#  Neutrino effective potential in a fermion and scalar background

José F<EMAIL_ADDRESS>
Laboratory of Theoretical Physics, Department of Physics
University of Puerto Rico, Río Piedras, Puerto Rico 00936
Sarira<EMAIL_ADDRESS>
Instituto de Ciencias Nucleares
Universidad Nacional Autónoma de Mexico
Circuito Exterior, C. U.
A. Postal 70-543, 04510 Mexico DF, Mexico

###### Abstract

We consider the neutrino self-energy in a background composed of a scalar
particle and a fermion using a simple model for the coupling of the form
$\lambda\bar{f}_{R}\nu_{L}\phi$. The results are useful in the context of Dark
Matter-neutrino interaction models in which the scalar and/or fermion
constitute the dark-matter. The corresponding formulas for models in which the
scalar particle couples to two neutrinos via a coupling of the form
$\lambda^{(\nu\nu\phi)}\bar{\nu}^{c}_{R}\nu_{L}\phi$ are then obtained as a
special case. In the presence of these interactions there can be new
contributions to the neutrino effective potential and index of refraction in
the context of neutrino collective oscillations in a supernova and in the
Early Universe hot plasma before neutrino decoupling. The formulas obtained
here can be used to estimate those effects and/or put limits on the model
parameters based on the contribution that such interactions can have in those
contexts. A particular feature of the results is that the contribution to the
neutrino self-energy or effective potential in a neutrino background due to
the $\nu\nu\phi$ coupling is proportional to the antineutrino-neutrino
asymmetry $(n_{\bar{\nu}}-n_{\nu})$, in contrast to the standard $Z$
contribution which is proportional to $(n_{\nu}-n_{\bar{\nu}})$. Therefore the
two contributions tend to cancel. If the cancellation is significant, it is
conceivable that the $O(1/m^{4}_{\phi})$ terms give the dominant contribution.
Alternatively, a limit can be set by requiring that the contribution of the
$\nu\nu\phi$ interaction to the neutrino effective potential does not cancel
the standard contribution in an appreciable way.

##  1 Introduction

The possible existence of complex scalars that interact with neutrinos via a
coupling of the form

$L_{int}=\frac{1}{2}\lambda^{(\nu\nu\phi)}\bar{\nu}^{c}_{R}\nu_{L}\phi+H.c\,,$
(1)

has been explored recently[1, 2, 3, 4, 5, 6, 7]. Such scalars may be produced
in terrestrial neutrino experiments, and constraints on their properties and
interactions have been obtained from particle physics, astrophysics, and
cosmology considerations in the works cited.

Here we note that such couplings would produce additional contributions beyond
the standard ones to the neutrino index of refraction and effective potential
when the neutrino propagates in a neutrino background. This occurs in the
environment of a supernova, where it is now well known that the effect leads
to the collective neutrino oscillations and related phenomena[8], and it can
occur also in the hot plasma of the Early Universe before the neutrinos
decouple[9]. Therefore, further constraints on the $\nu\nu\phi$ couplings
and/or the $\phi$ properties can be obtained by considering their effects in
those contexts.

In a different line of development, models in which neutrinos interact with a
scalar and a fermion via a coupling of the form

$L_{int}=\lambda\bar{f}_{R}\nu_{L}\phi+H.c.\,,$ (2)

have been considered recently in the context of Dark Matter-neutrino
interactions[10, 11]. Again, such interactions produce additional
contributions to the neutrino effective potential when the neutrino propagates
in a background of $\phi$ and $f$ particles.

In this paper we determine the effective potential of a neutrino or
antineutrino that propagates in such backgrounds. To be specific, we calculate
the neutrino self-energy in a background of scalars $\phi$ and fermions $f$
due to the interaction given in Eq. (2). From the self-energy, the neutrino
and antineutrino effective potential is then obtained. The corresponding
formulas for the case of the neutrino and scalar background, with the
couplings given in Eq. (1), are obtained as the special case in which
$f_{R}\rightarrow\nu^{c}_{R}$. Our motivation is to provide a uniform
treatment and present the results in sufficiently general way such that they
can be applied to the situations described and also adapted to others not
considered here. For example, models in which sterile neutrinos have _secret_
gauge interactions of the form
$\bar{\nu}_{s}\gamma^{\mu}\nu_{s}A^{\prime}_{\mu}$ have been considered by
several authors[12]. The formulas we obtain for the neutrino self-energy and
effective potential, including the case of an anisotropic background that we
consider, can be applied in the context of such models when a sterile neutrino
propagates in a background of sterile neutrinos and $A^{\prime}$ bosons, with
minor modifications.

In principle the scalar-neutrino couplings can be of the form
$\lambda^{(\nu\nu\phi)}_{ij}\bar{\nu}^{c}_{Ri}\nu_{Lj}$, involving various
neutrino species. In writing Eq. (1) we are taking into account only the
diagonal neutrino coupling and assuming the presence of only one scalar field.
In the general case with more neutrino species in the background and non-
diagonal neutrino-$\phi$ couplings, the density matrix formalism[8, 9] will
come into play, which is outside the scope of the present work. It is worth
mentioning that, despite the simplification we are making by considering only
one neutrino type, the results reveal some interesting and potentially
important features that can serve as a guide for considering the more general
and realistic cases. A particular noteworthy feature is that the contribution
to the neutrino self-energy or effective potential in the neutrino background
due to this coupling is proportional to antineutrino-neutrino asymmetry
$(n_{\bar{\nu}}-n_{\nu})$, in contrast to the standard $Z$ contribution which
is proportional to $(n_{\nu}-n_{\bar{\nu}})$. Therefore the two contributions
tend to cancel. If the cancellation is significant, it is conceivable that the
$O(1/m^{4}_{\phi})$ terms give the dominant contribution. As an example
application, we use these results to indicate the constraint on
$\lambda^{(\nu\nu\phi)}/m_{\phi}$ that can be placed based on the contribution
that such interactions can have in the context of neutrino collective
oscillations in a supernova.

In Section 2, we review in a generic way the conventions that we use to
determine the neutrino effective potential in a medium from the calculation of
the self-energy. The calculation of the self-energy of the neutrino
propagating in the background of fermions and scalars is undertaken in Section
3. We consider various cases separately, depending on whether the background
is isotropic or not, and various limits of the momentum distribution functions
of the background particles. In Section 4, we use these results to calculate
the effective potential under different physical conditions for a $\nu\phi$
background. We briefly summarize our results in Section 5.

##  2 Dispersion relation and effective potential

We consider the situation in which a neutrino propagates and interacts with
the $f$ and $\phi$ background particles as illustrated in Fig. 1. For the
purpose of determining the background contribution to the neutrino self-energy
and dispersion relation we treat the neutrino as a massless chiral particle.

Figure 1:  Neutrino self-energy diagram in a background of fermions$f$ and
scalars $\phi$.

Denoting the neutrino momentum by $k^{\mu}=(\omega,\vec{\kappa})$, and the
background contribution to the neutrino self-energy by $\Sigma^{(T)}$, the
dispersion relations are obtained by solving

$(k\hbox to0.0pt{\hss/}-\Sigma^{(T)})\psi_{L}=0\,.$ (3)

The chirality of the neutrino interactions imply that $\Sigma^{(T)}$ is of the
form

$\Sigma^{(T)}=V^{\mu}(\omega,\vec{\kappa})\gamma_{\mu}\,.$ (4)

The dispersion relations are then given by the solutions of

$\omega-V^{0}=\pm|\vec{\kappa}-\vec{V}|\,,$ (5)

which in general is an implicit equation for $\omega(\vec{\kappa})$. We
consider two cases, according to whether the distribution functions are
isotropic or not.

###  2.1 Isotropic case

In this case $V^{\mu}$ is of the form

$V_{\mu}=a(\omega,\kappa)k_{\mu}+b(\omega,\kappa)u_{\mu}\,,$ (6)

where $u^{\mu}$ is the velocity four-vector of the background. In the frame in
which the background is at rest, $u^{\mu}=(1,\vec{0})$, which we adopt from
now on in this case. In terms of $a$ and $b$, Eq. (5) becomes

$(1-a)\omega-b=\pm(1-a)\kappa\,.$ (7)

As already mentioned, in the most general case this is an implicit equation
for $\omega(\vec{\kappa})$. On the other hand, for small $a$ and $b$, the
perturbative solutions around the vacuum solutions
$\omega_{\pm}(\vec{\kappa})=\pm\kappa$ are to the lowest order

$\omega_{\pm}(\vec{\kappa})=\pm\kappa+b(\pm\kappa,\kappa)\,.$ (8)

The neutrino and antineutrino dispersion relations,

$\displaystyle\omega_{\nu}(\vec{\kappa})$ $\displaystyle=$
$\displaystyle\omega_{+}(\vec{\kappa})\,,$
$\displaystyle\omega_{\bar{\nu}}(\vec{\kappa})$ $\displaystyle=$
$\displaystyle-\omega_{-}(-\vec{\kappa})\,,$ (9)

are then

$\omega^{(\nu,\bar{\nu})}(\vec{\kappa})=\kappa+V^{(\nu,\bar{\nu})}_{eff}(\vec{\kappa})\,,$
(10)

where we have introduced the effective potentials $V^{(\nu,\bar{\nu})}_{eff}$,
which in the present case are given explicitly by

$\displaystyle V^{(\nu)}_{eff}(\vec{\kappa})$ $\displaystyle=$ $\displaystyle
b(\kappa,\kappa)\,,$ $\displaystyle V^{(\bar{\nu})}_{eff}(\vec{\kappa})$
$\displaystyle=$ $\displaystyle-b(-\kappa,\kappa)\,.$ (11)

###  2.2 Anisotropic case

In this case the $V^{\mu}$ cannot be decomposed in the form given in Eq. (6).
We use Eq. (5) and assume that the equation can be solved perturbatively
around the vacuum solutions $\omega_{\pm}(\vec{\kappa})=\pm\kappa$. The
solutions obtained in this way are then

$\displaystyle\omega_{+}(\vec{\kappa})$ $\displaystyle=$
$\displaystyle\kappa+V^{0}(\kappa,\vec{\kappa})-\hat{\kappa}\cdot\vec{V}(\kappa,\vec{\kappa})\,,$
$\displaystyle\omega_{-}(\vec{\kappa})$ $\displaystyle=$
$\displaystyle-\kappa+V^{0}(-\kappa,\vec{\kappa})+\hat{\kappa}\cdot\vec{V}(-\kappa,\vec{\kappa})\,.$
(12)

Identifying the neutrino and antineutrino dispersion relations by Eq. (2.1),
they are given as in Eq. (10), where the effective potential in this case are

$\displaystyle V^{(\nu)}_{eff}(\vec{\kappa})$ $\displaystyle=$ $\displaystyle
V^{0}(\kappa,\vec{\kappa})-\hat{\kappa}\cdot\vec{V}(\kappa,\vec{\kappa})\,,$
$\displaystyle V^{(\bar{\nu})}_{eff}(\vec{\kappa})$ $\displaystyle=$
$\displaystyle-V^{0}(-\kappa,-\vec{\kappa})+\hat{\kappa}\cdot\vec{V}(-\kappa,-\vec{\kappa})\,.$
(13)

Of course in the isotropic case, in which $V^{\mu}$ can be decomposed as in
Eq. (6), Eq. (2.2) reduces to Eq. (2.1).

