# Time delays in anisotropic systems

Ulf Saalmann  Jan M. Rost  Max-Planck-Institut für Physik komplexer Systeme,
Nöthnitzer Str. 38, 01187 Dresden, Germany

###### Abstract

Scattering properties and time delays for general (non-symmetric) potentials
in terms of the respective S-matrices are discussed paradigmatically in one
dimension and in comparison to symmetric potentials. Only for the latter the
Wigner and Smith time delays coincide. Considering asymmetric potentials also
reveals that only one version of S-matrices used in the literature (the one
with reflection coefficients on the diagonal) generalizes to the asymmetric
case. Finally, we give a criterion how to identify a potential with intrinsic
symmetry which behaves like an asymmetric one if it is merely offset from the
scattering center.

##  I Introduction

Time delays related to scattering phases [1, 2] have been discussed for a long
time in transport problems [3, 4]. More recently, they have been addressed in
acoustics [5], electromagnetics [6], and from a fundamental perspective of
quantum trajectories [7], and since about a decade in the context of photo-
ionization by ultra-short laser pulses. Experimentally, photo-ionization time
delays have been extracted from streaking the momenta of electrons released by
a short XUV pulse with a moderate IR field [8] or so called RABBIT
measurements aiming at the same time-delay information of the released
electron wave-packet by using IR sidebands of the XUV photo-ionizing pulse
train [9, 10]. The link of the photo-ionization time delay to the Wigner-Smith
time delay from scattering theory, as well as the delays in general emerging
from these setups have been a source of ongoing debate [11, 12, 13]. This is
not surprising since the setups are quite intricate and become even more
cumbersome, if the long-range Coulomb interaction comes into play, which is
the case for almost all experiments performed.

Recent experimental advances have made it possible to measure time delays
originating in photo-ionizing molecules [14, 15, 16, 17, 18, 19], that is from
anisotropic potentials. This success motivates to ask for the theoretical
foundation of time delays and their formulation for general interactions,
since almost always time delay and S-matrices are discussed in the context of
single-centered, often spherically-symmetric potentials [20].

In the following we elucidate basic properties of time delays in the simplest
setting which is general enough to be sensitive to the properties of
anisotropic (and isotropic, parity-respecting) potentials. Since
characteristic features (such as the difference between proper and partial
time delays) can only be uncovered in a system with at least two independent
scattering channels, we do not investigate photo-ionization but scattering in
one dimension from a generic short-range potential, a scenario which provides
two scattering channels. Such type of scattering is relevant in complex media,
in wave-guides, or generally, for transport problems.

Additional motivation is provided by the fact, that symmetric potentials in 1D
hide subtleties of scattering and related time delays in at least two aspects:
(i) Two different versions of the S-matrix $\boldsymbol{S}$ are pursued in the
literature [21, 22, 23, 24], which have different eigenvalues. Yet, both
fulfill the criteria for S-matrices, derived from overarching principles of
flux conservation and time-reversal invariance (for a real potential), namely
that $\boldsymbol{S}$ is unitary and symmetric. However, only the version
which is a symmetric matrix with respect to incoming and outgoing channels
[21] remains symmetric in case of anisotropic potentials. (ii) Furthermore,
without symmetric interaction, the two commonly used formulations of time
delay, namely partial time delays and proper time delays do not agree,
prompting the question what their respective meaning is.

Scattering in one dimension was mostly theoretically investigated [22, 23, 24,
25] long before time delays have become popular, however, to the best of our
knowledge never with a discussion or even a focus on situations where the
scattering potential is not symmetric.

##  II Scattering in 1D

For our context a potential $V(x)$ is short range if at large distances $x$
the solutions of the time-independent Schrödinger equation
$[-\mathrm{d}^{2}/\mathrm{d}x^{2}+2V(x)-2E]\psi(x)=0$ are free waves,
$\psi(|x|\gg 1)\propto e^{\pm\mathrm{i}kx}$ with $k\,{=}\,\sqrt{2E}>0$, see
also App. 1. We will use atomic units $e\,{=}\,\hbar\,{=}\,m_{\rm
e}\,{\equiv}\,1$ and consider for convenience a particle of mass $m_{\rm e}$,
unless stated otherwise.

###  II.1 The S-matrix and its parameterization

There are two channels in an 1D scattering scenario. Most easily [26] they are
described by reflection ($r$) and transmission ($t$) amplitudes for incoming
waves from the left or the right side. Asymptotically those wave function
read, with $k\,{=}\,\sqrt{2E}$ and
$\\{\lim_{x\to{-}\infty}\psi(x,E),\;\lim_{x\to{+}\infty}\psi(x,E)\\}$,

$\displaystyle\psi_{\rm l}(x,E)$
$\displaystyle=\\{\mathrm{e}^{{+\mathrm{i}kx}}{+}r_{\rm
l}(E)\,\mathrm{e}^{{-\mathrm{i}kx}},\;t_{\rm
l}(E)\mathrm{e}^{{+\mathrm{i}kx}}\\}$ (1a) $\displaystyle\psi_{\rm r}(x,E)$
$\displaystyle=\\{t_{\rm
l}(E)\,\mathrm{e}^{{-\mathrm{i}kx}},\;\mathrm{e}^{{-\mathrm{i}kx}}{+}r_{\rm
r}(E)\mathrm{e}^{{+\mathrm{i}kx}}\\}.$ (1b)

By means of the four reflection and transmission amplitudes in Eq. (1) one
gets immediately the scattering matrix

$\boldsymbol{S}(E)=\left(\\!\\!\begin{array}[]{cccc}r_{\rm l}(E)&t_{l}(E)\\\
t_{\rm r}(E)&r_{\rm r}(E)\end{array}\\!\\!\right)\,.$ (2)

This S-matrix connects the amplitudes for incoming ($a_{\rm l,r}$) and
outgoing ($b_{\rm l,r}$) waves [27]

$\left(\\!\\!\begin{array}[]{cccc}b_{\rm l}\\\ b_{\rm
r}\end{array}\\!\\!\right)=\boldsymbol{S}\left(\\!\\!\begin{array}[]{cccc}a_{\rm
l}\\\ a_{\rm r}\end{array}\\!\\!\right).$ (3)

For both channels (l, r) particle-flux conservation guarantees
$|r(E)|^{2}+|t(E)|^{2}=1$. Furthermore, for real potentials $V(x)$ the
complex-conjugated channels $\psi_{\rm l,r}^{*}$ are valid solutions as well.
Comparing $[\psi_{\rm l}^{*}{-}r_{\rm l}^{*}\psi_{\rm l}]/t_{\rm l}^{*}$ with
$\psi_{\rm r}$ implies $t_{\rm l}=t_{\rm r}\equiv t$ and $r_{\rm l}/t_{\rm
l}={-}r_{\rm r}^{*}/t_{\rm r}^{*}$. Those four conditions render the
scattering matrix unitary
($\boldsymbol{S}^{\dagger}\boldsymbol{S}\,{=}\,\boldsymbol{1}$) and symmetric
($\boldsymbol{S}^{{\scriptscriptstyle\mathsf{T}}}{=}\,\boldsymbol{S}$).

Note that the diagonal elements of $\boldsymbol{S}$ connect the in- and
outgoing channels on the same side (l$\to$l, r$\to$r). There is another
version in the literature [22, 23, 24] that mixes the channels on the diagonal
(l$\to$r, r$\to$l). If the scattering potential is symmetric, the channel-
mixing version fulfills all symmetries required for an S-matrix, namely that
$\boldsymbol{S}$ is unitary and symmetric. However, if the potential does not
have parity, this channel-mixing version is no longer symmetric and therefore
is not a proper S-matrix. Since the overwhelming majority of published work
discusses only symmetric potentials, this shortcoming of the channel-mixing
S-matrix has not been pointed out.

