obtained
by conditionally averaging the flow around low-velocity streaks. To this end,
low-velocity streaks were identified by finding local minima of $u$ at
$y^{*}=15$. For each streak, a local frame of reference was positioned with
its origin at the wall; the $y$-axis aligned with the local minimum of $u$;
and the $z$-axis pointing towards the closest local maximum of $u$. Then, the
conditional average flow was computed by averaging $[u,v,w]$ over a window of
size $\pm h$. The resulting conditionally-averaged flow in the $(y,z)$-plane
is shown in figure 6. This process was repeated for the informative and
residual velocity fields using the same streaks previously identified for $u$.
The conditionally-averaged informative and residual velocities are shown in
figures 6 and 6, respectively.

$\Delta z^{*}$$y^{*}$(a)$\Delta z^{*}$(b)$\Delta z^{*}$(c) Figure 6:
Conditionally-averaged flow in the $(y,z)$-plane centred about low-velocity
streaks. (a) Original field, (b) informative field, and (c) residual field.
The contours range from (blue) $-5u_{\tau}$ to (red) $5u_{\tau}$. The arrows
display the in-plane velocity components ($v_{\square}$ and $w_{\square}$).

The conditional average velocity is shown in figure 6, which captures the
structure of the low-/high-velocity streak pair and the accompanying roll
characteristic of wall-bounded turbulence. The informative velocity (figure 6)
is dominated by streak motions, although these are smaller than the streaks of
the entire field. The informative wall-normal velocity is present mostly
within the streaks, while the informative spanwise component is active close
to the wall in the interface of the streak. Conversely, figure 6 shows that
the residual velocity contains the large-scale streaks and the remaining
spanwise motions. The emerging picture is that the informative component of
the velocity contributing to the wall shear stress consists of smaller near-
wall streaks collocated with vertical motions (i.e., sweeps and ejections),
and spanwise velocity at the near-wall root of the roll. This informative
structure is embedded within a larger-scale streak-roll structure of residual
velocity, which bears no information about the wall-shear stress.

We close this section by analysing the mappings
$\tau_{x,+}=\mathcal{F}_{u}(u_{I})$, $\tau_{x,+}=\mathcal{F}_{v}(v_{I})$,
$\tau_{x,+}=\mathcal{F}_{w}(w_{I})$ obtained from the constraints
$H(\tau_{x,+}|u_{I})=0$, $H(\tau_{x,+}|v_{I})=0$, and $H(\tau_{x,+}|w_{I})=0$,
respectively. The mapping are depicted in figure 7 at the wall-normal position
where the energy for $u_{I}$, $v_{I}$, and $w_{I}$ is maximum, namely,
$y^{*}\approx 8$, $19$, and $6$, respectively (see Appendix C). Figure 7
reveals an almost linear relationship between $u_{I}$ and $\tau_{x,+}$ within
the range of $0\leq\tau_{x,+}^{*}\leq 2$. Negative values of $u_{I}$ align
with $\tau_{x,+}^{*}<1$, while positive values of $u_{I}$ correspond to
$\tau_{x,+}^{*}>1$. This is clear a manifestation of the proportionality
between streak intensity and $\tau_{x}$, such that higher streamwise
velocities translate into higher wall shear stress by increasing $\partial
U/\partial y$. However, the process saturates and a noticeable change in the
slope occurs for larger values of $\tau_{x,+}$, leading to $u_{I}$ values
which are relatively independent of $\tau_{x,+}$. This finding indicates that
$u_{I}$ provides limited information about high values of $\tau_{x,+}$ at the
timescale $\Delta T^{*}=25$. In other words, minor uncertainties in $u_{I}$
result in significant uncertainties in $\tau_{x,+}$ after $\Delta T$.

The effect of $\Delta T$ on $\mathcal{F}_{u}(u_{I})$ is also analysed in
figure 7. The main effect of decreasing $\Delta T^{*}$ is to decrease the
slope of $\mathcal{F}_{u}(u_{I})$ for $u_{I}^{*}>5$. This result reveals that
there exists a time horizon beyond which it is not possible to predict extreme
events of wall shear stress from local fluctuations. Hence, extreme values of
the wall shear stress can be attributed to almost instantaneous high
fluctuations of the streamwise velocity. The latter is in agreement with
Guerrero _et al._ , (2020), who linked extreme positive wall shear stresses
with the presence of high-momentum regions created by quasi-streamwise
vortices.

The mapping of $v_{I}$ is shown in figure 7, which demonstrates again a nearly
linear, albeit negative, relationship between $v_{I}$ and $\tau_{x,+}$ in the
range $0\leq\tau_{x,+}^{*}\leq 2$. Positive values of $v_{I}$ are indicative
of $\tau_{x,+}^{*}<1$, whereas negative values imply $\tau_{x,+}^{*}>1$. Note
that changes in the value of $\tau_{x,+}$ encompass either $u_{I}>0$ and
$v_{I}<0$ or $u_{I}<0$ and $v_{I}>0$, revealing a connection between the
dynamics of $\tau_{x,+}$ and the well-known sweep and ejection motions in
wall-bounded turbulence (Wallace _et al._ ,, 1972; Wallace,, 2016). The
mappings also show that excursions into large wall shear stresses are caused
by sweeps. Analogous to $u_{I}$, the value of $v_{I}$ remains approximately
constant for $\tau_{x,+}^{*}>2$. Beyond that threshold, $v_{I}$ provides no
information about $\tau_{x,+}$.