##  3 Self-energy in the $f\phi$ background

In this section we consider a neutrino propagating in a background of the
fermion $f$ and complex scalar $\phi$ due to the interaction term in Eq. (2).
The starting point is the calculation of the self-energy via the diagram shown
in Fig. 1, from which the dispersion relations are determined as indicated in
Section 2.

The contribution of the diagram in Fig. 1 to the neutrino self-energy is given
by

$-iR\Sigma
L=|\lambda|^{2}\int\,\frac{d^{4}p}{(2\pi)^{4}}i\Delta^{(\phi)}_{F}(p-k)RiS^{(f)}_{F}(p)L\,,$
(14)

where

$\displaystyle iS^{(f)}_{F}(p)$ $\displaystyle=$ $\displaystyle
iS^{(\bar{\nu})}_{F0}(p)-\Gamma_{f}(p)\,,$ $\displaystyle
i\Delta^{(\phi)}_{F}(p)$ $\displaystyle=$ $\displaystyle
i\Delta^{(\phi)}_{F0}(p)+\Gamma_{\phi}(p)\,,$

with

$\displaystyle\Gamma_{f}(p)$ $\displaystyle=$ $\displaystyle p\hbox
to0.0pt{\hss/}2\pi\delta(p^{2}-m^{2}_{f})\eta_{f}(p)\,,$
$\displaystyle\Gamma_{\phi}(p)$ $\displaystyle=$ $\displaystyle
2\pi\delta(p^{2}-m^{2}_{\phi})\eta_{\phi}(p)\,.$ (16)

Using the label $x$ to stand for either $f$ or $\phi$, the functions
$\eta_{x}(p)$ are given by

$\eta_{x}(p)=\theta(p^{0})f_{x}(p^{0},\vec{p})+\theta(-p^{0})f_{\bar{x}}(-p^{0},-\vec{p})\,,$
(17)

where $f_{x,\bar{x}}(E_{p},\vec{p})$ are the momentum distribution functions
of the background particles and antiparticles, which may not necessarily be
isotropic. For an isotropic thermal background, the distribution functions are

$\displaystyle f_{f,\bar{f}}(E_{f})$ $\displaystyle=$
$\displaystyle\frac{1}{e^{\beta E_{f}\mp\alpha_{f}}+1}\,,$ $\displaystyle
f_{\phi,\bar{\phi}}(E_{\phi})$ $\displaystyle=$
$\displaystyle\frac{1}{e^{\beta E_{\phi}\mp\alpha_{\phi}}-1}\,.$ (18)

Discarding the pure vacuum contribution to $\Sigma$ and denoting by
$\Sigma^{(f,\phi)}$ the background-dependent part, we write

$\Sigma^{(f,\phi)}=\Sigma^{(f)}+\Sigma^{(\phi)}\,,$ (19)

where

$\displaystyle\Sigma^{(f)}$ $\displaystyle=$
$\displaystyle-|\lambda|^{2}\int\frac{d^{4}p}{(2\pi)^{3}}\frac{p\hbox
to0.0pt{\hss/}}{(p-k)^{2}-m^{2}_{\phi}}\delta(p^{2}-m^{2}_{f})\eta_{F}(p,\alpha_{f})\,,$
(20) $\displaystyle\Sigma^{(\phi)}$ $\displaystyle=$
$\displaystyle|\lambda|^{2}\int\frac{d^{4}p}{(2\pi)^{3}}\frac{(p\hbox
to0.0pt{\hss/}+k\hbox
to0.0pt{\hss/})}{(p+k)^{2}-m^{2}_{f}}\delta(p^{2}-m^{2}_{\phi})\eta_{B}(p,\alpha_{\phi})\,.$
(21)

We will evaluate $\Sigma^{(f,\phi)}$ in the following limiting cases. In the
first case, to which we refer as the _high temperature_ limit, we assume that
the $f$ and $\phi$ backgrounds are isotropic, and also extremely relativistic
so that their respective masses $m_{f,\phi}$ and chemical potentials
$\alpha_{f,\phi}$ can be taken to be zero in Eqs. (20) and (21). In the second
case we also assume the isotropic thermal distribution functions, but with the
temperature and chemical potentials being such that the integrand in Eqs. (20)
and (21) can be approximated by expanding them in powers of

$\Delta\equiv m^{2}_{\phi}-m^{2}_{f}\,.$ (22)

We refer to this as the _heavy background_ limit. This also includes the case
in which either $f$ or $\phi$ is massless, provided that the other particle is
heavy enough. Finally we will consider a variant of the heavy background case,
in which the distribution functions are not isotropic. By taking
$f_{R}=\nu^{c}_{R}$ we can apply the results of any case, depending on whether
$\phi$ is a light or heavy particle in the sense specified above, to the model
in which $\phi$ couples to $\bar{\nu}^{c}_{R}\nu_{L}$ and consider situations
in which a neutrino propagates in a neutrino background. Following the
discussion in Section 2, each $\Sigma^{(x)}$ is of the form

$\Sigma^{(x)}=V^{(x)\mu}\gamma_{\mu}\,,$ (23)

and

$V_{\mu}=V^{(f)}_{\mu}+V^{(\phi)}_{\mu}\,.$ (24)

When the background is isotropic each $V^{(x)\mu}$ can be decomposed as

$V^{(x)}_{\mu}=a^{(x)}k_{\mu}+b^{(x)}u_{\mu}\,.$ (25)

###  3.1 High temperature limit

Here we assume that $T$ and $\kappa$ are sufficiently large such that $f$ and
$\phi$ can be taken to be massless. As we have already stated above, we then
approximate the integrals in Eqs. (20) and (21) by setting $m_{f,\phi}$ and
chemical potentials $\alpha_{f,\phi}$ to zero. In this case

$\displaystyle\Sigma^{(f)}$ $\displaystyle=$
$\displaystyle|\lambda|^{2}\int\frac{d^{4}p}{(2\pi)^{3}}\delta(p^{2})\frac{p\hbox
to0.0pt{\hss/}}{(p+k)^{2}}\frac{1}{e^{|\beta p\cdot u|}+1}\,,$ (26)
$\displaystyle\Sigma^{(\phi)}$ $\displaystyle=$
$\displaystyle|\lambda|^{2}\int\frac{d^{4}p}{(2\pi)^{3}}\delta(p^{2})\frac{(p\hbox
to0.0pt{\hss/}+k\hbox to0.0pt{\hss/})}{(p+k)^{2}}\frac{1}{e^{|\beta p\cdot
u|}-1}\,.$ (27)

We will restrict ourselves to the dominant contribution in powers of $T$. In
this case the term with the factor of $k\hbox to0.0pt{\hss/}$ in the integrand
of Eq. (27) can be discarded and we can write

$\Sigma^{(x)}=|\lambda|^{2}I^{(x)}_{\mu}\gamma^{\mu}\,,$ (28)

where

$I^{(f,\phi)}_{\mu}=\int\frac{d^{4}p}{(2\pi)^{3}}\delta(p^{2})\frac{p_{\mu}}{(p+k)^{2}}\frac{1}{e^{|\beta
p\cdot u|}\pm 1}\,.$ (29)

The integrals in Eq. (29) are the same ones evaluated by Weldon for chiral
fermions in a non-Abelian gauge theory[13] and also used in a previous
discussion of the $\nu\nu\phi$ interaction model in Ref. [14]. Borrowing from
those calculations, the results for $\Sigma^{(f,\phi)}$ can be written in the
form of Eq. (25) with

$\displaystyle a^{(\phi)}=2a^{(f)}$ $\displaystyle=$
$\displaystyle-\frac{|\lambda|^{2}T^{2}}{24\kappa^{2}}\left[1-\frac{\omega}{2\kappa}I\right]$
$\displaystyle b^{(\phi)}=2b^{(f)}$ $\displaystyle=$
$\displaystyle\frac{|\lambda|^{2}T^{2}}{24\kappa}\left[\frac{\omega}{\kappa}-\frac{1}{2}\left(\frac{\omega^{2}}{\kappa^{2}}-1\right)I\right]\,,$
(30)

where

$I\equiv\ln\left|\frac{\omega+\kappa}{\omega-\kappa}\right|\,.$ (31)

The total background-dependent part of the self-energy is then

$\Sigma^{(f\phi)}=a^{(f\phi)}k\hbox to0.0pt{\hss/}+b^{(f\phi)}u\hbox
to0.0pt{\hss/}\,,$ (32)

with

$\displaystyle a^{(f\phi)}$ $\displaystyle=$
$\displaystyle-\frac{M^{2}}{\kappa^{2}}\left[1-\frac{\omega}{2\kappa}I\right]$
$\displaystyle b^{(f\phi)}$ $\displaystyle=$
$\displaystyle\frac{M^{2}}{\kappa}\left[\frac{\omega}{\kappa}-\frac{1}{2}\left(\frac{\omega^{2}}{\kappa^{2}}-1\right)I\right]\,,$
(33)

where

$M^{2}=\frac{|\lambda|^{2}T^{2}}{16}\,.$ (34)

The corresponding dispersion relations in this case are given by the solutions
of Eq. (7), with the coefficients $a,b$ given by Eq. (3.1). They are similar
to the dispersion relations obtained by Weldon[13] for a chiral fermion in a
non-Abelian gauge theory, which in particular have $\omega(\kappa\rightarrow
0)=\pm M$, and other interesting features[15, 16].