For the unitary and symmetric S-matrix, three real parameters are sufficient
to define reflection and transmission amplitudes in the general form

$\displaystyle t(E)$
$\displaystyle=\cos(\alpha(E))\,\mathrm{e}^{{\mathrm{i}\beta(E)}}$ (4a)
$\displaystyle r_{\rm l,r}(E)$
$\displaystyle=\mathrm{i}\,\sin(\alpha(E))\,\mathrm{e}^{{\mathrm{i}[\beta(E)\pm\gamma(E)]}}.$
(4b)

The basis $\psi_{\rm l,r}$ of incoming waves from the left and the right is
only one of infinitely many choices. The one which diagonalizes the S-matrix
with a suitable unitary transformation $\boldsymbol{U}_{s}$ stands out and has
the physical meaning, that the ratio of incoming waves from the right and left
is not changed for the outgoing waves upon scattering. Due to the properties
of the S-matrix, its diagonal representation
$\boldsymbol{D}[\mathrm{e}^{{\mathrm{i}\boldsymbol{s}(E)}}]=\boldsymbol{U}_{\\!s}{\\!\\!}^{\dagger}\\!(E)\,\boldsymbol{S}(E)\,\boldsymbol{U}_{\\!s}(E)$
contains the eigenvalues given by pure phases
$\mathbfit{s}(E)=(s_{1}\;s_{2})^{\scriptscriptstyle\mathsf{T}}$. They read in
terms of reflection and transmission amplitudes or in their parameterization
(4)

$\displaystyle\mathrm{e}^{{\mathrm{i}\,s_{1,2}(E)}}$
$\displaystyle=\mbox{{\Large$\frac{r_{\rm l}+r_{\rm
r}}{2}$}}\pm\sqrt{\Big{[}\mbox{{\Large$\frac{r_{\rm l}-r_{\rm
r}}{2}$}}\Big{]}^{2}+t^{2}}$ (5a)
$\displaystyle=\mathrm{e}^{{\mathrm{i}[\beta+\chi_{\pm}]}}$ (5b)
$\displaystyle\chi_{\pm}$
$\displaystyle\equiv\mbox{atan}(\pm\sqrt{1{-}\sin^{2}\\!\alpha\cos^{2}\\!\gamma},\sin\alpha\cos\gamma).$
(5c)

If the potential is symmetric, i.e., $V(-x)=V(x)$, then $r_{\rm l}=r_{\rm r}$
and therefore $\gamma(E)=0$.

We close this section with a note on the channel-mixing S-matrix [22, 23, 24]:
Even if it is a valid S-matrix (i. e. for the case of a symmetric potential
where it is symmetric) its eigenvalues differ from those of the proper
S-matrix (2) while the time delays, to be discussed below, agree. Only the
time delays must agree since they are observables based on Hermitian
operators. The S-matrix itself is not Hermitian and therefore not an
observable. It provides a description of scattering, whose parameterization
can be done in different ways, as long as they are consistent with the
fundamental properties of a collision process.

###  II.2 Partial time delays

As it has become clear from its definition, the S-matrix connects the two
input with the two output channels. This means that a wave-packet sent from
one side splits upon scattering and leaves the interaction region towards both
sides. Any change in the incoming configuration will lead to a different
partition of outgoing waves to the left and the right. Time delays, however,
are meaningful for the channels which diagonalize the S-matrix and thereby
keep the ratio of waves entering and leaving the scattering region from the
left and the right the same. Consequently, partial time delays are defined as

$\widetilde{\tau}_{\\!j}(E)=\mbox{{\Large$\frac{\mathrm{d}}{\mathrm{d}E}$}}s_{\\!j}(E)$
(6)

with $s_{\\!j}$ given in Eqs. (5).

###  II.3 Proper time delays

Another, potentially more intuitive way of defining a time delay is to
consider the dwell time $\tau$, i. e. the reduced or increased time it takes a
wave-packet to traverse the region of interaction $|x|\leq X$ compared to the
traversal time $\tau_{0}$ of a free wave-packet,

$\displaystyle\tau$
$\displaystyle=\mbox{{\Large$\int$}}\\!\mathrm{d}t\mbox{{\Large$\int\limits_{\scriptscriptstyle|x|\leq
X}$}}\\!\\!\\!\mathrm{d}x\,\rho(x,t)-\tau_{0}$ (7a)
$\displaystyle=\mbox{{\Large$\int$}}\\!\mathrm{d}t\,t\;\big{[}J({+}X,t)-J({-}X,t)\big{]}-\tau_{0}\,,$
(7b)

where $\tau_{0}$ are the corresponding integrals for the free wave-packet
involving $\rho_{0}$ and $J_{0}$, respectively. The expression (7b) follows
from (7a) by means of the continuity equation and contains the time-dependent
current $J$ (or $J_{0}$) of the wave-packet at the left and right boundary of
the scattering region. Note that in 1D the current and the current density are
the same. If $X$ is chosen sufficiently large for the asymptotic description
(1) to be valid, it is obvious that $\tau$, defined spectrally by
$\boldsymbol{\psi}(E)$, can be calculated by means of the reflection and
transmission coefficients or the S-matrix for any given wave-packet. Without
details, which can be found elsewhere [28, 24], the result is

$\tau=\mbox{{\Large$\int$}}\\!\mathrm{d}E\,\boldsymbol{\psi}^{*}(E)\boldsymbol{Q}(E)\boldsymbol{\psi}(E),$
(8)

with Smith’s life-time matrix [2]

$\boldsymbol{Q}(E)=-\mathrm{i}\,\boldsymbol{S}^{\dagger}(E)\mbox{{\Large$\frac{\mathrm{d}}{\mathrm{d}E}$}}\boldsymbol{S}(E),$
(9)

where the time delay $\tau$ appears now as an expectation value of
$\boldsymbol{Q}(E)$ with the state $\boldsymbol{\psi}(E)$. Indeed,
$\boldsymbol{Q}$ is a Hermitian matrix since $\boldsymbol{S}$ is unitary.
Diagonalizing the life-time matrix $\boldsymbol{Q}$

$\boldsymbol{D}[\boldsymbol{q}(E)]=\boldsymbol{U}_{\\!q}{\\!\\!}^{\dagger}\\!(E)\,\boldsymbol{Q}(E)\,\boldsymbol{U}_{\\!q}(E),$
(10)

yieds real eigenvalues which are called proper time delays

$\overline{\tau}_{\\!j}(E)=q_{\\!j}(E).$ (11)

Since we consider a 2$\times$2 problem, they correspond to the minimal and
maximal dwell time $\tau$ in Eq. (8) with the respective eigenvectors
corresponding to the combination of incoming and outgoing waves that minimize
or maximize the time delay.

We have already introduced three different bases for the scattering channels,
waves coming in from the left and the right to define the S-matrix in the
first place, and the two bases which diagonalize the S-matrix giving partial
time delays and the one which diagonalizes Smith’s life-time matrix whose
eigenvalues are the proper time delays. A fourth basis is often introduced if
one wants to use a basis of real functions, namely linear combinations
$\\{\cos(kx),\sin(kx)\\}$ of left- and right-traveling waves which have even
and odd parity, respectively. For completeness, we also give the S-matrix in
the parity basis

$\boldsymbol{S}_{\rm
p}(E)=\mathrm{e}^{{\mathrm{i}\beta}}\sin\\!\alpha\left(\\!\\!\begin{array}[]{cccc}\mathrm{i}\,\cos\gamma{+}\cot\\!\alpha&\sin\gamma\\\
\sin\gamma&\mathrm{i}\,\cos\gamma{-}\cot\\!\alpha\end{array}\\!\\!\right)$
(12)

with details of the derivation given in App. 2.

Note that the S-matrix is always defined with respect to the center of
incoming and outgoing waves. A potential with $V(-x)=V(x)$ we have called
symmetric. There can be, however, the case, that the potential is symmetric
about about a point $x_{\rm cen}\neq 0$, which we call intrinsically
symmetric. Scattering from such a potential will have formally a full S-matrix
in any generic basis including the parity basis (12). This prompts the
question, if and how one could tell from experimental time delays, if the
potential has intrinsic symmetry or not. We will come back to this question
later.