The mapping of $w_{I}$ presents two maxima ($\pm\Delta z_{w}^{\max}$) due to
the spanwise symmetry of the flow. The results for each maximum, shown in
figure 7, are antisymmetric with respect to $w_{I}$. Similarly to $u_{I}$ and
$v_{I}$, there is an almost linear relationship between $w_{I}$ and
$\tau_{x,+}$ in the range $0\leq\tau_{x,+}^{*}\leq 2$. For $+\Delta
z_{w}^{\max}$, negative values of $w_{I}$ indicate $\tau_{x,+}^{*}<1$, whereas
positive values are linked to $\tau_{x,+}^{*}>1$. The opposite is true for
$-\Delta z_{w}^{\max}$. Low values of $\tau_{x,+}$ are connected to low
$u_{I}$ and positive (negative) values of $w_{I}$ for $+\Delta z_{w}^{\max}$
($-\Delta z_{w}^{\max}$). This outcome is consistent with the conditional
average flow from figure 6, where it was shown that the information transfer
between $w_{I}$ and $\tau_{x,+}$ is mediated through the bottom part of the
roll structure that accompanies high/low velocity streaks. The saturation of
the influence of $w_{I}$ to intense values of the wall shear stress is again
observed for $\tau_{x,+}^{*}\gtrsim 2$.

$\tau_{x,+}^{*}$$u_{I}^{*}$(a)

$\tau_{x,+}^{*}$$v_{I}^{*}$(b)

$\tau_{x,+}^{*}$$w_{I}^{*}$(c)

Figure 7: Mapping functions of the informative contributions of (a)
streamwise, (b) wall-normal, and (c) spanwise velocity fluctuations to the
streamwise wall shear stress for $\Delta T^{*}=25$. (a) Also shows the effect
of the time lag $\Delta T$ in the mapping $\tau_{x}=\mathcal{F}(u_{I})$. Line
colors correspond to (dark blue) $\Delta T^{*}\approx 25$; (cyan) $\Delta
T^{*}\approx 10$; and (light blue) $\Delta T^{*}\approx 5$. In (c), solid line
corresponds to $+\Delta z^{\max}_{w}$ and dashed line to $-\Delta
z^{\max}_{w}$.

The information provided by the mappings can be embedded into the
instantaneous coherent structures. In figure 3, the $u_{I}(\bm{x},t)$
structures are coloured by the local value of $\partial\mathcal{F}/\partial
u_{I}$. As discussed above, this metric serves as a measure of the uncertainty
in the wall shear stress as a function of $u_{I}$. Low values of
$\partial\mathcal{F}/\partial u_{I}$ are associated with low uncertainty in
$\tau_{x,+}$. This implies that small changes in $u_{I}$ result in small
changes in $\tau_{x,+}$. On the other hand, high values of
$\partial\mathcal{F}/\partial u_{I}$ are associated with high uncertainty in
$\tau_{x,+}$, such that small variations in $u_{I}$ result in large changes in
$\tau_{x,+}$. Interestingly, figure 3 shows that low-speed streaks –associated
with ejections– are connected to low uncertainty values for $\tau_{x}$ along
their entire wall-normal extent. On the contrary, the high-speed streaks of
$u_{I}$, linked to extreme events, carry increasing uncertainty in $\tau_{x}$
(indicated by the light yellow colour) as they move further away from the
wall.

###  3.2 Reduced-order modelling: reconstruction of the wall-shear stress from
$u$

We evaluate the efficacy of the informative and residual components of the
streamwise velocity fluctuations for the prediction of the wall shear stress
in the future. Two scenarios are considered. In the first case, we devise a
model for the point-wise reconstruction of $\tau_{x}$, using pointwise data of
$u$. In the second scenario, the spatially two-dimensional wall-shear stress
is predicted using $u$ data from a wall-parallel plane located at given
distance from the wall. The data is extracted from a simulation with the same
set-up and friction Reynolds number as in §3 but in a smaller computational
domain ($\pi h\times 2h\times\pi/2h$).

For the first case, we use long-short term memory (LSTM) recurrent neural
networks (Sak _et al._ ,, 2014) to predict the future of the wall shear stress
$\tau_{x,+}=\tau_{x}(x_{0},z_{0},t+\Delta T)$, where $x_{0}$ and $z_{0}$ are
fixed, and the time lag is $\Delta T^{*}=25$. Note that all the points
$[x_{0},z_{0}]$ are statistically equivalent and can be used to train the
model. We employ LSTM networks, as they are widely employed for time signal
forecasting (Yu _et al._ ,, 2019). Three neural networks are trained: one
using as input the original velocity $u(x_{0}+\Delta
x_{u}^{\max},y_{\text{ref}},z_{0},t)$, the second one using
$u_{I}(x_{0}+\Delta x_{u}^{\max},y_{\text{ref}},z_{0},t)$ and the third one
using $u_{R}(x_{0}+\Delta x_{u}^{\max},y_{\text{ref}},z_{0},t)$, where
$y_{\text{ref}}^{*}\approx 10$. In the three cases, the LSTMs consist of a
single layer with 64 neurons. For the training, the first 4800 samples (out of
6000) of the temporal data are used to predict the future 1200 time instants.
The Adam algorithm (Kingma and Ba,, 2017) is used to find the optimum
solution. The predictions are represented by

$\displaystyle\tilde{\tau}_{x,+}$
$\displaystyle=\text{LSTM}\left(u(x_{0}+\Delta
x^{\max}_{u},y_{\text{ref}},z_{0},t)\right),$ (21)
$\displaystyle\tilde{\tau}_{x,+}^{I}$
$\displaystyle=\text{LSTM}_{I}\left(u_{I}(x_{0}+\Delta
x^{\max}_{u},y_{\text{ref}},z_{0},t)\right),$ (22)
$\displaystyle\tilde{\tau}_{x,+}^{R}$
$\displaystyle=\text{LSTM}_{R}\left(u_{R}(x_{0}+\Delta
x^{\max}_{u},y_{\text{ref}},z_{0},t)\right).$ (23)

Figure 8 displays the temporal evolution of $\tau_{x,+}$ and the reconstructed
signals from the LSTMs. The prediction based on $u_{I}$ greatly improves
compared to that obtained with the original field, $u$, yielding almost a
perfect agreement with the actual wall shear stress, except for a small
mismatch in the extreme values of $\tau_{x,+}$. This can be appreciated in the
peak around $t^{*}=530$, and is explained by the high uncertainty in $u_{I}$
about high values of $\tau_{x,+}$, as discussed in §3.1. Conversely, the model
based on the residual component of $u$ is unable to predict the wall shear
stress, yielding an almost constant value close to the time-average of
$\tau_{x}$. This result further emphasises that the most accurate reduced-
order models are those based on input variables containing the maximum amount
of information about the output, as recently shown by Lozano-Durán and Arranz,
(2022); Yuan and Lozano-Durán, (2024).