###  3.2 Heavy background

In this case we assume that either $f$ of $\phi$ is a heavy particle, in the
sense that its mass is sufficiently larger than $T$, $\omega$ and $\kappa$,
such that the denominators in the integrand in Eqs. (20) and (21) can be
expanded in the form

$\displaystyle\frac{1}{(p+k)^{2}-m^{2}_{f}}$ $\displaystyle=$
$\displaystyle\frac{1}{\Delta}-\frac{2p\cdot k+k^{2}}{\Delta^{2}}\,,$
$\displaystyle\frac{1}{(p-k)^{2}-m^{2}_{\phi}}$ $\displaystyle=$
$\displaystyle-\frac{1}{\Delta}+\frac{2p\cdot k-k^{2}}{\Delta^{2}}\,,$ (35)

where $\Delta$ is defined in Eq. (22). Thus, $\Sigma^{(f,\phi)}$ are given as
in Eq. (23) with

$\displaystyle V^{(f)}_{\mu}(\omega,\vec{\kappa})$ $\displaystyle=$
$\displaystyle|\lambda|^{2}\left(\frac{1}{\Delta}+\frac{k^{2}}{\Delta^{2}}\right)L^{(f)}_{\mu}-\frac{2|\lambda|^{2}}{\Delta^{2}}L^{(f)}_{\mu\nu}k^{\nu}\,,$
$\displaystyle V^{(\phi)}_{\mu}(\omega,\vec{\kappa})$ $\displaystyle=$
$\displaystyle|\lambda|^{2}\left(\frac{1}{\Delta}-\frac{k^{2}}{\Delta^{2}}\right)L^{(\phi)}_{\mu}-\frac{2|\lambda|^{2}}{\Delta^{2}}L^{(\phi)}_{\nu}k^{\nu}k_{\mu}$
(36)
$\displaystyle\mbox{}+|\lambda|^{2}\left(\frac{1}{\Delta}-\frac{k^{2}}{\Delta^{2}}\right)L^{(\phi)}k_{\mu}-\frac{2|\lambda|^{2}}{\Delta^{2}}L^{(\phi)}_{\mu\nu}k^{\nu}\,,$

where

$(L^{(x)},L^{(x)}_{\mu},L^{(x)}_{\mu\nu})=\int\,\frac{d^{4}p}{(2\pi)^{3}}\delta(p^{2}-m^{2}_{x})\eta_{x}(p)\left\\{1,p_{\mu},p_{\mu}p_{\nu}\right\\}\,.$
(37)

In order to obtain the effective potential from Eq. (2.2), we use the fact
that, for $\omega=\kappa$, we have

$k^{\mu}=\kappa n^{\mu}\,,$ (38)

where

$n^{\mu}=(1,\hat{\kappa})\,,\qquad n^{2}=0\,.$ (39)

From Eq. (2.2) the effective potentials are then given by

$V^{(\nu,\bar{\nu})}_{eff}(\vec{\kappa})=\pm\frac{|\lambda|^{2}}{\Delta}\left(L^{(f)}_{1}+L^{(\phi)}_{1}\right)-\frac{2|\lambda|^{2}}{\Delta^{2}}\left(L^{(f)}_{2}+L^{(\phi)}_{2}\right)\,,$
(40)

where the upper(lower) sign corresponds to the neutrino(antineutrino),
respectively. In Eq. (40) the coefficients $L^{(x)}_{n}$ are defined as

$\displaystyle L^{(x)}_{1}$ $\displaystyle\equiv$ $\displaystyle
L^{(x)}_{\mu}n^{\mu}\,,$ $\displaystyle L^{(x)}_{2}$ $\displaystyle\equiv$
$\displaystyle L^{(x)}_{\mu\nu}n^{\mu}n^{\nu}\,,$ (41)

and using the definitions in Eq. (37), the following formula follows

$L^{(x)}_{n}=\frac{1}{2}\int\,\frac{d^{3}p}{(2\pi)^{3}}E^{n-1}_{x}\left(1-\hat{\kappa}\cdot\vec{v}_{\vec{p}}\right)^{n}\left(f_{x}+(-1)^{n}f_{\bar{x}}\right)\,.$
(42)

For isotropic momentum distributions functions we can set $\vec{p}\rightarrow
0$ and $p^{i}p^{j}\rightarrow\frac{1}{3}{\vec{p}}^{\,2}$ in the integrands of
$L^{(x)}_{1}$ and $L^{(x)}_{2}$, respectively, and the formulas reduce in that
case to

$\displaystyle L^{(x)}_{1}$ $\displaystyle=$ $\displaystyle J^{(x)}_{1}\,,$
$\displaystyle L^{(x)}_{2}$ $\displaystyle=$
$\displaystyle\frac{4}{3}J^{(x)}_{2}-\frac{1}{3}m^{2}_{x}J^{(x)}_{0}\,,$ (43)

where

$J^{(x)}_{n}=\frac{1}{2}\int\,\frac{d^{3}p}{(2\pi)^{3}}E^{n-1}_{x}\left(f_{x}+(-1)^{n}f_{\bar{x}}\right)\,.$
(44)

Using Eq. (3.2), the formulas given in Eq. (40) for the neutrino and
antineutrino effective potential then reduce to

$V^{(\nu,\bar{\nu})}_{eff}(\vec{\kappa})=\pm\frac{|\lambda|^{2}}{\Delta}\left(J^{(f)}_{1}+J^{(\phi)}_{1}\right)+\frac{2|\lambda|^{2}\kappa}{3\Delta^{2}}\left(m^{2}_{f}J^{(f)}_{0}+m^{2}_{\phi}J^{(\phi)}_{0}-4J^{(f)}_{2}-4J^{(\phi)}_{2}\right)\,,$
(45)

in the isotropic case. Similarly to Eq. (40), in Eq. (45) the the upper(lower)
sign corresponds to the neutrino(antineutrino), respectively. Introducing the
number densities

$n_{x,\bar{x}}=g_{x}\int\,\frac{d^{3}p}{(2\pi)^{3}}f_{x,\bar{x}}\,,$ (46)

we have

$J^{(x)}_{1}=\frac{1}{2g_{x}}(n_{x}-n_{\bar{x}})\,,$ (47)

where

$g_{f}=2\,,\quad g_{\phi}=1\,.$ (48)

In anticipation to the application of these formulas in the next section we
must remember that for chiral neutrinos

$g_{\nu}=1\,.$ (49)

It is straightforward to obtain explicit formulas for the integrals
$J^{(x)}_{0,2}$ in a number of cases. For example,

$\displaystyle
J^{(x)}_{n}=\frac{1}{2g_{x}}\left(n_{x}+(-1)^{n}n_{\bar{x}}\right)\left\\{\begin{array}[]{cl}m^{n-1}_{x}&(\mbox{NR
limit})\\\\[12.0pt] \frac{1}{2}(n+1)!\beta^{1-n}&\mbox{(Maxwell-Boltzman ER
limit)}\end{array}\right.$ (52)

in the non-relativistic (NR) or the Maxwell-Boltzman extremely-relativistic
(ER) limits.

##  4 $\nu\phi$ background

In this section we use the results obtained in Section 3 to discuss the
effects of the interaction given in Eq. (1) on a neutrino propagating in a
neutrino-$\phi$ background. It is useful to remember that in the relevant
diagram for this case, as shown in Fig. 2, the internal fermion that
corresponds to the $f_{R}$ fermion line in Fig. 1 is the antineutrino
$\nu^{c}_{R}$. As a result of this, the formulas obtained in Section 3 can be
adapted to this case provided we identify $f,\bar{f}\rightarrow\bar{\nu},\nu$
in the labels of the various physical quantities that refer to the background
particles, such as the chemical potentials, particle number densities. In
addition, we have to remember that for chiral neutrinos we must use
$g_{\nu}=1$ in the formula for $J^{(\bar{\nu})}_{1}$ given in Eq. (47), as
stated in Eq. (49).

Figure 2: Neutrino self-energy diagram in a background of neutrinos and
scalars $\phi$.

###  4.1 High temperature limit

Denoting the background-dependent part of the self-energy in this case by
$\Sigma^{(\bar{\nu}\phi)}$ and writing

$\Sigma^{(\bar{\nu}\phi)}=a^{(\bar{\nu}\phi)}k\hbox
to0.0pt{\hss/}+b^{(\bar{\nu}\phi)}u\hbox to0.0pt{\hss/}\,,$ (53)

the coefficients are given by the same expressions of Eq. (3.1), with

$M^{2}\equiv\frac{1}{16}|\lambda^{(\nu\nu\phi)}|^{2}T^{2}\,.$ (54)

These are the results used also in a previous discussion of the $\nu\nu\phi$
interaction model in Ref. [14]. As already mentioned in Section 3.1, the
dispersion relation for the neutrino in this case is similar to the dispersion
relation obtained by Weldon[13] for a chiral fermion in a non-Abelian gauge
theory. However, this case is not the one that is relevant to the situations
where $m_{\phi}$, although assumed to be below the electroweak breaking scale,
is assumed to be $\sim GeV$, such as those considered in Ref. [1]. Thus we
turn the attention to the case in which $\phi$ is sufficiently heavy such that
its background density is negligible.

###  4.2 Heavy $\phi$ limit

For illustrative purposes we consider in some detail the isotropic case. From
Eq. (2.1), neglecting the $\phi$ background terms and putting $m_{\nu}=0$, the
$\nu\phi$ background contribution to the effective potentials is then

$\left(V^{(\nu,\bar{\nu})}_{eff}(\vec{\kappa})\right)_{\nu\nu\phi}=\pm\frac{|\lambda^{(\nu\nu\phi)}|^{2}}{m^{2}_{\phi}}J^{(\bar{\nu})}_{1}-\frac{8|\lambda^{(\nu\nu\phi)}|^{2}\kappa}{3m^{4}_{\phi}}J^{(\bar{\nu})}_{2}\,.$
(55)

The full background-dependent effective potential is then

$V^{(\nu,\bar{\nu})}_{eff}(\vec{\kappa})=\left(V^{(\nu,\bar{\nu})}_{eff}(\vec{\kappa})\right)_{\nu\nu\phi}+\left(V^{(\nu,\bar{\nu})}_{eff}(\vec{\kappa})\right)_{Z}\,,$
(56)

where $\left(V^{(\nu,\bar{\nu})}_{eff}\right)_{Z}$ is the _standard_ neutrino-
background contribution from the $Z$ exchange and tadpole diagrams

$\left(V^{(\nu,\bar{\nu})}_{eff}(\vec{\kappa})\right)_{Z}=\pm
4\sqrt{2}G_{F}J^{(\nu)}_{1}\,.$ (57)

Eq. (57) follows, from example, from the formulas obtained in Ref. [17]. As
shown there, the contribution of those two diagrams to the neutrino self-
energy can be written in the form of Eqs. (23) and (25), with

$\displaystyle\left(b^{(Z)}\right)_{tadpole}\ $ $\displaystyle=$
$\displaystyle 4\sqrt{2}G_{F}\sum_{f}X_{\nu}J^{(f)}_{1}\,,$
$\displaystyle\left(b^{(Z)}\right)_{Z-exchange}$ $\displaystyle=$
$\displaystyle 2\sqrt{2}G_{F}J^{(\nu)}_{1}\,,$ (58)

where the sum runs over all the fermion species in the background and $X_{f}$
is the (vector) neutral-current coupling of the fermion. In Eq. (57) we are
including only the neutrino background contribution, and using
$X_{\nu}=\frac{1}{2}$. Remembering Eqs. (47) and (49),

$J^{(\bar{\nu})}_{1}=-J^{(\nu)}_{1}=\frac{1}{2}(n_{\nu}-n_{\bar{\nu}})\,,$
(59)

we then have

$V^{(\nu,\bar{\nu})}_{eff}(\vec{\kappa})=\pm\left(2\sqrt{2}G_{F}-\frac{|\lambda^{(\nu\nu\phi)}|^{2}}{2m^{2}_{\phi}}\right)(n_{\nu}-n_{\bar{\nu}})-\frac{8|\lambda^{(\nu\nu\phi)}|^{2}\kappa}{3m^{4}_{\phi}}J^{(\bar{\nu})}_{2}\,.$
(60)

###  4.3 Discussion

In a core collapse supernova, neutrinos are trapped in the neutrino sphere and
slowly diffuse out. These neutrinos have self-interactions that lead what is
known as collective neutrino oscillations[8]. These phenomena can also occur
in the hot plasma of the Early Universe[9] before the neutrinos decouple. In
the presence of the type of neutrino-scalar interaction we have considered,
the collective oscillations could be modified.