##  III Symmetry and time delays

###  III.1 Symmetric potentials and the Wigner-Smith time delay

For a symmetric potential, $V(-x)=V(x)$, the S-matrix is diagonal in the
parity basis (12) since $\gamma=0$. The eigenphases reduce to

$\boldsymbol{s}_{\rm
sym}(E)=\left(\\!\\!\begin{array}[]{cccc}\alpha(E)+\beta(E)\\\
\pi-\alpha(E)+\beta(E)\end{array}\\!\\!\right).$ (13)

Also the life-time matrix becomes diagonal in this basis with the eigenvalues

$\boldsymbol{q}_{\rm
sym}(E)=\left(\\!\\!\begin{array}[]{cccc}\alpha^{\prime}(E)+\beta^{\prime}(E)\\\
-\alpha^{\prime}(E)+\beta^{\prime}(E)\end{array}\\!\\!\right),$ (14)

where there prime denotes derivation with respect to $E$. Hence, partial and
proper time delays ($j{=}1,2$) agree for symmetric potentials

$\widetilde{\tau}_{\\!j}(E)=\overline{\tau}_{\\!j}(E).$ (15)

The equivalence holds similarly for spherical potentials in 3D, with the
eigen-basis given by spherical harmonics. It is this equivalence which has led
to the notion of “Wigner-Smith time delays”.

###  III.2 Asymmetric potentials

For arbitrary potentials the equivalence (15) does not hold any longer. With
the eigenvalues from Eqs. (5) we obtain

$\widetilde{\tau}_{1,2}=\mbox{{\Large$\frac{\mathrm{d}}{\mathrm{d}E}$}}s_{1,2}=\beta^{\prime}\mp\frac{\cos\\!\alpha\,\cos\gamma\,\alpha^{\prime}-\sin\\!\alpha\,\sin\gamma\,\gamma^{\prime}}{\sqrt{1{-}\cos^{2}\\!\gamma\,\sin^{2}\\!\alpha}}$
(16a) and for the proper time delays from diagonalizing $\boldsymbol{Q}$
$\overline{\tau}_{1,2}=q_{1,2}=\beta^{\prime}\pm\sqrt{\alpha^{\prime}{}^{2}+\sin^{2}\\!\\!\alpha\,\gamma^{\prime}{}^{2}},$
(16b)

where again we have dropped the energy dependence and indicate energy
derivatives with a prime. Obviously, these expression simplify to Eqs. (13)
and (14) for symmetric potentials ($\gamma{=}0$, $\gamma^{\prime}{=}0$).

Although proper and partial time delays differ, the time delay averaged over
the channels is the same for both

$\mbox{{\Large$\frac{1}{2}$}}\mbox{{$\sum$}}_{j=1,2}\widetilde{\tau}_{\\!j}(E)=\mbox{{\Large$\frac{1}{2}$}}\mbox{{$\sum$}}_{j=1,2}\overline{\tau}_{\\!j}(E)=\mbox{{\Large$\frac{\mathrm{d}}{\mathrm{d}E}$}}\beta(E)\,.$
(17)

This result generalizes to higher dimensions (2D, 3D) where the sum runs
formally over infinitely many channels, cf. App. 3.

Since the eigenchannels of $\boldsymbol{S}$ differ from those of
$\boldsymbol{Q}$, the natural question arises: What is the dwell time
$\tau_{\\!j}$ of a (shape-conserving) eigenchannel $j$ of $\boldsymbol{S}$?
Indeed the expectation value of $\boldsymbol{Q}$ in an eigenfunction
$\boldsymbol{\psi}_{\\!j}$ of the partial time delay produces this partial
time delay,

$\displaystyle\tau_{\\!j}\equiv\boldsymbol{\psi}_{\\!j}^{*}\boldsymbol{Q}\boldsymbol{\psi}_{\\!j}=-\mathrm{i}\boldsymbol{\psi}_{\\!j}^{*}\boldsymbol{S}^{\dagger}\boldsymbol{S}^{\prime}\boldsymbol{\psi}_{\\!j}=\widetilde{\tau}_{\\!j}$
(18)

as expected and shown in App. 4.

Figure 1: Upper raw: Time delays for a symmetric ($V_{1}$, a) and an
asymmetric ($V_{2}$, b) potential as a function of energy $E$, see Eq. (19)
and insets. Partial time delays $\widetilde{\tau}_{\\!j}$ (dashed lines)
according to Eq. (6) are compared to proper time delays
$\overline{\tau}_{\\!j}$ (solid lines) given in Eq. (11). The averages of both
$[\widetilde{\tau}_{1}{+}\widetilde{\tau}_{2}]/2$ and
$[\overline{\tau}_{1}{+}\overline{\tau}_{2}]/2$ are identical (thin gray
lines). Blue and red refer to the first and second channel, respectively.
Bottom raw: The corresponding parameters $\alpha$ (solid-blue line), $\beta$
(green-dotted), $\gamma$ (red-dashed) as a function of energy $E$, cf. Eqs.
(4).

###  III.3 Generic examples

We illustrate the time-delay behavior for generic symmetric and asymmetric
potentials with two examples in Fig. 1. The potentials are defined by

$V_{1,2}(x)=-V_{0}\sum_{j=-2}^{+2}f_{\\!j}^{[1,2]}\,\mathrm{e}^{{-[x/d-2j]^{2}}}$
(19)

with $V_{0}=2$ eV and $d=1$ Å, to be specific. The prefactors are
$f_{\\!j}{}^{[1]}\,{=}\,1$ and $f_{\\!j}{}^{[2]}\,{=}\,1{+}j/3$ for the
symmetric and asymmetric potential, respectively.

One can see in Fig. 1a that for the symmetric potential partial and proper
time delays are identical for all energies $E$, as stated in Sect. III.1
above.

However, time delays differ for the asymmetric potential (Fig. 1b), where the
proper time delays form an envelope for the partial time delays. This is to be
expected as the eigenvalues of $\boldsymbol{Q}$ are the minimal and maximal
dwell times. No scattering states (and therefore not even a shape-conserving
one) can fall below or exceed those values. Nevertheless, both time delays can
agree at certain energies $E_{=}$, i. e.,
$\widetilde{\tau}_{\\!j}(E_{=})=\overline{\tau}_{\\!j}(E_{=})$, if

$\alpha^{\prime}=-\sin\\!\alpha\,\cos\\!\alpha\,\cot\\!\gamma\,\gamma^{\prime},$
(20)

as can be easily derived from Eqs. (16). Similarly one can find those energies
$E_{\times}$ where the partial time delays cross, i. e.
$\widetilde{\tau}_{\\!1}(E_{\times})=\widetilde{\tau}_{\\!2}(E_{\times})$
which requires

$\alpha^{\prime}=\tan\\!\alpha\,\tan\\!\gamma\,\gamma^{\prime}.$ (21)

For completeness we also present the energy dependence of the S-matrix
parameters ($\alpha,\beta,\gamma$) in Figs. 1c,d, which confirm that
$\gamma(E)=0$ for a symmetric potential, but finite for an asymmetric one.

##  IV The dependence of time delays on spatial properties

Time delay is not immune to shifting the potential in a coordinate system,
which may be surprising given its relative character, that is a delay relative
to free motion at a given energy. Yet, scattering and subsequently time delay
define a coordinate system, in particular a scattering center through incoming
and outgoing waves and the S-matrix. The location of the potential relative to
the scattering center will have an influence on the time delay. A very loose
analogy is the echo of an object placed at some distance in front of a
reflecting wall, which plays the role of the scattering center: The echo one
receives will depend on the object as well as on its distance to the wall.
Similarly, for angular differential cross sections parameterized with partial
waves, the amplitudes of the partial waves depend on the location of the
origin of the coordinate system relative to the target. We will elucidate the
dependence of the time delays on spatial properties of the scattering scenario
below with examples.

###  IV.1 Position of the potential with respect
to the scattering state

The results so far render time delay a useful observable, if carefully
assessed in a specific physical situation. What makes time delay, however,
quite cumbersome is the fact that it depends also on the location of the
potential with respect to the incoming wave-packet [5]. While it is natural
(albeit not necessary) to place an intrinsically symmetric potential at the
origin rendering it symmetric, no obvious choice exists a priori for an
asymmetric potential. In a realistic situation time delays are extracted from
asymptotic electron wave-packets. Those wave-packets have a clear origin, but
it is the location of the potential (e. g. the molecule) with respect to that
origin which matters for the time delays. This location is not uniquely
“defined” and could be difficult to determine.