$t^{*}$$\tilde{\tau}_{x}^{*}$ Figure 8: Temporal reconstruction of the wall
shear stress using LSTMs trained with (dotted orange) $u$, (red) $u_{I}$, and
(blue) $u_{R}$ as inputs, respectively. The black dashed line correspond to
the actual wall shear stress.

Next, we reconstruct the spatially varying wall shear stress
$\tau_{x}(x,z,t+\Delta T)$ using as input either the informative or residual
component of $u(x,y_{\text{ref}},z,t)$ in the plane $y_{\text{ref}}^{*}\approx
10$ and $\Delta T^{*}=25$. We utilise a fully convolutional neural network
(FCNN, Long _et al._ ,, 2015) with the same architecture as in Guastoni _et
al._ , (2021). Two FCNNs are trained using as input either
$u_{I}(x,y_{\text{ref}},z,t)$ or $u_{R}(x,y_{\text{ref}},z,t)$. A total of
12000 snapshots are used, split into training (70%) and validation (30%). The
predictions are denoted by

$\displaystyle\tilde{\tau}_{x,+}^{I}(x,z,t)=\tilde{\tau}_{x}^{I}(x,z,t+\Delta
T)$ $\displaystyle=\text{FCNN}_{I}\left(u_{I}(x,y_{\text{ref}},z,t)\right),$
(24)
$\displaystyle\tilde{\tau}_{x,+}^{R}(x,z,t)=\tilde{\tau}_{x}^{R}(x,z,t+\Delta
T)$ $\displaystyle=\text{FCNN}_{R}\left(u_{R}(x,y_{\text{ref}},z,t)\right).$
(25)

The results are shown in figure 9 for a given time instant. Figure 9 contains
a snapshot of the wall shear stress to be reconstructed. The reconstruction
using the original field is not included, although the results are
qualitatively similar to those shown in figure 8. Figures 9 and 9 display the
reconstructed wall shear stress using Eq. (24) and Eq. (25), respectively. The
instantaneous $\tilde{\tau}_{x,+}^{I}$ field closely reconstructs
$\tau_{x,+}$, whereas $\tilde{\tau}_{x,+}^{R}$ shows no correlation with
$\tau_{x,+}$. The absolute error between the reconstructed and original
fields, $\epsilon_{\tau_{x}}=|\tau_{x,+}-\tilde{\tau}_{x,+}|$, is included in
figures 9 and 9 for the models based on the informative and residual
components, respectively. The error associated with the informative field is
limited to points where $\tau_{x,+}^{*}>2$. This finding aligns with the
analysis in §3.1, where it was shown that $u_{I}$ lacks the information about
intense $\tau_{x}$ events for the considered time lag. Conversely, the error
in the residual field is significant across all values of $\tau_{x}$,
underlying the fact that the residual field of the aIND provides no relevant
information about the target field, both pointwise and globally. In summary,
the cases investigated here demonstrate that $u_{I}$ carries the information
needed for constructing reduced-order models of $\tau_{x}$, while the residual
component has no practical use for reduced-order modelling.

$\tau_{x}^{*}$$\epsilon_{\tau_{x}}^{*}$(a)(b)(c)(d)(e)$z/h$$z/h$$x/h$$x/h$
Figure 9: Reconstruction of the instantaneous wall shear stress using fully
convolutional neural networks. (a) Reconstructed wall shear stress
($\tilde{\tau}_{x,+}^{I}$) using as input $u_{I}$; (b) original field
$\tau_{x,+}$; (c) Reconstructed wall shear stress ($\tilde{\tau}_{x,+}^{R}$)
using as input $u_{R}$; Absolute error $\epsilon_{\tau_{x}}$ for (d)
$\tilde{\tau}_{x,+}^{I}$; and (e) $\tilde{\tau}_{x,+}^{R}$.

###  3.3 Control: wall-shear stress reduction with opposition control

We investigate the application of the IND to opposition control in a turbulent
channel flow (Choi _et al._ ,, 1994; Hammond _et al._ ,, 1998). Opposition
control is a drag reduction technique based on blowing and sucking fluid at
the wall with a velocity opposed to the velocity measured at some distance
from the wall. The hypothesis under consideration in this section is that the
informative component of the wall-normal velocity is more impactful for
controlling the flow compared to the residual component.

Figure 10 shows a schematic of the problem setup for opposition control in a
turbulent channel flow. The channel is as in §3.2 but the wall-normal velocity
at the wall is replaced by $v(x,0,z,t)=f(v(x,y_{s},z,t))$, where $y_{s}$ is
the distance to the _sensing_ plane, and $f$ is a user-defined function. In
the original formulation by Choi _et al._ , (1994), $f\equiv-v(x,y_{s},z,t)$,
hence the name of _opposition_ control. Here, we set $y_{s}^{*}\approx 14$,
which is the optimum wall distance reported in previous works (Chung and
Talha,, 2011; Lozano-Durán and Arranz,, 2022). Two Reynolds numbers are
considered, $\mathit{Re}_{\tau}=180$ and $395$.