A striking and surprising result is that the standard contribution
$\left(V^{(\nu,\bar{\nu})}_{eff}\right)_{Z}$ on one hand, and
$\left(V^{(\nu,\bar{\nu})}_{eff}\right)_{\nu\nu\phi}$ on the other, have
opposite sign. Namely, they are proportional to
$J^{(\nu)}_{1}=\frac{1}{2}(n_{\nu}-n_{\bar{\nu}})$ and
$J^{(\bar{\nu})}_{1}=\frac{1}{2}(n_{\bar{\nu}}-n_{\nu})$, respectively. Thus,
they tend to cancel. A limit can be set by requiring that the contribution of
the $\nu\nu\phi$ interaction to the neutrino self-energy does not cancel the
standard contribution in an appreciable way. In symbols we can express this as

$\left(V^{(\nu)}_{eff}\right)_{\nu\nu\phi}<\left(V^{(\nu)}_{eff}\right)_{Z}\,,$
(61)

which requires

$m_{\phi}>\frac{\lambda^{(\nu\nu\phi)}}{g}m_{Z}\,.$ (62)

Therefore, if $\lambda^{(\nu\nu\phi)}\sim O(1)$, this would imply $m_{\phi}$
must be larger than $\sim 3m_{Z}$. In order to have $m_{\phi}\sim O(GeV)$, a
small coupling $\lambda^{(\nu\nu\phi)}\sim 10^{-2}-10^{-3}$ would be required.

Another possibility is that the first term in Eq. (60), proportional to the
neutrino-antineutrino asymmetry is small, either because the factor in
parenthesis and/or because the asymmetry is small. In such a case, the
$O(1/m^{4}_{\phi})$ term in effective potential

$V^{(\nu,\bar{\nu})}_{eff}(\vec{\kappa})=-\frac{8|\lambda^{(\nu\nu\phi)}|^{2}\kappa}{3m^{4}_{\phi}}J^{(\bar{\nu})}_{2}\,.$
(63)

would give the dominant contribution. For example, taking the Maxwell-Boltzman
limit of the neutrino background from Eq. (52), this would give

$V^{(\nu,\bar{\nu})}_{eff}(\vec{\kappa})=-\frac{2|\lambda^{(\nu\nu\phi)}|^{2}\kappa}{m^{4}_{\phi}}\frac{(n_{\nu}+n_{\bar{\nu}})}{\beta}\,,$
(64)

which is the same for the neutrino and antineutrino, and independent of the
neutrino-antineutrino asymmetry in the background.

Although we have focused above on the case of isotropic thermal distributions
of the neutrino background, similar considerations apply to the case of
anisotropic distributions as well.

##  5 Conclusions

The existence of scalars that interact with neutrinos via couplings of the
form $\lambda^{(\nu\nu\phi)}\bar{\nu}^{c}_{R}\nu_{L}\phi$ would produce
additional contributions to the neutrino effective potential beyond the
standard ones when the neutrino propagates in a neutrino background. This can
occur in the environment of a supernova, and in the Early Universe hot plasma
before neutrino decoupling.

Motivated by studying the possible effects of such interactions in those
contexts, we have considered a simple model in which neutrino interacts with a
scalar and a fermion, and calculated the effective potential for a neutrino
propagating in a background of those particles. The results are useful in the
context of Dark Matter-neutrino interaction models in which the scalar and/or
fermion constitute the dark-matter, and are also applicable to the situations
mentioned in which the fermion background is a neutrino background. We
obtained the expressions for the neutrino effective potential that can be
applied to different situations and background conditions.

As a specific application we considered the case of the neutrino background,
with the scalar particle being sufficiently heavy so that their background
density is negligible. A noteworthy result in this case is that the effective
potential experienced by the propagating neutrino in this background has the
opposite sign to the effective potential due to the standard contribution from
the Z-boson. By assuming that the standard model contribution dominates over
the $\nu\nu\phi$ contribution, for example in the neutrino sphere of a core
collapse supernova, we have obtained limit on the parameter
$\lambda^{(\nu\nu\phi)}/m_{\phi}$.

Although for definiteness we have restricted ourselves to consider the
diagonal scalar-neutrino coupling, the results already reveal some potentially
important features, for example the sign difference mentioned above, that can
serve as a guide for considering more realistic or complicated models
involving, for example, more than one scalar particle or off-diagonal neutrino
couplings.

## References

  * [1] Jeffrey M. Berryman, André de Gouvea, Kevin J. Kelly and Yue Zhang, _Lepton-Number-Charged Scalars and Neutrino Beamstrahlung_ , arxiv:1802.00009 (jan 2018)
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# Sparse sensor reconstruction of vortex-impinged airfoil wake with machine
learning

Yonghong Zhong[1,∗], Kai Fukami[1], Byungjin An[2], Kunihiko Taira[1]
1\. Department of Mechanical and Aerospace Engineering,
University of California, Los Angeles, CA 90095, USA
2\. Fundamental Technologies, R&D Department, Ebara Corporation, Tokyo
144-8510, Japan
Corresponding author<EMAIL_ADDRESS>
###### Abstract

Reconstruction of unsteady vortical flow fields from limited sensor
measurements is challenging. We develop machine learning methods to
reconstruct flow features from sparse sensor measurements during transient
vortex-airfoil wake interaction using only a limited amount of training data.
The present machine learning models accurately reconstruct the aerodynamic
force coefficients, pressure distributions over airfoil surface, and two-
dimensional vorticity field for a variety of untrained cases. Multi-layer
perceptron is used for estimating aerodynamic forces and pressure profiles
over the surface, establishing a nonlinear model between the pressure sensor
measurements and the output variables. A combination of multi-layer perceptron
with convolutional neural network is utilized to reconstruct the vortical
wake. Furthermore, the use of transfer learning and long short-term memory
algorithm combined in the training models greatly improves the reconstruction
of transient wakes by embedding the dynamics. The present machine-learning
methods are able to estimate the transient flow features while exhibiting
robustness against noisy sensor measurements. Finally, appropriate sensor
locations over different time periods are assessed for accurately estimating
the wakes. The present study offers insights into the dynamics of vortex-
airfoil interaction and the development of data-driven flow estimation.

_K_ eywords Vortex-airfoil interaction, Machine learning, Flow reconstruction

##  1 Introduction

Vortex-airfoil interaction is ubiquitous around fluid-based systems, including
aircraft [1, 2, 3, 4], wind turbines [5], and pumps [6, 7]. Such interactions
can cause unsteady loading, fatigue, and structural damage to these systems.
For analyzing vortex-airfoil interactions, it is useful to assess the state of
the flow from sparse measurements for understanding the governing dynamics
[8], prediction of flow disturbance [9], and performing the wake flow control
[10]. However, it is challenging to identify vortical structures during the
vortex-airfoil interactions from sparse measurements due to its strong
nonlinear dynamics and the high-degree of freedom required to describe the
vortical flows.

A number of studies have examined sparse state estimation for aerodynamics. In
particular, linear techniques have been studied over the last several decades.
For instance, gappy proper orthogonal decomposition [11] has been considered
to obtain dominant flow features from spatially incomplete and sparse data
sets [12]. Focusing on the characterization of flows and boundary layers near
body surface, the applications of four-dimensional variational method [13],
linear stochastic estimation [14], and Kalman filters [15] have also been
explored. However, these techniques are constrained by their linear
formulations, which poses challenges when the applications involve strongly
nonlinear dynamics.

To overcome such limitations, nonlinear machine learning approaches have been
considered as a promising approach in analyzing fluid flows from sparse
information. Nonlinear machine learning techniques have shown to be useful in
estimating and modeling high-dimensional flow [16]. For example, Pawar et al.
[17, 18] applied a physics-guided machine-learning framework to estimate the
lift coefficient of a variety of airfoils. Hui et al. [19] utilized a signed
distance function-assisted convolutional neural network (CNN) to predict the
pressure distribution over an airfoil surface. For flow field reconstructions,
Erichson et al. [20] proposed a shallow decoder based on multi-layer
perceptron (MLP) for a circular cylinder wake, the sea surface temperature,
and forced isotropic turbulence. Fukami et al. [21] proposed a CNN-based
method to reconstruct the global turbulent flow field from sparse sensors that
can be in motion or change in numbers. In addition to the aforementioned
efforts, there are various machine-learning-based flow reconstruction
techniques based on super-resolution analysis [22, 23, 24].

However, there are issues with utilizing nonlinear machine learning techniques
for estimating unsteady fluid flows from limited sensor measurements. The most
outstanding issue is the computational costs for using machine learning models
are expensive. For neural network-based models with low-dimensional inputs to
high-dimensional outputs, an enormous number of interior parameters (weights)
are required. To determine the internal parameters, generally, thousands of
flow (or sensor) snapshots are required, which causes a large computational
burden in terms of both training costs and data storage. In our case, if a
variety of unsteady flow fields is needed to be accurately reconstructed,
storage and computing costs can rise significantly if the problem is
approached naively. From this aspect, it is crucial to develop a method that
can qualitatively reconstruct a flow field with a small amount of training
data and a reduced number of tuning parameters. In addition, generalizable
models promote a reduction in cost. Most machine learning models can only be
used for specific flow fields, for example, a single model trained with a
laminar flow may not be applicable to use to reconstruct turbulent flow
fields. In fact, the data used for testing needs to be similar to the training
data to achieve accurate results. If we need to consider different flows over
a vast parameter space, it is almost impossible to perform experiments or
simulations for each and every case. In this regard, the diversity of the
training data needs to be considered so that a single model can effectively
predict unsteady flow fields over a large range of parameters.