One should note, however, that general time delays in the absence of
particular spectral features have only become of interest with the advent of
ultra-short laser pulses. Before, time delays were mostly discussed in
relation to a resonance. At the resonance energy, the time delays are
drastically enhanced for all channels sensitive to the resonance, see Fig. 2b,
where time delays are shown for the potential

$V_{4}(x)=\mathrm{e}^{{-[x/d-1/2]^{2}}}\mbox{atan}\big{(}2\sin(2x/d)\big{)}.$
(22)

In such a situation the location of the potential plays a subdominant role.
This is probably the reason, that the difference between various time delay
definitions and the dependence of those time delays on the location of the
potential has seen little attention to date.

Figure 2: Upper row: Time delays for a displaced potential ($V_{3}$, a) and a
potential with a resonance ($V_{4}$, b) as a function of energy $E$, see Eqs.
(23), (22) and insets. See caption of Fig. 1 for line styles.
Bottom row: Corresponding parameters $\gamma$ (red-dashed) and
$\gamma{-}2E\gamma^{\prime}$ (gray-dot-dashed) as a function $E$. Note that
for the symmetric potential (panel c) $\gamma\sim\sqrt{E}$ and thus
$\gamma{-}2E\gamma^{\prime}\sim\mbox{const}$.

However, the consequence of shifting the potential can be significant and is
particularly dramatic for a symmetric potential. We shift the one defined in
Eq. (19) according to

$V_{3}(x)=V_{1}(x-\delta x)$ (23)

which turns it from a symmetric one into one with only intrinsic symmetry. The
result is shown for a particular displacement of $\delta x=2d$ in Fig. 2a. Not
only do proper and partial time delays no longer agree, also the partial time
delays differ substantially from the ones where the location of the potential
is chosen such that it becomes symmetric as in Fig. 1a. Yet, the intrinsic
symmetry of the potential is still reflected in a more subtle relation as one
can see in Fig. 2c, namely that $\gamma(E)\,{-}\,2E\gamma^{\prime}(E)=0$ holds
for all energies $E$.

Figure 3: Time delays for a symmetric ($V_{1}$, a) and an asymmetric ($V_{2}$,
b) potential as a function a displacement $\delta x$ with respect to the
scattering center, see Eq. (19) and insets. The energy is $E=2$ eV in both
cases. See caption of Fig. 1 for line styles.

This relation, which is not fulfilled for a generically asymmetric potential,
see Fig. 2d, can be understood from (4) and the following consideration
regarding the effect on the time delays when shifting the potential by $\delta
x$, see Fig. 3. A little thought reveals that shifting the potential’s
position by $\delta x$ will only change $\gamma$ according to

$\displaystyle\gamma_{\\!\delta x}(E)$ $\displaystyle=\gamma_{0}(E)+2k\delta
x,$ (24a) such that $\displaystyle\gamma^{\prime}_{\\!\delta x}(E)$
$\displaystyle=\gamma^{\prime}_{0}(E)+2\delta x/k,$ (24b)

where $\gamma_{0}(E)$ refers to some reference position of the potential.

Partial and proper time delays can still be obtained from Eqs. (16) with
$\gamma$ and $\gamma^{\prime}$ replaced by $\gamma_{\\!\delta x}$ and
$\gamma^{\prime}_{\\!\delta x}$, respectively. The difference
$\Delta\overline{\tau}=\overline{\tau}_{1}-\overline{\tau}_{2}$ grows for
large displacement $\delta x$ linearly with a slope of $2/k$. Taking into
account $\hbar$, the slope defines a velocity, which (multiplied with $\delta
x$) represents the increased/reduced time a particle needs to reach/leave a
displaced potential.

The minimal “gap” $\Delta\overline{\tau}_{\rm min}=2\alpha^{\prime}$ of the
proper time delays occurs according to Eqs. (16b) and (24b) for
$\gamma^{\prime}_{\delta x_{\rm min}}=0$, realized with $\delta x_{\rm
min}=-k\gamma^{\prime}_{0}/2$. The resulting

$\gamma_{\\!\delta x_{\rm min}}=\gamma_{0}-k^{2}\gamma^{\prime}_{0}$ (25)

varies for generically asymmetric potentials with energy in the expression for
the partial-time-delay difference

$\Delta\widetilde{\tau}_{\rm
min}=2\frac{\cos\\!\alpha\,\cos(\gamma_{0}{-}k^{2}\gamma^{\prime}_{0})\,\alpha^{\prime}}{\sqrt{1{-}\sin^{2}\\!\alpha\,\cos^{2}(\gamma_{0}{-}k^{2}\gamma^{\prime}_{0})}}$
(26)

at the minimal gap of the proper time delays. Figure 3 illustrates this for a
particalur energy.

For potentials with intrinsic symmetry the reference in Eqs. (24) can be
chosen such that $\gamma_{0}\,{=}\,0$ (by making $x_{\rm cen}\,{=}\,0$), which
entails $\gamma^{\prime}_{0}\,{=}\,0$. Once this potential is offset from the
center, $\gamma$ becomes finite and energy-dependent. Yet, as follows directly
from (24), $\gamma_{\delta x}{-}\,2E\gamma^{\prime}_{\delta
x}\,{=}\,\mbox{const}$ for all energies $E$. For a generically asymmetric
potential (not possessing an $x_{\rm cen}$) this is not possible.

We may conclude that the interplay of partial and proper time delays reveal
the symmetry of the potential, despite their sensitivity to its location: If
the minimal gap of the proper time delays coincides at all energies with the
maximal gap of the partial time delays, the underlying potential has intrinsic
symmetry.

###  IV.2 Reflection-less potentials

Apparently, time delays are quite sensitive to the quantum-mechanical
interference of transmitted and reflected waves. Hence, one would expect a
radically different behavior, if reflection is suppressed. This can be double-
checked by investigating the time delays of a reflection-less potential. It is
well known [29] that certain potentials show perfect transmission
$|t(E)|^{2}=1$ for all energies $E$, e.g.,

$V_{5}(x)=\mbox{{\Large$\frac{1}{d^{2}\cosh^{2}(x/d)}$}}\,.$ (27)

In Fig. 4 we show partial and proper time delays for $V_{5}(x)$ and its
shifted version $V_{6}(x)=V_{5}(x-\delta x)$ with $\delta x=3$ Å. As can be
seen, all time delays agree even for the displaced potential. This reveals
that subtle interference effects due to the position of the potential as well
as the difference of proper and partial time delays are of pure quantum nature
and vanish in a (semi-)classical setting, as provided by the potential free of
reflections which behaves as a classical system would do (full transmission
and zero reflection). Of course, in such a situation only a single channel is
left and therefore even in full quantum mechanics, no interference can occur.
This lets the proper and partial time delays collapse to a single time delay
which is identical to the average one. All of this follows directly from the
parameters (4) in this case. It is $\alpha\,{=}\,0$, thus
$\alpha^{\prime}\,{=}\,0$ and $\gamma$ being irrelevant.

Figure 4: Time delays for a reflection-less potential ($V_{5}$, a) and the
same one displaced ($V_{6}$, b) , cf. Eq. (27) and see insets, as a function
of energy $E$. See caption of Fig. 1 for line styles.

##  V Conclusion

We have discussed scattering and ensuing time delays for generic potentials
(without symmetry), paradigmatically in one dimension. For potentials
symmetric to the origin, the standard case almost exclusively discussed also
in 3D with centrally-symmetric potentials, proper and partial time delays
agree which has given the notion of Wigner-Smith time delays. This is not the
case for generic potentials whose location relative to the collision origin
have significant influence on the various time delays. With asymmetric
potentials, one also notices a widespread use of a definition of the S-matrix
which is only symmetric (as it should be from general considerations) for
symmetric potentials and therefore not suitable to describe scattering for
generic potentials. Finally, we have provided a criterion which allows one to
identify from the time delays an intrinsically symmetric potential (symmetric
with respect to some position $x_{\rm cen}$) located at an arbitrary position.

## Acknowledgment

We dedicate this analysis to Ravi Rau, who has always been an inspiration and
role model to view a problem from different perspectives and to articulate it
in a clear and transparent manner. We wish him many productive years to come.