$y_{s}$$v(x,y_{s},z)$$v(x,0,z)\equiv-v(x,y_{s},z)$ Figure 10:  Schematic of
the opposition control. The contour corresponds to the instantaneous vertical
velocity on a $(x,y)$-plane, the green line depicts the instantaneous velocity
at the sensing plane, $y_{s}$, and the yellow line depicts the velocity at the
wall imposed by the classical opposition control technique. Colormap ranges
from (blue) $v^{*}=-1.8$ to (red) $1.8$.

We split $v(x,y_{s},z,t)$ into its informative ($v_{I}$) and residual
($v_{R}$) components to $\tau_{x}(x,z,t)$. Three controllers are investigated.
In the first case, the function of the controller $f$ is such that it only
uses the informative component of $v(x,y_{s},z,t)$, namely
$f(v(x,y_{s},z,t))\equiv-v_{I}(x,y_{s},z,t)$. In the second case, the
controller uses the residual component $f(v(x,y_{s},z,t))\equiv-
v_{R}(x,y_{s},z,t)$. Finally, the third controller follows the original
formulation $f(v(x,y_{s},z,t))\equiv-v(x,y_{s},z,t)$.

Note that this is a more challenging application of the IND due to the dynamic
nature of the control problem. When the flow is actuated, the dynamics of the
system change, and the controller should re-compute $v_{I}$ (or $v_{R}$) for
the newly actuated flow. This problem is computationally expensive, and we
resort to calculating an approximation. The control strategy is implemented as
follows:

  1. 1.

A simulation is performed with $f\equiv-v(x,y_{s},z,t)$, corresponding to the
original version of opposition control.

  2. 2.

The informative term ($v_{I}$) of $v(x,y_{s},z,t)$ related to the wall shear
stress $\tau_{x}(x,z,t)$ is extracted for $\Delta T=0$.

  3. 3.

We find an approximation of the controller, such that
$\tilde{v}_{I}=f(v)\approx-v_{I}$. To obtain this approximation, we solve the
minimisation problem

$\arg\min_{\tilde{v}_{I}}\quad\|v_{I}-\tilde{v}_{I}\|^{2}+\gamma\frac{I(\tau_{x};\tilde{v}_{R})}{H(\tau_{x})}$
(26)

where $\gamma=0.75$. The approximated informative term is modelled as a feed-
forward artificial neural network with 3 layers and 8 neurons per layer.

  4. 4.

Two new simulations are conducted using either $\tilde{v}_{I}$ or
$\tilde{v}_{R}=v-\tilde{v}_{I}$ for opposition control.

Figure 11 summarises the drag reduction for the three scenarios, namely:
$f\equiv-v(x,y_{s},z,t)$, $f\equiv-\tilde{v}_{I}(x,y_{s},z,t)$, and
$f\equiv-\tilde{v}_{R}(x,y_{s},z,t)$. The original opposition control achieves
a drag reduction of approximately $22\%$ and $24\%$ for
$\mathit{Re}_{\tau}=180$ and $\mathit{Re}_{\tau}=395$, respectively. The
values show a marginal dependency on $\mathit{Re}_{\tau}$, in agreement with
previous studies (Iwamoto _et al._ ,, 2002). Opposition control based on
$\tilde{v}_{I}$ yields a moderate increase in drag reduction with a $24\%$ and
$26\%$ drop for each $\mathit{Re}_{\tau}$, respectively. Conversely, the drag
reduction is only up to $7\%$ for the control based on the estimated residual
velocity, $\tilde{v}_{R}$. Note that $v_{I}$ is the component of $v$ with the
highest potential to modify the drag. Whether the drag increases or decreases
depends on the specifics of the controller. On the other hand, the residual
component $v_{R}$ is expected to have a minor impact on the drag. As such, one
might anticipate a 0% drag reduction by using $v_{R}$. However, the
approximation $\tilde{v}_{R}$ retains some information from the original
velocity for intense values of the latter, which seems to reduce the drag on
some occasions. Simulations using $f\equiv-k\tilde{v}_{R}$ –with $k$ adjusted
to $f\sim\|v(x,y_{s},z,t)\|^{2}$– were also conducted, yielding no additional
improvements in the drag reduction beyond 8%.

$\mathit{Re}_{\tau}$$1-u_{\tau,c}^{2}/u_{\tau}^{2}$ Figure 11: Drag reduction,
computed as $1-u_{\tau,c}^{2}/u_{\tau}^{2}$, where $u_{\tau,c}$ is the
friction velocity for the controlled case. Each colour corresponds to a
different controller: (black) $f\equiv-v$; (blue) $f\equiv-\tilde{v}_{I}$; and
(orange) $f\equiv-\tilde{v}_{R}$.

Figures 12 and 12 show the wall-normal velocity in the sensing plane for the
controlled cases at $\mathit{Re}_{\tau}=180$ with $f\equiv-\tilde{v}_{I}$ and
$f\equiv-\tilde{v}_{R}$, respectively. Larger velocity amplitudes are observed
in figure 12 compared to figure 12, indicating that higher Reynolds stresses
are expected, which aligns with a larger average wall shear stress. On the
other hand, figures 12 and 12 display the negative wall-normal velocity
imposed at the boundary for the cases with $f\equiv-\tilde{v}_{I}$ and
$f\equiv-\tilde{v}_{R}$, respectively. The informative component,
$\tilde{v}_{I}$, closely resembles the original velocity but with smaller
amplitudes at extreme events of $v$. This appears to play a slightly
beneficial role in drag reduction. Conversely, figure 12 shows that the
estimated residual component is negligible except for large values of $v$.
This is responsible for the smaller reduction in the mean drag. Although not
shown, similar flow structures are observed for $\mathit{Re}_{\tau}=395$, and
the same discussion applies.