In this study, we aim to develop machine learning methods that reconstruct
dominant wake features from limited sensor measurements and a small set of
training data sampled over a vast parameter space. Because the disturbance
vortex can be of any size, strength, or position from the airfoil, a very
large parameter space is needed to be explored to capture the complex vortex-
impinged airfoil wake dynamics. In this case, the amount of data can be
tremendously large. Instead of naively training machine learning models with
all parameter combinations, we develop models that are trained with a few
cases in the parameter space and use the models to estimate unseen cases. For
the machine learning methods, we choose a multi-layer perceptron (MLP) to
model the nonlinear relationship between the low-dimensional sensors inputs
and the outputs, including the lift coefficient, drag coefficient, and surface
pressure coefficient. Moreover, combining the convolutional neural networks
and MLP allows the reconstruction of the vorticity field over time with modest
computational costs. The transfer learning and long-short term memory further
help in incorporating the dynamics of the transient flow, which reduces the
required training data and improves the flow estimation. The current model is
robust for a variety of wake scenarios separate from the training data. We
also assess the influence of sensor numbers and placement on flow estimation.

The present paper is organized as follows. The problem setup and data
compilation are discussed in section 2. Flow physics of vortex-airfoil wake
interactions are presented in section 3. Machine learning techniques utilized
in this study are introduced in section 4. Results and discussion of machine
learning-based flow reconstruction are presented in section 5. Concluding
remarks are provided in section 6.

##  2 Data compilation

Figure 1: a) The size and position of the vortical disturbance, and 8 uniform
sensors are distributed on the airfoil surface; b) The velocity profile of the
disturbance vortex.  Figure 2: Randomly distributed 50 training cases (blue)
and 10 test cases (red). Example test cases shown: a)
$(u_{\theta\rm{max}}/u_{\infty},R/c,y_{0}/c)=(0.14,0.15,0.15)$, b)
$(u_{\theta\rm{max}}/u_{\infty},R/c,y_{0}/c)=(0.78,0.99,0.18)$, c)
$(u_{\theta\rm{max}}/u_{\infty},R/c,y_{0}/c)=(-0.80,0.61,0.08)$, d)
$(u_{\theta\rm{max}}/u_{\infty},R/c,y_{0}/c)=(0.35,0.95,-0.15)$.

The present objective is to develop a robust machine-learning model for highly
disturbed flows around an airfoil from sparse pressure sensors and limited
training data. Here, we consider transient flow over a NACA 0012 airfoil at an
angle of attack of $\alpha=12^{\circ}$ experiencing various types of vortical
disturbances at a chord-based Reynolds number $Re\equiv
u_{\infty}c/{\nu}_{\infty}=400$ and a Mach number $M_{\infty}\equiv
u_{\infty}/a_{\infty}=0.1$. Here, $u_{\infty}$ is the free-stream velocity,
$c$ is the chord length, ${\nu}_{\infty}$ is the kinematic viscosity, and
$a_{\infty}$ is the freestream sonic speed. The simulated flows have been
verified and validated with previous studies [25, 26, 27].

The compressible flow solver CharLES [28] is used to simulate the transient
flows over the airfoil. For the present vortex-airfoil interaction problem, a
single vortical disturbance is initially introduced upstream of the airfoil.
This disturbance vortex is given as a compressible Taylor vortex [29],
described by

$u_{\theta}=u_{\theta\rm{max}}{\dfrac{r}{R}}{\rm
exp}\left[\dfrac{1}{2}\left({1-\dfrac{r^{2}}{R^{2}}}\right)\right],$ (1)

where $R$ is the radius, and $u_{\theta\rm{max}}$ is the maximum rotational
velocity of the vortex, as shown in figure 1. The vortex is initially
introduced at $(x_{0},y_{0})$ with $x_{0}=-2c$.

The present vortex-airfoil interaction problem exhibits a variety of flow
patterns, as shown in figure 2. A strong disturbance vortex produces strong
unsteadiness in the flow field, and the larger the vortex is, the larger the
region it influences. Apart from the radius and the strength, a vortex can
either hit the airfoil at the leading edge and thus incite large fluctuations
or pass through the airfoil without causing dramatic changes to the flow or
aerodynamic characteristics. Detailed discussion on the flow is offered in
section 3. The present study examines whether the flow field generated over
the wide parameter space can be recovered with the machine-learning model
trained with only a very few cases.

In the present study, we choose eight sensors distributed on both sides of the
airfoil surface to capture the vortex passing around an airfoil, as shown in
figure 1. These sensors are labeled $1$ to $8$, with the respective
$x$-locations of the sensors being
$(0.00,0.26,0.48,0.72,0.99,0.23,0.46,0.71)c$. Three parameters that describe
the disturbance vortex are maximum rotational velocity ($u_{\theta\rm{max}}$),
the radius ($R$), and the initial vertical location ($y_{0}$). The training
data sets are comprised of
$u_{\theta\rm{max}}/u_{\infty}\in[-0.9,-0.7,-0.5,-0.3,-0.1,0.1,0.3,0.5,0.7,\\\
0.9]$, $R/c\in[0.125,0.25,0.5,0.75,1]$, and $y_{0}/c\in[-0.3,-0.1,0,0.1,0.3]$,
respectively. Here, the positive value of $u_{\theta\rm{max}}/u_{\infty}$
indicates a counterclockwise rotation. The maximum rotational velocity of the
vortex $u_{\theta\rm{max}}$ covers a range from $0.1u_{\infty}$ to
$0.9u_{\infty}$. The choices for the vortex radius $R$ and $y_{0}$ are
carefully determined so that vortices can pass over or below the airfoil while
significantly influencing the airfoil wake. In section 5, we consider 25, 50,
and 100 training cases out of the vast combinations of parameters, then test
the models with untrained cases. Parameter combinations of test cases are
randomly chosen over the aforementioned ranges. Note that the training data is
a small proportion compared to the whole combinations of parameters. There are
no test cases overlapping with the training cases.

For each case, we collect 500 snapshots of the flow field for
$u_{\infty}t/c\in[0.85,5.1]$, which reflects the process from the vortex
approaching the airfoil to moving away from the tailing edge. Here,
$u_{\infty}t/c=0$ refers to the initial time at which the vortex is at
$x_{0}/c=-2$. The snapshots at $u_{\infty}t/c=[0,0.85]$ are not used in the
present analysis to remove the start-up period of the simulation. For a single
parameter set $(u_{\theta\rm{max}}/u_{\infty},R/c,y_{0}/c)$, the data sizes of
aerodynamic force coefficients, pressure over surface, and two-dimensional
vorticity field data amount to approximately 1MB, 15MB, and 500MB,
respectively. If we use 100 training cases with all 500 snapshots of two-
dimensional wake data, the training data size becomes approximately 50GB for a
single machine learning model, which is quite large with respect to storage
and computation.

##  3 Flow physics

The present vortex-impinged airfoil wake exhibits rich dynamics influenced by
the vortex velocity, size, and position. In this section, we present the flow
physics induced by a variety of vortex disturbances.

Figure 3: Effect of the largest rotational velocity of vortical disturbance.
a) lift coefficients, b) drag coefficients, and c) vorticity fields for
vortical disturbances of $(R/c,y_{0}/c)=(0.5,0.1)$ and
$u_{\theta\rm{max}}/u_{\infty}=-0.7,-0.3,0.3$, and $0.7$.

The maximum rotational velocity of the vortex disturbance is one of the most
important characteristics affecting the vortex-airfoil interaction. Here, we
investigate the influence of vortex largest velocity on $C_{L}$, $C_{D}$, and
vorticity fields when $(R/c,y_{0}/c)=(0.5,0.1)$. As depicted in figure 3a) and
b), a positive (counterclockwise) vortex generally induces a transient
increase in $C_{L}$ and $C_{D}$ when it impinges on the leading edge of the
airfoil. A secondary negative peak is then introduced when the center of the
vortical disturbance passes the center of the airfoil. A similar but reversed
trend is observed for a negative (clockwise) vortex. The initial decrease in
lift is followed by the vortex tail-induced lift increase.

For a positive vortex with two different magnitudes of the vortex rotational
velocity, the first peaks of $C_{L}$ are reached at nearly the same time, as
presented in figure 3a). However, the magnitude difference causes the temporal
shift for the secondary peak — the peak with $u_{\theta_{\rm
max}}/u_{\infty}=0.7$ is reached at $u_{\infty}t/c\approx 2.6$ while that with
$u_{\theta_{\rm max}}/u_{\infty}=0.3$ is achieved at $u_{\infty}t/c\approx
3.0$. This is because a stronger positive vortex produces a stronger
interaction with the pre-existing negative vorticity on the suction side of
the airfoil, forming a large negative vortex that detaches from the airfoil
afterward. Similar to the positive disturbance cases, larger fluctuation
induced by a stronger negative vortex gives rise to an earlier secondary peak.
For $C_{D}$, we observe a similar trend of the time history to the $C_{L}$ for
the positive disturbance, while the magnitudes of variation are much smaller
than $C_{L}$.

Figure 4: Effect of vortex size. a) lift coefficients, b) drag coefficient,
and c) vorticity fields for vortical disturbances of
$(u_{\theta\rm{max}}/u_{\infty},y_{0}/c)=(0.3,0.1)$ and $R/c=0.125,0.25,0.5$,
and $0.75$.

The dependence of the flow field response on the vortex size is also examined,
as shown in figure 4. We choose the same vortex strength and vertical position
as $(u_{\theta\rm{max}}/u_{\infty},y_{0}/c)=(0.3,0.1)$ for comparison. The
$C_{L}$ and $C_{D}$ histories experience the same trend of first increasing
and then decreasing among different vortex sizes. By increasing the vortex
size, the first peaks of $C_{L}$ and $C_{D}$ appear earlier because a vortex
with a larger radius encounters the airfoil earlier.

The changes in the vorticity fields caused by the different sizes of vortices
are also presented in figure 4c). When a small-size vortical disturbance
($R/c=0.125$) impinges on the airfoil, the whole vortex passes over the
suction side of the airfoil and induces mild fluctuation in the flow field. As
the size of the vortex becomes larger, the vortex splits into two structures
which advects over the suction side and the pressure side. The positive
vorticity around the trailing edge is rolled up and interacts with the wakes,
thus affecting the evolution of the wake region.