##  A Appendix

###  1 Coulombic systems

Long-range Coulomb potentials lead to infinite time delays if referenced with
free motion [2]. (Note, that finite values, reported in experiments [8, 9],
are the consequence of the measurement [11].) Therefore, free motion must be
replaced by motion in a pure Coulomb potential as reference [11]. Then time
delays are finite and measure the (short-range) deviation from a pure Coulomb
interaction. In 3D, many of the ideas and techniques presented here carry over
to Coulomb systems with asymptotic wavefunctions
$\exp(\pm[\mathrm{i}kr+\frac{1}{k}\ln(2kr)])$, i. e., amended by the
logarithmic Coulomb phase. However, dynamics with Coulomb interaction differs
substantially in 1D and 3D and disguises the general analogy between 1D and 3D
scattering.

###  2 Analogy to 3D scattering in a basis
of spherical harmonics

The treatment described so far is specific to 1D, but has a clear relation to
the 3D situation: The continuous solid angle $\\{\vartheta,\varphi\\}$ in 3D
gets replaced in 1D by two discrete directions $d\,{=}\,{-}1$ (left) and
$d\,{=}\,{+}1$ (right), respectively. The relation between 1D and 3D becomes
transparent with the commonly-used spherical-harmonics basis $Y_{\ell
m}(\vartheta,\varphi)$ with $\ell\,{=}\,0\ldots\infty$ and
$m\,{=}\,{-}\ell\ldots{+}\ell$ for the angular degrees of freedom. In 1D one
can use two ($m\,{=}\,0,1$) “angular” functions
$y_{m}(d)=d^{m}\\!/\\!\sqrt{2}$, which are orthonormal $\sum_{d=\pm
1}y_{m}(d)y_{m^{\prime}}(d)=\delta_{mm^{\prime}}$. Whereas the description
with a finite $\ell_{\rm max}$ is approximate but numerically accurate in 3D,
the description in 1D in terms of the $y_{\\!m}$ is exact.

Instead of the traditional form (1) in the Cartesian coordinate $x$, one can
write the two continuum states ($j{=}1,2$) asymptotically in terms of the
$y_{m}(d)$ and real radial functions $\phi_{mj}(r)$ in the discrete “angle”
$d\,{=}\,\mbox{sgn}(x)$ and the radial distance $r\,{=}\,|x|$, respectively,
as

$\displaystyle\psi_{j}(r,d,E)$
$\displaystyle=\sum_{m=0,1}y_{m}(d)\phi_{mj}(r,E)$ (28)
$\displaystyle\left.{\phi^{\infty}_{01}(r,E)\atop\phi^{\infty}_{12}(r,E)}\right\\}$
$\displaystyle=\cos(kr)\pm\cos\alpha\cos(kr{+}\beta)$
$\displaystyle\qquad\pm\cos\gamma\sin\alpha\sin(kr{+}\beta)$ (28a)
$\displaystyle\left.{\phi^{\infty}_{11}(r,E)\atop\phi^{\infty}_{02}(r,E)}\right\\}$
$\displaystyle=-\sin\gamma\sin\alpha\cos(kr{+}\beta),$ (28b)

where we have omitted the dependence of $\alpha$, $\beta$ and $\gamma$ on the
energy $E{=}k^{2}\\!/2$ and used the notation
$\phi^{\infty}(r,E)\equiv\lim_{r\to\infty}\phi(r,E)$. In general, in both
functions ($j\,{=}\,1,2$) the two “angular” channels ($m\,{=}\,0,1$) couple,
as can be seen in Eqs. (28). For symmetric potentials, where $\gamma=0$ [cf.
Eqs. (4)], this is not the case leading to the simplification

$\displaystyle\left.{\phi^{\infty}_{01}(r,E)\atop\phi^{\infty}_{12}(r,E)}\right\\}$
$\displaystyle=\cos(kr)\pm\cos(kr{\pm}\alpha{+}\beta)$ (29a)
$\displaystyle\left.{\phi^{\infty}_{11}(r,E)\atop\phi^{\infty}_{02}(r,E)}\right\\}$
$\displaystyle=0,$ (29b)

where one can directly read off the eigenphases (13). Note, that radial wave-
packets built from either symmetric ($m{=}0$) or anti-symmetric ($m{=}1$)
states, given in Eq. (29a), will keep their symmetry throughout the scattering
process.

The asymptotic expressions of the radial functions (28) have the form

$\phi^{\infty}_{mj}(r,E)=a_{mj}\mathrm{e}^{{-\mathrm{i}kr}}+b_{mj}\mathrm{e}^{{+\mathrm{i}kr}},$
(30)

from which the S-matrix is directly obtained by means of Eq. (3). Since we
have used the parity basis $y_{m=0,1}$ we get $\boldsymbol{S}$ given in Eq.
(12).

In 3D the analogous form of Eqs. (28) is the most efficient way to calculate
the continuum functions at energy $E$, since an equivalent form of Eqs. (1) is
not available. The calculations can be done by means of the coupled-channel
renormalized Numerov method [30].

###  3 Sum of time delays

It is shown that the sum of proper ($\overline{\tau}_{\\!j}$) and partial
($\widetilde{\tau}_{\\!j}$) time delays are equal [3]. We will use repeatedly
the unitarity of $\boldsymbol{U}_{\\!s}$ as well as $\boldsymbol{S}$ and the
possibility to change the order of matrices under the trace. Therewith

$\displaystyle\sum_{j}\overline{\tau}_{\\!j}$
$\displaystyle=\sum_{j}q_{j}=\mathrm{tr}(\boldsymbol{U}_{\\!q}{\\!\\!}^{\dagger}\\!\boldsymbol{Q}\boldsymbol{U}_{\\!q})=\mathrm{tr}(\boldsymbol{Q})=-\mathrm{i}\,\mathrm{tr}\big{(}\boldsymbol{S}^{\dagger}\boldsymbol{S}^{\prime}\big{)}$
$\displaystyle=-\mathrm{i}\,\mathrm{tr}\Big{(}\boldsymbol{U}_{\\!s}\boldsymbol{D}[\mathrm{e}^{{-\mathrm{i}\boldsymbol{s}}}]\boldsymbol{U}_{\\!s}{\\!\\!}^{\dagger}\big{(}\boldsymbol{U}_{\\!s}\boldsymbol{D}[\mathrm{e}^{{\mathrm{i}\boldsymbol{s}}}]\boldsymbol{U}_{\\!s}{\\!\\!}^{\dagger}\big{)}^{\prime}\Big{)}$
$\displaystyle=-\mathrm{i}\,\mathrm{tr}\big{(}\boldsymbol{D}[\mathrm{e}^{{-\mathrm{i}\boldsymbol{s}}}]\boldsymbol{D}^{\prime}[\mathrm{e}^{{\mathrm{i}\boldsymbol{s}}}]\big{)}-\mathrm{i}\,\mathrm{tr}\big{(}\boldsymbol{U}_{\\!s}{\\!\\!}^{\dagger}\boldsymbol{U}_{\\!s}^{\prime}+(\boldsymbol{U}_{\\!s}{\\!\\!}^{\dagger})^{\prime}\boldsymbol{U}_{\\!s}\big{)}$
$\displaystyle=\mathrm{tr}\big{(}\boldsymbol{D}[\mathrm{e}^{{-\mathrm{i}\boldsymbol{s}}}]\boldsymbol{D}[\boldsymbol{s}^{\prime}\mathrm{e}^{{\mathrm{i}\boldsymbol{s}}}]\big{)}-\mathrm{i}\,\mathrm{tr}\big{(}(\boldsymbol{U}_{\\!s}{\\!\\!}^{\dagger}\boldsymbol{U}_{\\!s})^{\prime}\big{)}$
$\displaystyle=\sum_{\\!j}s^{\prime}_{\\!j}=\sum_{j}\widetilde{\tau}_{\\!j}.$
(31)

As above, $\boldsymbol{D}[]$ denotes a diagonal matrix.

Note that we have nowhere used the fact that we treat a 1D system with
2$\times$2 matrices. Thus Eq. (3) holds for any dimension.