(a)(b)$x/h$$x/h$$z/h$

(c)(d)$x/h$$x/h$$z/h$

Figure 12: (a) Wall normal velocity in the sensing plane, $v(x,y_{s},z)$ and
(b) the minus velocity imposed at the wall for the case
$f\equiv-\tilde{v}_{I}$. (c,d) Same as (a,b) but for the case
$f\equiv-\tilde{v}_{R}$. Contours range from (blue) $v^{*}=-0.9$ to (red)
$v^{*}=0.9$.

##  4 Conclusions

We have presented IND, a method for decomposing a flow field into its
informative and residual components relative to a target field. The
informative field contains all the information necessary to explain the target
variable, contrasting with the residual component, that holds no relevance to
the target variable. The decomposition of the source field is formulated as an
optimisation problem based on mutual information. To alleviate the
computational cost and data requirements of IND, we have introduced an
approximate solution, referred to as aIND. This approach still ensures that
the informative component retains the information about the target, by
minimising the mutual information between the residual and the target in a
pointwise manner.

The method has been applied to study the information content of the velocity
fluctuations in relation to the wall shear stress in a turbulent channel flow
at $\mathit{Re}_{\tau}=180$. Our findings have revealed that streamwise
fluctuations contain more information about the future wall shear stress than
the cross-flow velocities. The energy of the informative streamwise velocity
peaks at $y^{*}\approx 10$, slightly below the well-known peak for total
velocity, while the residual component peaks at $y^{*}\approx 30$. This
suggests that the peak observed in the total velocity fluctuations results
from both active and inactive velocities, with ‘active’ referring to motions
connected to changes in the wall shear stress. Further investigation of the
coherent structure of the flow showed that the informative velocity consists
of smaller near-wall high- and low-velocity streaks collocated with vertical
motions (i.e., sweeps and ejections). The spanwise informative velocity is
weak, except close to the wall within the bottom root of the streamwise rolls.
This informative streak-roll structure is embedded within a larger-scale
streak-roll structure from the residual velocity, which bears no information
about the wall-shear stress for the considered time scale. We have also shown
that ejections propagate information about the wall stress further from the
wall than sweeps, while extreme values of the wall shear stress are attributed
to sweeps in close proximity to the wall.

The utility of IND for reduced-order modelling was demonstrated in the
prediction of the wall shear stress in a turbulent channel flow. The objective
was to reconstruct the 2-D wall shear stress in the future, after $\Delta
T^{*}=25$, by measuring the streamwise velocity in a wall-parallel plane at
$y^{*}\approx 10$ as input. The approach was implemented using a fully
convolutional neural network as the predictor. Two cases were considered,
using either the informative or the residual velocity component as input,
respectively. We have shown that the model can make accurate predictions when
the input is based on the informative component of the velocity. The main
discrepancies were localised in regions with high wall shear stress values.
This outcome aligns with our prior analysis, which indicated that extreme
wall-shear stress events are produced by short-time near-wall sweeps not
captured in the input plane. In contrast, the residual velocity component
offers no predictive power for wall shear stress, as it lacks any substantive
information relevant to the latter. This example in reduced-order modelling
reveals that models achieving the highest performance are those that utilise
input variables with the maximum amount of information about the output.

Finally, we have investigated the application of IND for drag reduction in
turbulent channel flows at $\mathit{Re}_{\tau}=180$ and
$\mathit{Re}_{\tau}=395$. The strategy implemented involved blowing/suction
via opposition control. To this end, the no-transpiration boundary condition
at the wall was replaced with the wall-normal velocity measured in the wall-
parallel plane at $y^{*}=14$. We explored the use of three wall-normal
velocities: the total velocity (i.e., as originally formulated in opposition
control), its informative component, and its residual component. The largest
reduction in drag was achieved using the informative component of $v$, which
performed slightly better than the total velocity for both Reynolds numbers.
The residual component was shown to yield the poorest results. The application
to drag reduction demonstrated here illustrates that the informative component
of $v$ contains the essential information needed for effective flow control.
This paves the way for using IND to devise enhanced control strategies by
isolating the relevant information from the input variables while disregarding
the irrelevant contributions.

We conclude this work by highlighting the potential of IND as a post-
processing tool for gaining physical insight into the interactions among
variables in turbulent flows. Nonetheless, it is also worth noting that the
approach relies on the mutual information between variables, which requires
estimating joint probability density functions. This entails a data-intensive
process that could become a constraint in cases where the amount of numerical
or experimental data available is limited. Future efforts will be devoted to
reducing the data requirements of aIND and extending its capabilities to
account for multi-variable and multiscale interactions among variables.

#### Acknowledgements

The authors acknowledge the Massachusetts Institute of Technology, SuperCloud,
and Lincoln Laboratory Supercomputing Center for providing HPC resources that
have contributed to the research results reported here.

#### Funding

This work was supported by the National Science Foundation under Grant No.
2140775 and MISTI Global Seed Funds and UPM. G. A. was partially supported by
the NNSA Predictive Science Academic Alliance Program (PSAAP; grant DE-
NA0003993).

##  Appendix A Numerical implementation

###  A.1 Solution for scalar variables using bijective functions

Here we provide the methodology to tackle the minimisation problem posed in
Eq. (10). For convenience, we write Eq. (10) again

$\arg\min_{\Phi_{I},\mathcal{F}}\quad\|\Phi-\Phi_{I}\|^{2}+\gamma_{I}I(\Phi_{R};\Phi_{I})\qquad\mathrm{s.t.}\quad\Psi_{+}=\mathcal{F}(\Phi_{I}).$
(27)

To solve Eq. (27), we note that there are 2 unknowns: $\Phi_{I}$ and the
function $\mathcal{F}$. If we assume that $\mathcal{F}$ is invertible, namely

$\Phi_{I}(t)=\mathcal{F}^{-1}(\Psi_{+}(t))\equiv\mathcal{B}(\Psi_{+}(t)),$
(28)

then, Eq. (27) can be recast as

$\arg\min_{\mathcal{B}}\quad\|\Phi-\mathcal{B}(\Psi_{+})\|^{2}+\gamma_{I}I(\Phi-\mathcal{B}(q_{+});\mathcal{B}(q_{+})),$
(29)

which can be solved by standard optimisation techniques upon the
parametrisation of the function $\mathcal{B}$.