Figure 5: Effect of vortex position. a) lift coefficients, b) drag
coefficients, and c) vorticity fields for vortical disturbances of
$(u_{\theta\rm{max}}/u_{\infty},R/c)=(-0.5,0.5)$ and $y_{0}/c=0.3,0$, and
$-0.3$.

In addition to the largest velocity and the size of the vortical disturbance,
the transient dynamics are also strongly influenced by whether the disturbance
vortex passes above or below the airfoil. Here, let we investigate three
vertical positions of $y_{0}/c={-0.3,0,0.3}$ with a negative vortical
disturbance $(u_{\theta\rm{max}}/u_{\infty},R/c)=(-0.5,0.5)$, as shown in
figure 5. For $y_{0}/c=0.3$, the disturbance passes over the airfoil, where a
large portion of negative disturbance passes through the suction side of the
airfoil, introducing a large jump in $C_{D}$ as the first peak. For
$y_{0}/c=0$, the negative vortical disturbance is split into two parts as it
passes around the airfoil. At $u_{\infty}t/c=2.55$, the large positive
vorticity attached on the pressure side of the airfoil produces the second
peak in $C_{L}$. For the case of $y_{0}/c=-0.3$ where the disturbance passes
below the airfoil, the variation is mostly dominated by the interaction along
the pressure side of the airfoil, and the drop and the increment of $C_{L}$
and $C_{D}$ occur at the same time.

##  4 Methods

We develop machine-learning models to estimate aerodynamic characteristics
that cover a variety of force and wake dynamics from sparse sensors.
Constructing a robust model suitable for the vast parameter space in figure 2
is challenging. To estimate different types of nonlinear wake responses from
limited training data, we consider several strategies with regard to machine-
learning model design and training methods for reproducing the transient
dynamics. For all machine-learning models used in the current study, three-
fold cross-validations are performed, ensuring the convergence of the
estimations in terms of data distribution.

Figure 6:  Overview of the present estimation problems. The inputs are
pressure sensor measurements on the airfoil surface, outputs are a) $C_{D}$ or
$C_{L}$, b) $C_{P}$, and c) vorticity field. $C_{L}$, $C_{D}$, and $C_{P}$ are
estimated using separate multi-layer perceptron models, vorticity field is
estimated using the combination of multi-layer perception and convolutional
neural network.

An overview of the present machine-learning-based estimation approaches is
shown in figures 6. The input is the sensor measurements
${\bm{s}}^{n\rm{\Delta}t}$ spanning over $n\rm{\Delta}t$. We first consider a
multi-layer perceptron (MLP) to build the relationship between the sensor
measurements and the aerodynamic force coefficients over time. Since the
degrees of freedom of the input and output are ${\cal O}(1)$, we can easily
employ a fully-connected neural network to construct such a relationship.
Similarly, we also use a multi-layer perceptron (MLP) to estimate the pressure
distribution over the airfoil surface. However, MLP can be challenging to use
for problems with high degrees of freedom due to its fully-connected structure
[21, 30]. To access the two-dimensional vorticity flow field (the degree of
freedom $\approx{\cal O}(10^{3}-10^{4})$), a model which can effectively
extract spatial information with a manageable computational cost is required.
To address this point, we incorporate a two-dimensional convolutional neural
network (CNN) to provide qualitative estimations while maintaining a low
computational cost. When the MLP is coupled with the CNN, the machine-learning
model can reconstruct the flow field from a limited number of sensor
measurements. Moreover, due to the transient nature of the current vortex-
airfoil interaction problem, accounting for the dynamics into model
construction aids in accurate estimation. For this reason, the long short-term
memory (LSTM) algorithm [31] assisted with transfer learning serves as an
effective method to estimate the flow fields from time traces. Hence, we embed
LSTM into the aforementioned MLP and MLP-CNN models. In what follows, we
introduce the algorithms of these machine learning methods.

Figure 7:  a) A minimum unit of perceptron. b) Two-dimensional convolutional
operation.

###  4.1 Multi-layer perceptron

In the present study, the input sensor measurements are first fed into a
multi-layer perceptron (MLP) [32]. For the estimation of aerodynamic forces
(section 5.1) and pressure distribution over the airfoil surface (section
5.2), the MLP ${\cal M}$ is used as a function approximator between the input
sensor measurements ${\bm{s}}$ and the output variables ${\bm{q}}$ such that
${\bm{q}}\approx{\cal M}({\bm{s}})$. For the estimation of the two-dimensional
vorticity field $\omega$ (section 5.3), MLP plays a role of a nonlinear
function mapping the low-dimensional sensor information
${\bm{s}}\in\mathbb{R}^{n_{\bm{s}}}$ to the high-dimensional variable in the
model. In addition, we incorporate LSTM [31] into the machine-learning models
to capitalize on the dynamical information of sensors.

In MLP, the input at layer $(l-1)$ is multiplied by weights $\bm{W}$, then
linearly combined, and passed through a nonlinear activation function
$\varphi$ as an output to the next layer $(l)$,

${q}^{(l)}_{i}=\varphi(\sum_{j}{W}_{ij}^{(l)}{q}^{(l-1)}_{j}+b_{i}^{(l)}),$
(2)

where $b$ is a bias added at each layer as illustrated in figure 7a). We
utilize the ReLU function [33] for $\varphi$, which is known to be effective
for addressing the vanishing gradient problems in deep neural networks. For
determining the weights $W$, the Adam algorithm [34] is utilized. In the
present model training, early stopping [35] with $20$ training epochs is also
applied to avoid overfitting the machine-learning model.

###  4.2 Convolutional neural network

Since the full flow field estimation requires a large number of spatial grid
points (high spatial degrees of freedom), the computational burden is
substantial for the direct application of MLP to the full flow field
reconstruction [36, 37]. To address this issue, we combine MLP and a two-
dimensional convolutional neural network (CNN) [38]. The CNN enables
regression while greatly reducing computational costs through filter sharing.
The two-dimensional convolutional operation is illustrated in figure 7b),
whose internal procedure is expressed as

$q^{(l)}_{ijg}=\varphi\left(\sum_{l=1}^{F}\sum_{p=0}^{H-1}\sum_{q=0}^{H-1}h^{(l)}_{pqlg}q^{(l-1)}_{i+p-C,j+q-C,l}+b_{g}^{(l)}\right),$
(3)

where $C=\lfloor H/2\rfloor$, $H$ is the width and height of the filter, $F$
is the number of input channels, $g$ is the number of output channels, $b$ is
the bias, and $\varphi$ is the activation function. The input sensor
measurements ${\bm{s}}\in\mathbb{R}^{n_{\bm{s}}}$ are transformed to a high-
dimensional representation $\hat{\bm{q}}\in\mathbb{R}^{n_{\hat{\bm{q}}}}$
through the MLP for the wake estimation. This representation
$\hat{\bm{q}}\in\mathbb{R}^{n_{\hat{\bm{q}}}}$ is then reshaped into a two-
dimensional matrix form $\hat{\bm{q}}\in\mathbb{R}^{n_{\hat{x}}\times
n_{\hat{y}}}$ so that the data can be managed with a two-dimensional CNN, as
illustrated in figure 6a). Through the CNN process in equation 3 and
upsampling operation, the present model extracts the relationship between the
input sensors and the vorticity field ${\omega}\in\mathbb{R}^{n_{x}\times
n_{y}}$. As with the MLP training, we apply the ReLU function [33] as the
nonlinear activation function, the Adam algorithm [34] for updating filters,
and early stopping [35] to prevent overfitting.

###  4.3 Long short-term memory-assisted transfer learning

To improve the present estimation, we also utilize the long short-term memory
(LSTM) algorithm [31]. LSTM is one of the recurrent neural network methods,
which is suitable for predicting temporal behaviors from time-series data.
Since LSTM can hold the time-series data as memory inside the function
referred to as cell, the implementation of LSTM can greatly help with the
present problem that is dependent on past flow states due to its transient
nature.

An LSTM layer is constructed by four functions; a cell $C$, an input gate $d$,
an output gate $o$, and a forget gate $g$. These functions play important
roles in deciding how past information is incorporated to predict the output
variables. The input gate $d$ determines how much of the current information
from the input of cell $e_{t}$ is used for prediction,

$\displaystyle d_{t}$
$\displaystyle=\sigma(W_{d}\cdot[\tilde{q}_{t-1},e_{t}]+\beta_{d}),$ (4)

where $q$ is the output of cell, $W$ and $\beta$ represent the weights and the
bias, respectively, for each gate denoted by its subscript; the subscripts $t$
and $t-1$ represent the time indices, and $\sigma$ is the sigmoid function.
Here, the concatenation of two inputs in a model is denoted as $[{m},{n}]$. In
parallel, the LSTM also considers how much of the past information is kept
from the cell state at the previous cell state $C_{t-1}$ using the forget gate
$g$,

$\displaystyle g_{t}$
$\displaystyle=\sigma(W_{g}\cdot[\tilde{q}_{t-1},e_{t}]+\beta_{g}).$ (5)

With the temporal cell state at the current time step,

$\displaystyle\widetilde{C}_{t}$
$\displaystyle=\tanh({W_{c}\cdot[\tilde{q}_{t-1},e_{t}]+\beta_{c}}),$ (6)

and the previous cell state $C_{t-1}$, the current cell state $C_{t}$ is
determined by balancing the input gate $d$ and the forget gate $g$,

$\displaystyle C_{t}$ $\displaystyle=g_{t}C_{t-1}+d_{t}\widetilde{C}_{t}.$ (7)

Note that the sigmoid functions used for the input and the output gates play
important roles in avoiding gradient vanishing problems. At the output of the
LSTM layer, the amount of information at the cell state $C_{t}$ being
leveraged for short-term prediction (i.e., the output at the next step
${\tilde{q}}_{t}$) is assessed using the output gate $o$ with

$\displaystyle o_{t}$
$\displaystyle=\sigma(W_{o}\cdot[\tilde{q}_{t-1},e_{t}]+\beta_{o}),$ (8)
$\displaystyle{\tilde{q}}_{t}$ $\displaystyle=o_{t}\tanh(\it{C_{t}}).$ (9)

With this formulation, the LSTM is able to predict the variable at the next
step ${\tilde{q}}_{t}$ while considering the long-term memory influence with
the concept of cell state $C$.

Figure 8: Long short-term memory-assisted transfer learning.

Here, we combine the high-dimensional representation of the input measurements
obtained through the MLP $\tilde{\bm{q}}^{n\Delta t}$ with two previous time
sequences extracted by LSTMs $\\{\tilde{\bm{q}}^{(n-1)\Delta
t},\tilde{\bm{q}}^{(n-2)\Delta t}\\}$ such that
${\tilde{\bm{q}}=[\tilde{\bm{q}}^{n\Delta t}+\tilde{\bm{q}}^{(n-1)\Delta
t}+\tilde{\bm{q}}^{(n-2)\Delta t}}]$, as illustrated in figure 8. This
combined vector $\tilde{\bm{q}}$ with three time steps is then provided to the
MLP layer of the force estimation and the $C_{p}$ estimation, or the two-
dimensional CNN layer of the vorticity reconstruction task.