###  4 Dwell time for a scattering eigenchannel

We assume that $\boldsymbol{v}_{\\!j}$ is an eigenvector of the scattering
matrix $\boldsymbol{S}\boldsymbol{v}_{\\!j}=\boldsymbol{v}_{\\!j}s_{\\!j}$.
Therefrom follows

$\displaystyle s^{\prime}_{\\!j}$
$\displaystyle=\boldsymbol{v}_{\\!j}{}^{\prime}\boldsymbol{S}\boldsymbol{v}_{\\!j}+\boldsymbol{v}_{\\!j}\boldsymbol{S}\boldsymbol{v}^{\prime}_{\\!j}+\boldsymbol{v}_{\\!j}\boldsymbol{S}^{\prime}\boldsymbol{v}_{\\!j}$
$\displaystyle=s_{j}[\boldsymbol{v}_{\\!j}{}^{\prime}\boldsymbol{v}_{\\!j}+\boldsymbol{v}_{\\!j}\boldsymbol{v}^{\prime}_{\\!j}]+\boldsymbol{v}_{\\!j}\boldsymbol{S}^{\prime}\boldsymbol{v}_{\\!j}=\boldsymbol{v}_{\\!j}\boldsymbol{S}^{\prime}\boldsymbol{v}_{\\!j}$
(32)

since $\boldsymbol{v}_{\\!j}\boldsymbol{v}_{\\!j}\,{=}\,1$ and thus the term
in brackets vanishes. Therewith we can calculate the expectation value of
$\boldsymbol{Q}$ for the scattering eigenstate

$\displaystyle\tau_{\\!j}$
$\displaystyle\equiv\boldsymbol{v}_{\\!j}\boldsymbol{Q}\boldsymbol{v}_{\\!j}=-\mathrm{i}\boldsymbol{v}_{\\!j}\boldsymbol{S}^{\dagger}\boldsymbol{S}^{\prime}\boldsymbol{v}_{\\!j}$
$\displaystyle=-\mathrm{i}\;s_{\\!j}\boldsymbol{v}_{\\!j}\boldsymbol{S}^{\prime}\boldsymbol{v}_{\\!j}=-\mathrm{i}\;s_{\\!j}s^{\prime}_{\\!j}=\widetilde{\tau}_{\\!j},$
(33)

where in the 2nd line we have used Eq. (4).

###  5 Matrices, eigenvalues and eigenvectors
in paramterized form

For completeness, the matrices $\boldsymbol{S}$ and $\boldsymbol{Q}$ and their
eigenforms are given in terms of the parametrization (4)

$\displaystyle\boldsymbol{S}$
$\displaystyle=\left(\\!\\!\begin{array}[]{cccc}\mathrm{i}\mathrm{e}^{{\mathrm{i}[\beta+\gamma]}}\sin\alpha&\mathrm{e}^{{\mathrm{i}\beta}}\cos\alpha\\\
\mathrm{e}^{{\mathrm{i}\beta}}\cos\alpha&\mathrm{i}\mathrm{e}^{{\mathrm{i}[\beta-\gamma]}}\sin\alpha\end{array}\\!\\!\right)$
(34) $\displaystyle s_{1,2}$
$\displaystyle=\mathrm{e}^{{\mathrm{i}[\beta+\mbox{atan}(\pm\sqrt{1-\cos^{2}\\!\gamma\sin^{2}\\!\alpha},\cos\gamma\sin\alpha)]}}$
(34c) $\displaystyle\boldsymbol{v}_{\\!1,2}$
$\displaystyle=\frac{1}{N_{s}}\left(\\!\\!\begin{array}[]{cccc}-\sin\alpha\sin\gamma\pm\sqrt{1-\cos^{2}\\!\gamma\sin^{2}\\!\alpha}\\\
\cos\alpha\end{array}\\!\\!\right)$ (34f)

with $N_{s}$ ensuring normalization. Note that the eigenvectors of
$\boldsymbol{S}$ can be chosen real (which is generally not the case for
unitary matrices). And further

$\displaystyle\boldsymbol{Q}$
$\displaystyle=\left(\\!\\!\begin{array}[]{cccc}+\sin^{2}\\!\alpha\,\gamma^{\prime}&\mathrm{e}^{{-\mathrm{i}\gamma}}[\alpha^{\prime}-\mathrm{i}\,\eta]\\\
\mathrm{e}^{{+\mathrm{i}\gamma}}[\alpha^{\prime}+\mathrm{i}\,\eta]&-\sin^{2}\\!\alpha\,\gamma^{\prime}\end{array}\\!\\!\right)$
(35) $\displaystyle\qquad\eta\equiv\cos\alpha\sin\alpha\,\gamma^{\prime}$
$\displaystyle q_{1,2}$
$\displaystyle=\beta^{\prime}-\sqrt{\alpha^{\prime}{}^{2}+\sin^{2}\\!\alpha\gamma^{\prime}{}^{2}}$
(35c) $\displaystyle\boldsymbol{w}_{\\!1,2}$
$\displaystyle=\frac{1}{N_{q}}\left(\\!\\!\begin{array}[]{cccc}\pm\sqrt{\alpha^{\prime}{}^{2}+\sin^{2}\\!\alpha\,\gamma^{\prime}{}^{2}}+\sin^{2}\\!\alpha\,\gamma^{\prime}\\\
\mathrm{e}^{{\mathrm{i}\gamma}}[\alpha^{\prime}+\mathrm{i}\,\cos\alpha\sin\alpha\,\gamma^{\prime}]\end{array}\\!\\!\right)$
(35f)

again with the normalization $N_{q}$ not explicitly given. The unitary
matrices referred to in the text are
$\boldsymbol{U}_{\\!s}\,{=}\,(\boldsymbol{v}_{\\!1}\;\boldsymbol{v}_{\\!2})$
and
$\boldsymbol{U}_{\\!q}\,{=}\,(\boldsymbol{w}_{\\!1}\;\boldsymbol{w}_{\\!2})$.

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# Physics guided machine learning using simplified theories

Suraj Pawar  School of Mechanical & Aerospace Engineering, Oklahoma State
University, Stillwater, OK 74078, USA.  Omer San<EMAIL_ADDRESS>School of
Mechanical & Aerospace Engineering, Oklahoma State University, Stillwater, OK
74078, USA.  Burak Aksoylu  Texas A&M University-San Antonio, Department of
Mathematical, Physical, and Engineering Sciences, San Antonio, TX 78224, USA.
Adil Rasheed  Department of Engineering Cybernetics, Norwegian University of
Science and Technology, 7465 Trondheim, Norway.  Department of Mathematics and
Cybernetics, SINTEF Digital, 7034 Trondheim, Norway.  Trond Kvamsdal
Department of Mathematical Sciences, Norwegian University of Science and
Technology, 7491 Trondheim, Norway.  Department of Mathematics and
Cybernetics, SINTEF Digital, 7034 Trondheim, Norway.

###### Abstract

Recent applications of machine learning, in particular deep learning, motivate
the need to address the generalizability of the statistical inference
approaches in physical sciences. In this letter, we introduce a modular
physics guided machine learning framework to improve the accuracy of such
data-driven predictive engines. The chief idea in our approach is to augment
the knowledge of the simplified theories with the underlying learning process.
To emphasise on their physical importance, our architecture consists of adding
certain features at intermediate layers rather than in the input layer. To
demonstrate our approach, we select a canonical airfoil aerodynamic problem
with the enhancement of the potential flow theory. We include features
obtained by a panel method that can be computed efficiently for an unseen
configuration in our training procedure. By addressing the generalizability
concerns, our results suggest that the proposed feature enhancement approach
can be effectively used in many scientific machine learning applications,
especially for the systems where we can use a theoretical, empirical, or
simplified model to guide the learning module.