However, by imposing bijectivity we constrain the feasible $\Phi_{I}(t)$
solutions that satisfy $H(\Psi_{+}|\Phi_{I})=0$ and could lead to lower
$\|\Phi-\Phi_{I}\|^{2}$ values than in the more lenient case, where
$\mathcal{F}$ only needs to be surjective. To circumvent this limitation, we
recall that a surjective function with $N-1$ local extrema points (points
where the slope changes sign) can be split into $N$ bijective functions (see
figure 13(a)). In particular, we define

$\Phi_{I}(t)=\mathcal{B}_{i}(\Psi_{+}(t))\,|\,\Phi(t)\in[r_{i-1},r_{i})\quad\forall
i=1,\dots,N,$ (30)

where $r_{i}$ is the $i$th local extremum, such that $r_{i}>r_{i-1}$,
$r_{0}\rightarrow-\infty$, and $r_{N}\rightarrow\infty$.

Therefore, the final form of the minimisation equation is

$\arg\min_{\mathcal{B}_{i}}\quad\sum_{i}^{N}\|\Phi-\mathcal{B}_{i}^{0}(\Psi_{+};\Phi)\|^{2}+\gamma
I(\Phi-\mathcal{B}^{\cup}(q_{+});\mathcal{B}^{\cup}(q_{+})),$ (31)

being

$\displaystyle\mathcal{B}_{i}^{0}(\Psi_{+};\Phi)$
$\displaystyle=\begin{cases}\mathcal{B}_{i},&\text{if}\quad\Phi(t)\in[r_{i-1},r_{i})\\\
0&\text{otherwise}\end{cases}$ $\displaystyle\mathcal{B}^{\cup}(\Psi_{+})$
$\displaystyle=\sum_{i}^{N}\mathcal{B}_{i}^{0}(\Psi_{+};\Phi),$

where the extrema ($r_{i}$) are unknowns to be determined in the minimisation
problem, and $\gamma$ and $N$ are the only free parameters. Once the functions
$\mathcal{B}_{i}$ are computed, the informative component is obtained from

${\Phi}_{I}(t)={\mathcal{B}}_{j}(\Psi_{+}(t))\quad|\quad
j=\arg\min_{i}\quad\left(\Phi(t)-\mathcal{B}_{i}(\Psi_{+}(t))\right)^{2}$ (32)

at every time step.

$\mathcal{F}(\Phi_{I})$$\mathcal{B}_{1}(\Psi_{+})$$\mathcal{B}_{2}(\Psi_{+})$$\Phi_{I}$$\Psi_{+}$$r_{1}$
(a)

$a_{1},b_{1}$$w_{1}$$a_{2},b_{2}$$w_{2}$$x_{0}$$x_{2}$ (b)

Figure 13:  (a) Illustration of a surjective function,
$\mathcal{F}(\Phi_{I})$, and its decomposition into two bijective functions:
$\mathcal{B}_{1}(\Psi_{+})$ and $\mathcal{B}_{2}(\Psi_{+})$. (b) Example of a
DSF architecture with 2 hidden layers and 4 neurons per hidden layer. The
functions plotted within boxes are the activation functions acting on the
neurons. Adapted from Huang _et al._ , (2018).

We use feed-forward networks to find $\mathcal{B}_{i}$, as they are able to
approximate any Borel-measurable function on a compact domain (Hornik _et al._
,, 1989). In particular, we use the deep simgoidal flow (DSF) proposed by
Huang _et al._ , (2018), who proved that a feed-forward artificial neural
network is a bijective transformation if the activation functions are
bijective and all the weights are positive. The details of the DSF
architecture and the optimisation can be found in Appendix A.2.

One must emphasise that the current minimisation problem posed in Eq. (10)
differs from the classical flow reconstruction problem (e.g.: Erichson _et
al._ , (2020)) where the maximum reconstruction of $\Phi$ is sought. In those
cases, we look for a function $\mathcal{G}(\Psi_{+})$ that minimises
$\|\Phi-\mathcal{G}(\Psi_{+})\|^{2}$. If the result is a non-bijective
function, the constraint $H(\Psi_{+}|\Phi_{I})=0$ will not be satisfied.

###  A.2 Networks architecture and optimisation details

The present algorithm uses DSF networks to approximate bijective functions.
This network architecture is depicted in figure 13(b). The DSF is composed of
$L$ stacked sigmoidal transformations. Each transformation produces the
output,

$x_{l}=\sigma^{-1}\left(w_{l}^{T}\cdot\sigma(a_{l}\cdot
x_{l-1}+b_{l})\right)\quad l=1,\dots,L$ (33)

where $x_{l-1}$ is the input, $\sigma(y)=1/(1+e^{-y})$ is the logistic
function, $\sigma^{-1}$ is the inverse of $\sigma$, ${a}_{l}$ and ${b}_{l}$
are vectors with the weights and biases of the decoder part of the $l$-layer,
and $w_{l}$ is a vector with the weights of the encoder part of the $l$-layer
(see figure 13(b)). In addition, the weights for each layer have to fulfil
$0<w_{l,i}<1$, $\sum_{i}w_{l,i}=1$, and $a_{l,i}>0$, $i=1,\dots,M$, where $M$
is the number of neurons per layer. These constraints are enforced via the
softmax and the exponential activation functions for $w_{l}$ and $a_{l}$,
respectively. Namely:

$\displaystyle w_{l,i}$
$\displaystyle=\frac{\exp({w^{\prime}_{l,i}})}{\sum_{i=1}^{N}\exp({w^{\prime}_{l,i}})}$
$\displaystyle a_{l,i}$ $\displaystyle=\exp({a^{\prime}_{l,i}}).$

More details on the DSF architecture can be found in Huang _et al._ , (2018).