Moreover, we utilize the concept of transfer learning for the LSTM-assisted
network. Transfer learning can facilitate the training process by setting
appropriate initial weights [39]. The present strategy of the LSTM-assisted
transfer learning is graphically summarized in figure 8. In the present study,
the weights of pre-trained MLP ${\bm{w}}_{\cal M}$ are adopted as initial
weights of the second model ${\cal F}_{2}$ which has two sensor input gates
${\bm{s}}^{(n-1)\Delta t}$ and ${\bm{s}}^{n\Delta t}$. The high-dimensional
feature of input sensor measurements $\hat{\bm{q}}$ from the MLP part of the
model is merged with that from LSTM. Once the training for the second model
${\cal F}_{2}$ is completed, the optimized weights of the second model
${\bm{w}}_{{\cal F}_{2}}$ are repeatedly transferred to the third model ${\cal
F}_{3}$ which considers sensor measurements at three different time steps
${\bm{s}}^{(n-2)\Delta t}$, ${\bm{s}}^{(n-1)\Delta t}$, and ${\bm{s}}^{n\Delta
t}$. The weight optimizations through these operations are mathematically
expressed as

$\displaystyle{\bm{w}}_{{\cal F}_{1}}$ $\displaystyle={\rm
argmin}_{{\bm{w}}_{{\cal F}_{1}}}||{\bm{q}}-{{\cal F}_{1}}({\bm{s}}^{n\Delta
t};{\bm{w}})||_{2},$ (10) $\displaystyle{\bm{w}}_{{\cal F}_{2}}$
$\displaystyle={\rm argmin}_{{\bm{w}}_{{\cal F}_{2}}}||{\bm{q}}-{\cal
F}_{2}([{\bm{s}}^{n\Delta t},{\bm{s}}^{(n-1)\Delta t}];{{\bm{w}}_{{\cal
F}_{2}}}({\bm{w}}^{\prime}_{{\cal F}_{1}}))||_{2},$ (11)
$\displaystyle{\bm{w}}_{{\cal F}_{3}}$ $\displaystyle={\rm
argmin}_{{\bm{w}}_{{\cal F}_{3}}}||{\bm{q}}-{\cal F}_{3}([{\bm{s}}^{n\Delta
t},{\bm{s}}^{(n-1)\Delta t},{\bm{s}}^{(n-2)\Delta t}];{\bm{w}}_{{\cal
F}_{3}}({\bm{w}}^{\prime}_{{\cal F}_{2}}))||_{2},$ (12)

where ${\bm{w}}^{\prime}_{{\cal F}_{1}}$ denotes the weights assigned to the
common part of the first MLP-CNN model and the second model,
${\bm{w}}^{\prime}_{{\cal F}_{2}}$ represents the weights assigned to the
common part of the second MLP-LSTM-CNN model and the third model,
respectively. Since transfer learning can aid in the computational reduction
by enabling fast convergence of weights [40, 41], we can expect accurate flow
reconstruction with minimal training costs using transfer learning with LSTM.

##  5 Results and Discussions

###  5.1 Aerodynamic forces

Figure 9: Estimation of $C_{D}$ and $C_{L}$ obtained with MLP and MLP-LSTM
models for the case of
$(u_{\theta\rm{max}}/u_{\infty},R/c,y_{0}/c)=(-0.19,0.83,0.28)$.

Let us first present the machine-learning-based estimation of $C_{L}$ and
$C_{D}$ from the pressure sensor inputs. Here, we here prepare machine
learning models ${\cal F}$ for each coefficient such that ${C_{L}}={\cal
F}_{L}({\bm{s}}(t))$ and ${C_{D}}={\cal F}_{D}({\bm{s}}(t))$. The estimation
results for $C_{D}$ and $C_{L}$ are shown in figure 9. When training with only
50 training cases with each case having 50 snapshots, the model achieves a
qualitative estimation of $C_{D}$. Here, we denote the number of cases as
$n_{\rm case}$, and the number of snapshots per case as $n_{\rm ss}$. We also
quote the $L_{2}$ error norm $\varepsilon\equiv||{\bm{f}}_{\rm
Ref}-{\bm{f}}_{\rm ML}||_{2}/||{\bm{f}}_{\rm Ref}-\overline{\bm{f}}||_{2}$,
where ${\bm{f}}_{\rm Ref}$ and ${\bm{f}}_{\rm ML}$ are the reference and the
machine-learning-based estimation, respectively, of variable ${\bm{f}}$. Note
that this error is normalized by the fluctuation of a variable from its steady
state value $\overline{\bm{f}}$. For $C_{L}$ and $C_{D}$, the error is
measured over the time range $u_{\infty}t/c=[0.85,5.1]$ for each case.

The estimation results show that the positions of the peak and trough of
$C_{D}$ induced by the vortex-airfoil wake interaction are qualitatively
predicted, yet the exact values are off from the DNS result. Increasing the
number of training cases $n_{\rm{case}}$ improves the estimation performance.
Enhanced agreement between the estimation and DNS is also achieved when
increasing the number of snapshots to 500, as illustrated in figure 9a). The
enhancement in the data diversity leads to a drastic decrease in the
prediction error. In contrast to 50 training cases, utilizing 100 training
cases yields a $67\%$ deduction in test error. The reason why the expansion of
training cases is beneficial for prediction performance is that the machine
learning model can cover a larger parameter space, which assists in better
predicting unseen test cases. Yet, considering the vast parameter space, 100
training cases are very few.

To obtain an accurate reconstruction while using as little data as possible,
we then incorporate the transfer-learning-based LSTM into the model for
$C_{D}$, as shown in figure 9a). Due to the transient nature of the current
vortex-airfoil interaction problem, the present transfer-learning-based LSTM
is able to build a reliable connection between sensor input and output based
on historical information. For all three examples, using the MLP-LSTM model
gives rise to a 10% decrease in the estimation error.

Estimation for $C_{L}$ is presented in figure 9b). Enhancement in the
reconstruction of $C_{L}$ from increasing the amount of training data is also
shown in figure 9b). The new MLP-LSTM model reduces the test error to 0.215,
0.158, and 0.148 for $(n_{\rm case},n_{\rm ss})=(50,50)$, $(n_{\rm
case},n_{\rm ss})=(50,500)$ and $(n_{\rm case},n_{\rm ss})=(100,50)$,
respectively. Similar to the $C_{D}$ estimation, the transfer-learned-LSTM
architecture is also useful in the estimation of $C_{L}$. We also note that
the reconstruction for $C_{L}$ is usually better than $C_{D}$, which is due to
the variation of $C_{L}$ over time being much larger than that of $C_{D}$.

###  5.2 Surface pressure distribution

Figure 10: Machine-learning-based estimation of the pressure distribution over
an airfoil surface. The pressure coefficient $C_{p}$ at $u_{\infty}t/c=$
a)$~{}0.85$, b)$~{}3.82$, and c)$~{}4.67$. Results are shown for the case of:
$(u_{\theta\rm{max}}/u_{\infty},R/c,y_{0}/c)=(0.35,0.95,-0.15)$.

Next, we perform the MLP-based estimation of the pressure distribution $C_{p}$
over the airfoil surfaces. Representative snapshots of the vorticity field for
a test case when a vortical disturbance passes around the airfoil are shown in
the first row of figures 10. Similar to the aerodynamics forces $C_{L}$ and
$C_{D}$, the reconstruction performance is strongly influenced by the number
of cases $n_{\rm case}$, the number of snapshots per case $n_{\rm ss}$, as
well as whether transfer-learned LSTM is incorporated. When training the MLP-
LSTM model with 50 cases and 250 snapshots per case, a qualitative
reconstruction is achieved for $C_{p}$. As shown in the first row of figure
10, the estimated $C_{p}$ at both the upper and lower surfaces of the airfoil
are in agreement with the DNS. As we increase the number of cases from 50 to
100 without utilizing transfer-learned LSTM, this machine-learning model also
reconstructs $C_{p}$ in a reasonable manner, as shown in the second row of
figure 10. However, by comparing the reconstruction of $C_{p}$ of $(n_{\rm
case},n_{\rm ss})=(100,250)$ without LSTM against the results with LSTM
implemented, it is found that the use of transfer-learned LSTM greatly
improves reconstruction, achieving enhanced performance with only half of the
training data. In order to further improve the estimation performance,
increasing the number of snapshots from 250 to 500 for 100 training cases
achieves a similar performance as the results of $(n_{\rm case},n_{\rm ss},\rm
LSTM)=(50,250,\rm{Y})$, as shown in the last row of figure 10. Note that
although we are using 100 training cases and 500 snapshots per case, the
training data is still small compared to the broad parameter space of the test
cases.

###  5.3 Vorticity field

Figure 11: Reconstructed vorticity flow field with large region training and
windowed region training. Results shown for
$(u_{\theta\rm{max}}/u_{\infty},R/c,y_{0}/c)=(0.65,0.40,0)$.

We employ the present machine learning techniques to reconstruct the two-
dimensional vorticity field from sensor measurements on an airfoil using the
MLP-CNN model. Analogous to the results in section 4.2, the combination of MLP
and CNN is suitable to estimate the vortical flow from sensors. The
reconstruction of the spatially discretized vorticity field
${\omega}\in\mathbb{R}^{100\times 200}$ is summarized in figure 11. The
present model successfully captures the vortical disturbance at
$u_{\infty}t/c=0.85$. The location and the strength of the vortex are well
reconstructed. The interaction between the vortex disturbance and the flow
field around the airfoil at $u_{\infty}t/c=2.12$ is also reproduced well. This
is approximately the time at which $C_{L}$ and $C_{D}$ drop to their minimum
values, serving as an important dynamic transition point. However, the wakes
behind the trailing edge at $u_{\infty}t/c=5.10$ are not accurately
reconstructed because these wake structures are farther away from the airfoil
during this period. Sensors on the airfoil surface measure do not observe a
sizeable change in pressure, making it difficult to reconstruct far-field
wakes, which is expected.

Let us now focus on the critical near-wake region around an airfoil since this
region primarily determines the unsteady loading. Considering only the near-
field region enables us to greatly reduce the size of training data and the
associated computational costs. Results from training with a smaller region
are described in figure 11. The windowed training model also provides improved
estimations with lower error. The averaged $L_{2}$ error for the test case
reduces from 0.329 (large region training) to 0.261 (windowed region
training), which also shows the influence of the region size to estimate the
wake field with the modest computational cost.