Physics guided machine learning, feature engineering, trustworthy artificial
intelligence, generalizibility, hybrid neurophysical modeling

Advanced machine learning (ML) models like deep neural networks are powerful
tools for finding patterns in complicated datasets. However, as these
_universal approximators_ Hornik, Stinchcombe, and White (1989); Hagan and
Menhaj (1994); Hagan, Demuth, and Beale (1997); Vidal  _et al._ (2017) become
complex, the number of trainable parameters (weights) quickly explodes,
adversely affecting their interpretability, and hence their trustworthiness.
Using these models in combination with other traditional models compromises
the trustworthiness of the overall system. While such techniques have been
emerging in both scientific computing and ML fields Krasnopolsky and Fox-
Rabinovitz (2006); Lee and Oh (1997); Tang and Hsieh (2001); Marsland,
Shapiro, and Nehmzow (2002); Raissi, Perdikaris, and Karniadakis (2019); Zhu
_et al._ (2019); Swischuk  _et al._ (2019); Sekar  _et al._ (2019); Beck,
Flad, and Munz (2019); Rahman, Rasheed, and San (2018); Maulik  _et al._
(2019a, b); Pan and Duraisamy (2020); Mohan  _et al._ (2020); Muralidhar  _et
al._ (2020); Qian  _et al._ (2020); Pawar  _et al._ (2020a); Ahmed  _et al._
(2020a, b); Pawar  _et al._ (2020b); Pawar, Ahmed, and San (2020); Pawar and
San (2020); Ahmed  _et al._ (2020c), they offer many opportunities to fuse
topics in domain-specific knowledge, numerical linear algebra, and theoretical
computer science Golub  _et al._ (2006) towards improving the generalizability
of these models. In this letter, we focus on a modular approach for improving
generalizability by augmenting input features with simplified physics-based
models or theories. By the term of _generalizability_ , we refer to the
performance of the learning engine for unseen configurations. Such new
situations are pervasive in scientific applications. In many cases, there is a
grand challenge in making accurate predictions using data-driven methods based
strictly on statistical inference. For example, in fluid mechanics, potential
flow and boundary layer theories often provide simplistic models pertaining to
a certain level of approximations. A remaining task is how to combine these
models for making more generalizable learning architectures. Therefore, our
approach provides intermediate features based on approximate physics to
statistical models that can relate or fulfill the gap between these simplified
theories and more accurate models (i.e., experimental or high fidelity
computational fluid dynamics simulation data sets).

Figure 1: Physics guided machine learning (PGML) framework to train a learning
engine between processes $A$ and $B$: (a) a conceptual PGML framework which
shows different ways of incorporating physics into machine learning models.
The physics can be incorporated using feature enhancement of the ML model
based on the domain knowledge, embedding simplified theories directly into ML
models, and corrector approach in which the output of the ML model is
constrained using the governing equations of the system, and (b) an overview
of the typical neural network architecture for the PGML framework.

A wide variety of engineering tasks such as optimal control, design
optimization, uncertainty quantification requires the prediction of the
quantity of interest in real-time. In such scenarios, the partial differential
equation (PDE) based discretization methods are not feasible as they can take
days or weeks to simulate different configurations. Reduced order models
(ROMs) are the state-of-the-art models that construct the basis from the past
data, and then solve the governing equations after projecting them on the low
dimensional manifold Lucia, Beran, and Silva (2004); Taira  _et al._ (2020);
Fonn  _et al._ (2019). The main limitation of such projection based intrusive
ROMs is that it requires a complete description of the dynamics of the system,
and often this information is unavailable or inadequate for the desired
application. Recently model-free prediction using machine learning has proven
successful for many physical systems Pathak  _et al._ (2018); White, Ushizima,
and Farhat (2019); Geneva and Zabaras (2020). One of the main challenges with
these non-intrusive models is the prediction for unseen data and its
interpretation. Even though probabilistic machine learning methods like
Bayesian neural networks can give the uncertainty bound on the predicted state
Zhu  _et al._ (2019); Maulik  _et al._ (2020), the generalizability of non-
intrusive models is not on par with physics-based models. On the other hand,
the simplified models like the lumped-capacitance model for heat transfer
analysis, and Blasius boundary layer model in fluid mechanics are highly
generalizable to different conditions. Therefore, it is important to leverage
the knowledge of physical systems into learning, and to this end, we make
progress towards physics guided machine learning by embedding features from
simplified physical models into the neural network architecture. The proposed
framework is flexible enough to be applied to many physical systems and it has
great potential in scientific machine learning.

We now introduce different components of the physics guided machine learning
(PGML) framework as depicted in Figure 1(a). In supervised machine learning,
the input vector $\mathbf{x}\in\mathbb{R}^{m}$ is fed to the machine learning
model (for example, the neural network in our case), and the mapping from the
input vector to output vector $\mathbf{y}\in\mathbb{R}^{n}$ is learned through
training. The neural network is trained to learn the function $F_{\theta}$,
parameterized by $\theta$, that includes the weights and biases of each
neuron. The function $F_{\theta}$ should be such that it approximates the
known labels and the cost function is minimized. Usually, for the regression
problems, the cost function is the mean squared error between true and
predicted output, i.e.,
$C(\mathbf{x},\theta)=||\mathbf{y}-F_{\theta}(\mathbf{x})||_{2}$. The weights
and biases of the neural network are optimized using any gradient-descent
algorithm like stochastic gradient descent. In the PGML framework, the neural
network is augmented with the output of the simplified physics-based model.
The features extracted from simplified physics-based models can be either
combined with input features, or they can be merged into hidden layers along
with latent variables. During the training, the PGML framework is provided
with $(\mathbf{x},G(\mathbf{x}))$, where $G$ is the model based on simplified
theories for the problem at hand, and the parameters $\theta$ are optimized
based on the true output $\mathbf{y}$. The features from the simplified model
$G$ can also be fed at an intermediate hidden layer along with learned latent
representation.

The PGML framework allows us to extract the knowledge from the simplified
physics-based model and combine it with the latent variables of the system
discovered by the neural network at intermediate hidden layers. This
information from the physics-based model assists the neural network in
constraining its output to a manifold of physically realizable models. It
encourages the neural network to learn the physically consistent
representation of the quantity of interest drawn from complex distributions
such as pressure and velocity field of fluid simulations. Another advantage of
the PGML framework is that it brings interpretability to otherwise black-box
models. We highlight here that the PGML framework allows us to incorporate the
physics of the problem even during the prediction stage, and not just the
training as in the case of approaches like regularization based on governing
equations.

In a nutshell, we highlight that there might be a handful of simplified models
(e.g., similarity solutions, lumped parameter models, zone models, zero or
one-dimensional models, etc) to approximate or describe the underlying
physical processes in many disciples. For example, in aerodynamics, the use of
simplistic panel methods is a proven approach for analysis of inviscid flow
over bodies, especially for a smaller angle of attacks where the potential
flow theory becomes valid. The execution time for these simplified models is
significantly smaller compared to the run time needed for full-fledged CFD
simulations. Therefore, in the PGML method, we propose to fuse the knowledge
coming from such simplified theories in a statistical inference architecture.
This is accomplished by a feature enhancement procedure as described in Figure
1(b). Specifically, it constitutes a predictor-corrector philosophy where we
first run a simplified theory (computationally less demanding) to estimate an
intermediate prediction as an input for the overall ML architecture. We then
estimate the quantity of interest based on the enhancement procedure
abstracted by $G(\mathbf{x})$. We hypothesize that the predictive performance
and generalizability of statistical inference will be significantly improved
by the involvement of $G(\mathbf{x})$ as illustrated with the aerodynamic
performance prediction task.

_Results and Discussion_ — We demonstrate the PGML framework using the
aerodynamic performance prediction task. This is a problem relevant to many
applications such as predictive controlZha  _et al._ (2007), and design
optimizationLeGresley and Alonso (2000). The prediction of flow around an
airfoil is a high-dimensional, multiscale, and nonlinear problem that can be
solved using high-fidelity methods like computational fluid dynamics (CFD).
However, these methods are computationally intractable as the design space
exploration increases. In certain flow regimes, the simplified methods like
panel codes come with a non-negligible difference between the actual dynamics
and approximate models for real-world problems. The full-order CFD simulations
typically take extensive computational run time, thus limiting their use in
many inverse modelling methodologies that require a model run to be performed
in each iteration. To overcome these challenges, combining CFD models with
machine learning to build a data-driven surrogate model is gaining widespread
popularityZhang, Sung, and Mavris (2018); Bhatnagar  _et al._ (2019); Rajaram
_et al._ (2020). In this example, we exploit the relevant physics-based
features from panel methods through the PGML framework to enhance the
generalizability of data-driven surrogate models.