To compute the optimal weights and biases that yield the optimal
$\mathcal{B}_{i}$ that minimise Eq. (31), we use the Adam algorithm (Kingma
and Ba,, 2017). This minimisation process requires all operations to be
continuous and differentiable. To achieve that, we compute the mutual
information using a kernel density estimator; and the piecewise-defined
functions $\mathcal{B}_{i}^{0}$ are made $\mathcal{C}^{1}$ continuous by
applying the logistic function,

$\mathcal{B}_{i}^{0}(\Psi_{+};\Phi)=\mathcal{B}_{i}(\Psi_{+})\sigma(k(\Phi-\tilde{r}_{i-1}))\sigma(k(\tilde{r}_{i}-\Phi)),$

where the parameter $k>0$ can be chosen to control the steepness of the
function, and $\tilde{r}_{j}=r_{j}\pm\log(p/(p-1))/k$, which ensures
$\mathcal{B}_{i}^{0}=p\mathcal{B}_{i}$ at the boundaries. In the present
study, the first term in Eq. (10) is normalised with $\|\Phi\|^{2}$ and the
second term is normalised with $H(\Phi_{I},\Phi_{R})$. Under this
normalisation, $p=0.99$, $k=500$ and $\gamma=0.5$ were found to be suitable
values for the optimisation process. We fix the DSF architecture to 3 layers
with 12 neurons per layer.

##  Appendix B Analytical solution for Gaussian distributions

When $\Phi$ and $\Psi_{+}$ are jointly normal distributed functions, it is
possible to derive an analytical expression for $\Phi_{I}$. Consider the
distribution
$(\Phi,\Psi_{+})\sim\mathcal{N}([0,0],[\sigma_{\Phi},\sigma_{\Psi_{+}},\rho_{\Phi\Psi_{+}}])$,
where $\sigma_{(\cdot)}^{2}=\langle\cdot,\cdot\rangle=\|\cdot\|^{2}$ is the
variance and
$\rho_{\Phi\Psi_{+}}=\langle\Phi,\Psi_{+}\rangle/\sigma_{\Phi}\sigma_{\Psi_{+}}$
is the Pearson’s correlation coefficient. If we assume that $\Phi_{I}$ is also
jointly normal distributed, we can write $I(\Psi_{+};\Phi_{I})$ as (Cover and
Thomas,, 2006):

$I(\Psi_{+};\Phi_{I})=-\frac{1}{2}\log(1-\rho_{\Psi_{+}\Phi_{I}}^{2}),$ (34)

which is maximised for $\rho_{\Psi_{+}\Phi_{I}}=\pm 1$. On the contrary, the
constraint $I(\Psi_{+};\Phi_{R})=0$ implies $\rho_{\Psi_{+}\Phi_{R}}=0$; or
equivalently

$\langle\Psi_{+},\Phi_{R}\rangle=\langle\Psi_{+},\Phi-\Phi_{I}\rangle=\sigma_{\Psi_{+}}\sigma_{\Phi}\rho_{\Psi_{+}\Phi}-\sigma_{\Psi_{+}}\sigma_{\Phi_{I}}\rho_{\Psi_{+}\Phi_{I}}=0.$
(35)

From this we obtain
$\rho_{\Psi_{+}\Phi_{I}}=\rho_{\Psi_{+}\Phi}/|\rho_{\Psi_{+}\Phi}|$ and
$\sigma_{\Phi_{I}}=\sigma_{\Phi}|\rho_{\Psi_{+}\Phi}|$, which uniquely defines
$\Phi_{I}$:

$\Phi_{I}(t)=\frac{\sigma_{\Phi}\rho_{\Psi_{+}\Phi}}{\sigma_{\Psi_{+}}}\Psi_{+}(t)=\frac{\langle\Psi_{+},\Phi\rangle}{\sigma_{\Psi_{+}}^{2}}\Psi_{+}(t)$
(36)

Note that, $\Phi_{I}$ is fully determined from the constraints and no
minimisation problem is involved. Despite this, Eq. (36) is the optimal
solution to

$\arg\min_{m}\|\Phi-m\Psi_{+}\|^{2}\qquad m\in\mathbb{R}$ (37)

obtained by linear stochastic estimation (LSE, Adrian and Moin,, 1988; Encinar
and Jiménez,, 2019). Therefore, LSE is the optimal solution to Eq. (29) when
the variables are jointly normal distributed functions.

##  Appendix C Computation of $\Delta x^{\text{max}}$ for the turbulent
channel flow

The aIND requires the value of $\Delta\bm{x}_{\square}^{\max}=(\Delta
x_{\square}^{\max},\Delta z_{\square}^{\max})$ for each informative component
$\square=u$, $v$ and $w$. To that end, we calculate their relative energy as a
function of $\Delta x$, $\Delta z$ and the wall-normal distance:

$\displaystyle E_{I}^{u}(\Delta x,\Delta z,y)$
$\displaystyle=\frac{\|u_{I}^{2}\|}{\|u^{2}\|},$ $\displaystyle
E_{I}^{v}(\Delta x,\Delta z,y)$
$\displaystyle=\frac{\|v_{I}^{2}\|}{\|v^{2}\|},$ $\displaystyle
E_{I}^{w}(\Delta x,\Delta z,y)$
$\displaystyle=\frac{\|w_{I}^{2}\|}{\|w^{2}\|}.$

The parametric sweep is performed using data a channel flow at
$\mathit{Re}=180$ in a computational domain of size $\pi h\times
2h\times\pi/2$ in the streamwise, wall-normal, and spanwise direction,
respectively.