Figure 12: Dependence of the reconstruction accuracy on the present
enhancement methods with window training for vorticity wake problem. Results
are shown for the case
$(u_{\theta\rm{max}}/u_{\infty},R/c,y_{0}/c)=(0.72,0.64,-0.10)$.

Moreover, the enhancement in reconstructing the vorticity field can also be
achieved by increasing the amount of training data and utilizing transfer-
learned LSTM, as summarized in figure 12. For all time series, a qualitative
and insightful reconstruction of the vorticity field is achieved with as less
as 10 snapshots per case, as shown in the second column in figure 12. When we
apply the transfer-learned-LSTM to the same dataset, up to 33% reduction in
$L_{2}$ error is accomplished. Additionally, increasing the number of
snapshots to 50 or increasing the data diversity by using 100 training cases
produces further improvements.

It is worth noting that the reconstruction accuracy for the interaction
process is not uniform. For example, at $u_{\infty}t/c=1.70$ and
$u_{\infty}t/c=2.55$ when the center of the disturbance is near the airfoil,
the $L_{2}$ errors are relatively high. Due to the high level of interaction,
the complex morphological changes in the vorticity field result in an
increased error. However, this does not indicate that the machine-learning
model is not able to extract the crucial features of the flow field. Instead,
the errors are partially due to the modest displacement of vortical
structures. In addition, the reconstructed vorticity field at
$u_{\infty}t/c=5.10$ shows that the transfer-learned LSTM shows its
superiority in estimating the small fluctuation behind the trailing edge
compared to the enhancement of data amount or diversity. Based on the insights
gained from this study, we deduce that when the influence from the disturbance
is greater (strong disturbances with large sizes and the interactions around
the airfoil), the accuracy of the reconstruction is improved. Here again, the
transfer-learned LSTM greatly improves the estimation for the overall dynamic
process.

###  5.4 Influence on the sensor positions

Figure 13:  Dependence of the $L_{2}$ errors on the sensor positions. Cases 1
to 5 denote 8 uniform sensors, 3 leading edge sensors, 3 top surface sensors,
3 trailing edge sensors, and 3 bottom surface sensors, respectively. The
machine-learning model with the condition of $(n_{\rm case},n_{\rm
sss},\rm{LSTM})=(50,50,Y)$ is used.  Figure 14: Dependence of the
reconstruction accuracy on the location of sensors for the reconstructed
vorticity wake. $(n_{\rm case},n_{\rm sss},\rm{LSTM})=(50,50,Y)$ As a test
case, we use $(u_{\theta\rm{max}}/u_{\infty},R/c,y_{0}/c)=(0.65,0.40,0)$.

Next, let us examine the estimation performance of the machine-learning models
trained with different numbers and placements of the sensors. As shown in
figure 13, we consider the uses of 8 sensors (case 1), 3 sensors around the
leading edge (case 2), 3 sensors on the top surface (case 3), 3 sensors around
the trailing edge (case 4), and 3 sensors on the bottom surface (case 5),
respectively. With 8 sensors (case 1), the lowest $L_{2}$ error is achieved
compared to the other cases with 3 sensors, as expected. With 3 sensors, we
observe that Case 5 with the bottom surface sensors usually presents a lower
error than Cases 2 to 4 for the whole time range. This is likely because the
sensors on the pressure side may sense the vortical structures approaching an
airfoil easier and earlier than having sensors on the suction side.

We also assess the estimation performance over time in figure 13. Before the
vortical disturbance impinges on the airfoil ($u_{\infty}t/c<2$) and after the
vortex moves away from the trailing edge of the airfoil ($u_{\infty}t/c>4$),
we observe relatively low $L_{2}$ error. For $u_{\infty}t/c\in[2,4]$, due to
the complex interactions between the disturbance and the airfoil, the
estimation for this time period is more difficult than other times. However,
we note that the present model still achieves qualitative reconstructions even
for the strong vortex-airfoil wake interaction process, as depicted in figure
14. These reconstructed snapshots correspond to the moment
$u_{\infty}t/c=2.12$ for
$(u_{\theta\rm{max}}/u_{\infty},R/c,y_{0}/c)=(0.65,0.40,0)$. This implies that
monitoring not only the scalar error measurement but also the reconstructed
flow fields is essential for appropriate assessments of machine-learning-based
flow estimations. These results also provide practical insights into the
choice of sensor locations. It is recommended that sensors are placed on the
suction and pressure sides for the present problem.

###  5.5 Robustness against noisy sensor measurements

Figure 15: Machine-learning model robustness against noisy sensor
measurements. a) $C_{L}$ and $C_{D}$, $(n_{\rm case},n_{\rm
ss},\rm{LSTM})=(100,50,Y)$, b) $C_{P}$, $(n_{\rm case},n_{\rm
ss},\rm{LSTM})=(100,500,Y)$, and c) Vorticity field $\omega$, $(n_{\rm
case},n_{\rm ss},\rm{LSTM})=(100,100,Y)$. Figure 16: Reconstruction of $C_{L}$
and $C_{D}$ subjected to different levels of input noise. Results are shown
for $(u_{\theta\rm{max}}/u_{\infty},R/c,y_{0}/c)=(0.96,0.57,0.21)$.  Figure
17: Comparison of $C_{P}$ subjected to different levels of input noise.
Results are shown for
$(u_{\theta\rm{max}}/u_{\infty},R/c,y_{0}/c)=(0.35,0.93,-0.15)$.  Figure 18:
Comparison of estimated vorticity fields subject to different levels of input
noise.

Let us evaluate the machine-learning model robustness against the noisy sensor
measurements. We use the Gaussian noise $\bm{n}$ for the sensor input
$\bm{s}$. Hence, the estimated output is expressed as

${\bm{q}}_{n}={\cal F}({\bm{s}}+\bm{n})$ (13)

where ${\bm{q}}_{n}$ is the output of the model, ${\cal F}$ is the model
trained without noisy inputs, and $\gamma=\|{\bm{n}}\|/\|{\bm{s}}\|$ is the
magnitude of the noise.

The estimation performance of $C_{L}$, $C_{D}$, $C_{P}$, and vorticity field
with noisy inputs (pressure measurements) are considered herein, as shown in
figure 15. For all estimations, the error increases with the magnitude of the
input noise, as expected. The reconstructed $C_{L}$, $C_{D}$, $C_{P}$, and
vorticity field are also shown in figures 16-18. Regarding the estimated
$C_{L}$ and $C_{D}$ in figure 16, the reconstructed $C_{L}$ and $C_{D}$
present smooth curves without noisy input of $\gamma=0$. With increasing
$\gamma$, $C_{L}$ and $C_{D}$ have high fluctuations resulting in a larger
$L_{2}$ error but with the overall trend well reproduced.

The estimated $C_{P}$ from noisy pressure inputs is depicted in figure 17.
While the error solely increases with the noise magnitude, we find that the
error at $u_{\infty}t/c=2.55$ is larger than that at $u_{\infty}t/c=0.85$.
This is caused by the intense wake-vortex gust interaction at
$u_{\infty}t/c=2.55$ which induces rapid changes in the pressure distribution
on the airfoil surface. Although the error reports approximately 0.5 with
$\gamma=0.178$, the whole trend of the $C_{P}$ curve is well-estimated,
supporting the robustness of the present machine-learning model.

The reconstructed vorticity fields across the different levels of noisy inputs
are also exhibited in figure 18. We show two cases of the vortical
disturbance, $(u_{\theta\rm{max}}/u_{\infty},R/c,y_{0}/c)=(0.35,0.95,-0.15)$
and $(-0.60,0.91,-0.09)$. A large positive disturbance is introduced in the
former case, while a negative vortical gust travels over the airfoil in the
latter case. In the case of
$(u_{\theta\rm{max}}/u_{\infty},R/c,y_{0}/c)=(0.35,0.95,-0.15)$, the estimated
vorticity field retains the primary vortical features for $\gamma\leq 0.178$.
The estimated flow field deviates from the reference DNS field at
$\gamma=0.28$. For the case of
$(u_{\theta\rm{max}}/u_{\infty},R/c,y_{0}/c)=(-0.60,0.91,-0.09)$, spurious
negative structure attached to the trailing edge vortex emerges beyond
$\gamma=0.178$, albeit the overall flow is reconstructed well. At
$\gamma=0.28$, although the $L_{2}$ error norm is relatively high, the main
wake structures are nonetheless reconstructed. These results suggest that the
present machine-learning models that incorporate dynamics are robust against
noisy pressure measurements even with a small amount of training data.

##  6 Concluding remarks

High-fidelity machine-learning-based reconstructions are developed for
aerodynamic force coefficients, pressure distribution over the airfoil, and
two-dimensional vorticity flows that experience an impact with a disturbance
vortex. Such reconstruction using sparse sensor measurements and a modest
amount of training data is extremely challenging due to the strong
nonlinearities and the transient nature of flow fields which requires a vast
parameter space to be covered during the learning process. For accurate
reconstruction, we developed machine learning models that are suitable for
estimating the transient flow features. A multi-layer perceptron is chosen for
its ability in constructing the nonlinear relation between limited sensor
measurements and aerodynamic forces coefficient as well as pressure over the
airfoil surface. A convolutional neural network coupled with MLP addresses the
problem of estimating the vorticity fields with rich information in an
efficient way with the filtering process. To better capture dynamical features
in time, long short-term memory (LSTM)-assisted transfer learning is utilized
via passing information from the historical scenarios, which is embedded in
the aforementioned two model structures. Due to the transient nature of the
vortex-airfoil interaction problem, the use of LSTM greatly assists in the
improvement of the estimation with as few as 10 training snapshots.

The main contribution of the present study is how time-varying flows with a
vast parameter space are reconstructed accurately. For this study, the
parameter space is comprised of maximum rotational velocity, radius, and
position of the disturbance vortex. As shown in this paper, careful sampling
of training data and incorporation of dynamics into the machine-learning model
is important. Based on our study, we also showed that accurate reconstruction
of vortical structures is easier to accomplish for high-intensity interaction
processes between the vortical disturbance and the airfoil (strong vortex with
large size, interacting close to the airfoil). In addition, we accessed proper
sensor locations over different time periods. We expect that the present
machine-learning-based reconstruction method will be useful in predicting and
controlling flows associated with vortex-airfoil interactions in the future.

## Acknowledgments

YZ, KF and KT acknowledge Ebara Corporation for supporting this research. We
are thankful to Akira Goto, Motohiko Nohmi, Masashi Obuchi, and Hiroyoshi
Watanabe for enlightening discussions.

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