The training data for the neural network is generated using a series of
numerical simulations. The main goal of this work is to demonstrate how a
simplified model can be used in the PGML framework and therefore, to reduce
the computational complexity, we utilize XFOILDrela (1989) for the aerodynamic
analysis of different airfoils instead of full CFD simulation to get the most
accurate results for the forces on the airfoil. The lift coefficient data were
obtained for different Reynolds numbers between $1\times 10^{6}$ to $4\times
10^{6}$ and several angles of attacks in the range of $-20$ to $+20$. A total
of 168 sets of two-dimensional airfoil geometry were generated for training
the neural network. This includes NACA 4-digit, NACA210, NACA220, and NACA250
series, and each airfoil is represented by 201 points. The maximum thickness
of all airfoils in the training dataset was between 6% to 18% of the chord
length. We use the NACA23012 and NACA23024 airfoil geometry as the test
dataset to evaluate the predictive capability of the trained neural network.
The simplified model used to generate the physics-based feature corresponds to
the Hess-Smith panel methodHess (1990) based on potential flow theory. We
highlight that our testing airfoils are selected not only from a different
NACA230 series (i.e., not used in the training dataset) but also the maximum
thickness of 24% is well beyond the thickness ratio limit included in our
training data set.

Figure 2: The representative neural network architecture of the PGML framework
used in this study for aerodynamic forces prediction task. The latent
variables at the third hidden layers are augmented with the physical
parameters of the flow (i.e., the Reynolds number and angle of attack) and the
prediction from the Hess-Smith panel method (i.e., lift coefficient and
pressure drag coefficient denoted as $\bar{C_{L}}$ and $\bar{C_{DP}}$,
respectively).

The neural network architecture used in this study has four hidden layers with
20 neurons in each hidden layer. The physical parameters, i.e., the Reynolds
number and the angle of attack are concatenated at the third hidden layer
along with the latent variables at that layer. In the PGML model, we augment
the latent variables at the third layer with the lift coefficient and the
pressure drag coefficient predicted by the panel method along with physical
parameters of the flow (i.e., the Reynolds number and angle of attack).
Therefore, the third layer of the PGML neural network has 24 latent variables.
The representative neural network for the PGML framework to predict the
aerodynamic forces on an airfoil is displayed in Figure 2. We utilize ensemble
of neural networks to quantify the predictive uncertaintyTibshirani (1996);
Heskes (1997); Lakshminarayanan, Pritzel, and Blundell (2017). In this method,
a small number of neural networks are trained from different initialization
and the variance of the ensemble’s prediction is interpreted as the epistemic
uncertainty. This method is appealing due to its simplicity, scalability, and
strong empirical results of the uncertainty estimate that are as good as the
Bayesian neural networksLakshminarayanan, Pritzel, and Blundell (2017). We
train multiple neural network models using different values of the initial
weights and biases. The weights and biases of each model are initialized using
the Glorot uniform initializer and different random seed numbers are used to
ensure that the different values of weights and biases are assigned for each
model. The ensemble of all these models indicates the model uncertainty
estimate of the predicted lift coefficient. Figure 3 shows the actual and
predicted lift coefficient for the NACA23012 and NACA23024 airfoil geometry.
The reference _True_ performance is obtained by XFOIL. The ML corresponds to a
simple feed-forward neural network that uses the airfoil $x$ and $y$
coordinates as the input features, and the physical parameters of the flow are
concatenated at the third hidden layer along with the latent variables at that
layer.

Figure 3: Actual versus predicted lift coefficient ($C_{L}$) for NACA23012 and
NACA23024 airfoils at $Re=3\times 10^{6}$ using ML and PGML framework. The
dashed red curve represent the average of the predicted lift coefficient by
all ML models (i.e., testing runs with different initialization seeds). Both
airfoil geometries were not included in the training. Here we note that there
is a significant reduction of uncertainty in performance when we use the
proposed PGML framework, especially for the smaller angle of attacks where the
embedded simplistic physics-based model is valid. The physics embedding in
these tests are based on the utilization of the Hess-Smith panel method, which
is limited mostly for the angle of attacks between -10 and +10 degrees, which
further verifies the success of the proposed PGML framework.

As shown in Figure 3, we can see that the uncertainty in the prediction of the
lift coefficient by the PGML model is less than the ML model for both
NACA23012 and NACA23024 airfoils. The proposed PGML framework provides
significantly more accurate predictions with reduced uncertainty for the angle
of attacks between -10 and +10 degrees. This further illustrates the viability
of the proposed PGML framework, since the physics embedding considered here
employs constant source panels and a single vortex to approximate the
potential flow around the airfoil. We can also notice that the uncertainty is
higher for the angle of attacks outside the range of -10 to +10 degrees. This
finding is not surprising as the Hess-Smith panel method is a proven method
for analysis of inviscid flow over airfoil for the smaller angle of attacks
regime. We highlight that the maximum thickness of an airfoil included in the
training dataset is 18% of the chord length. Therefore, the uncertainty in the
prediction of the lift coefficient by the ML model is higher for the NACA23024
airfoil compared to the NACA23012 airfoil. The inclusion of physics-based
features from the panel method in the PGML model leads to a reduction in this
uncertainty estimate. These results clearly show the advantage of augmenting
the neural network model with simplified theories and the potential of the
PGML framework for physical systems. One of the important caveats with any
neural network is its design and the hyperparameters. The neural architecture
search and hyperparameter optimization are important processes for the success
of the PGML framework. If the network is shallow, then it has less expressive
capabilities and that deteriorates the prediction. On the other hand, if the
neural network architecture is very deep and there is no sufficient training
data, then the network fits the training data very well. However, its
generalizability is reduced and this is usually indicated by an increase in
the loss of the validation dataset after a few iterations of the training.
Some of the strategies to mitigate overfitting issues in the deep neural
network are using $L_{1}/L_{2}$ regularization, applying dropout, early
stopping, data augmentation, and $k-$fold cross-validation. Overall, our
findings suggest that the absorption of physical information into ML methods
improves the modeling uncertainty of the ML architectures.

_Concluding Remarks_ — The data-driven models derived from the data collected
from satellite measurements, internet of things (IoT) devices, and numerical
simulation are increasingly being applied for scientific applications in fluid
dynamics. While these data-driven models are successful, it might be vital to
complement them with physical laws that have been studied for many decades. To
this end, physics-informed machine learning approaches such as embedding soft
and hard constraints designed based on governing laws of the physical system,
have been proposed Raissi, Perdikaris, and Karniadakis (2019); Zhu  _et al._
(2019); Pan and Duraisamy (2020); Mohan  _et al._ (2020). In this work, we
illustrated how physics-based models derived from simplified approximations of
the physical system can be coupled within data-driven models like neural
networks. The PGML framework introduced in this study allows us to take
advantage of the generalizability of physics-based simplistic models and the
robustness of data-driven models. We demonstrated the proof-of-concept for the
aerodynamic performance analysis of airfoil geometry. The physics-based
features computed from the simplistic panel method are coupled with the latent
representation learned at the intermediate layers of the neural network. The
inclusion of these physics-based features assists the neural network model in
reducing the uncertainty of the lift coefficient prediction. Therefore, the
PGML framework is successful in improving the generalizability of data-driven
models.

Additionally, the PGML framework will also be useful for physical systems
where the data is scarce. For example, the generation of training data is
computationally expensive when dealing with larger problems such as the
optimization of a wind farm layout Samorani (2013) or the shape optimization
of a three-dimensional wing Epstein  _et al._ (2009). The simplified models
like the actuator disk theory can be used along with a velocity field
generated from few high-fidelity numerical simulations to build a physics-
guided neural network-based surrogate model of the wind farm that can be
coupled with any optimization algorithm. Another area where the PGML framework
may bear huge potential is the digital twin technologies for physical
systemsRasheed, San, and Kvamsdal (2020), where the digital replica of the
physical system is build for tasks such as real-time control, efficient
operation, and accurate forecast of maintenance schedules. In the present
study, we started with XFOIL to assess the feasibility of the proposed
framework. Despite its simplicity, this is the first step towards
demonstrating the PGML framework for problems in fluid dynamics. In our future
work we plan to extend the PGML approach to complex and high-dimensional
problems like prediction of terrain induced atmospheric boundary layer flows
or flows around bluff bodies immersed in fluid to illustrate the true
capability of the PGML framework.

This material is based upon work supported by the U.S. Department of Energy,
Office of Science, Office of Advanced Scientific Computing Research under
Award Number DE-SC0019290. O.S. gratefully acknowledges their support.

The data that supports the findings of this study are available within the
article. Implementation details and Python scripts can be accessed from the
Github repositoryPawar (2020).

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