Figure 14 displays $E_{I}^{u}$, $E_{I}^{v}$ and $E_{I}^{w}$ as functions of
$\Delta x$ and $\Delta z$. Note that, due to the symmetry of the flow,
$E_{I}^{u}(\Delta x,\Delta z,y)=E_{I}^{u}(\Delta x,-\Delta z,y)$ (similarly
for $E_{I}^{v}$ and $E_{I}^{w}$). For $E_{I}^{u}$ and $E_{I}^{v}$, the maximum
is always located at $\Delta z=0$, which is the plane displayed in figures 14
and 14. For the spanwise component, the maximum value of $E_{I}^{w}$ is offset
in the spanwise direction and its location varies with $y$. Figure 14 displays
the horizontal section that contains its global maximum, which is located at
$y^{*}\approx 6$. This offset is caused by the fact that $w$ motions travel in
the spanwise direction until they reach the wall and affect the wall shear
stress.

(a)$\Delta x/h$$y/h$(b)$\Delta x/h$$y/h$(c)$\Delta x/h$$\Delta z/h$ Figure 14:
Informative regions quantifying the relative energy contained in the
informative components $u_{I}$, $v_{I}$ and $w_{I}$ for $\Delta T^{*}=25$. (a)
$E_{I}^{u}$ and (b) $E_{I}^{v}$ in the $\Delta z=0$ plane; (c) $E_{I}^{w}$ in
the plane $y^{*}\approx 6$. In (a,b) black dashed line corresponds to $\Delta
x^{\max}(y)$.

Close to the wall, we find high values of $E_{I}^{u}$, with a peak vale of
approximately $60\%$ at $y^{*}\approx 8$, and $\Delta
x_{u}^{\max}(y)\approx-h$, following an almost linear relationship with $y$.
Farther from the wall ($y>0.2h$), $\Delta x_{u}^{\max}$ becomes more or less
constant, although it should be noted that, in this region, the values of
$E_{I}^{u}$ for a fixed $y$ are low and relatively constant. This may induce
to some numerical uncertainty in the particular value of $\Delta
x_{u}^{\max}$, but the overall results are not affected. In contrast, high
values of $E_{I}^{v}$ are located in a compact region further away from the
wall ($y^{*}\approx 19$). The values $\Delta x_{v}^{\max}(y)$ lies close to
$-1.2h$ in this region, following a negative linear relationship with $y$. As
before, $\Delta x_{v}^{\max}(y)$ remains relative constant in low $E_{I}^{v}$
regions. Finally, although not shown, $\Delta x_{w}^{\max}(y)$ and $\Delta
z_{w}^{\max}(y)$ lie in the interval $[-h,-0.7h]$ and $\pm[0.1h,0.2h]$,
respectively. Nevertheless, $E_{w}^{I}$ becomes negligible for $y>0.2h$.

##  Appendix D Validity of aIND of $u$ with respect to $\tau_{x}$

Figure 15 displays the mutual information between $u_{R}(x_{0},y_{0},z_{0})$
for $y_{0}^{*}\approx 10$, and $\tau_{x,+}(x_{0}-\Delta x_{u}^{\max}-\delta
x,z_{0}-\Delta z_{u}^{\max}-\delta z)$ as a function of $\delta\bm{x}=[\delta
x,\delta z]$, denoted as $I(u_{R};\tau_{x,+})(\delta\bm{x})$. The mutual
information is normalised by the total Shannon information of the wall shear
stress, $H(\tau_{x})$, such that
$I(u_{R};\tau_{x,+})(\delta\bm{x})/H(\tau_{x})=0$ means that $u_{R}$ contains
no information about the wall shear stress at $\delta\bm{x}$, and
$I(u_{R};\tau_{x,+})(\delta\bm{x})/H(\tau_{x})=1$ implies that $u_{R}$
contains all the information about $\tau_{x,+}(\delta\bm{x})$. Note that aIND
seeks to minimise $I(u_{R};\tau_{x,+})(\bm{0})$. The results show that value
of the $I(u_{R};\tau_{x,+})(\delta\bm{x})/H(\tau_{x})$ remains always low,
reaching a maximum of approximately 0.06 at $\delta x\approx-1.2h$ along the
streamwise direction. Hence, we can conclude that the residual term contains a
negligible amount of information about the wall shear stress at any point in
the wall and aIND is a valid approximation of IND. For the sake of
completeness, we also display in figure 15 the mutual information between
$u_{I}$ and the wall shear stress. Since $\tau_{x,+}=\mathcal{F}(u_{I})$, the
mutual information $I(u_{I};\tau_{x,+})(\delta\bm{x})$ has to be equal to
$H(\tau_{x})$ at $\delta\bm{x}=\bm{0}$, as corroborated by the results. For
larger distances, $I(u_{I};\tau_{x,+})(\delta\bm{x})$ decays following the
natural decay of $I(\tau_{x,+};\tau_{x,+})(\delta\bm{x})$, with values below
0.1 after $|\delta\bm{x}|\approx h$.

$\delta x/h$$\delta z/h$$I(\cdot,\cdot)/H(\tau_{x})$ Figure 15: Mutual
information between the streamwise wall shear stress and (blue circles) the
residual field, $I(u_{R};\tau_{x,+})(\delta\bm{x})$; (red circles) the
informative field$I(u_{I};\tau_{x,+})(\delta\bm{x})$. Dashed line correspond
to $I(\tau_{x,+};\tau_{x,+})(\delta\bm{x})$;

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