# Asymptotic Approximant for the Falkner-Skan Boundary-Layer equation

Elizabeth R. Belden1, Zachary A. Dickman1, Steven J. Weinstein1, Alex D.
Archibee2, Ethan Burroughs3, Nathaniel S. Barlow3 1 Department of Chemical
Engineering, Rochester Institute of Technology, Rochester, NY 14623  2
Department of Mechanical Engineering, Rochester Institute of Technology,
Rochester, NY 14623  3 School of Mathematical Sciences, Rochester Institute of
Technology, Rochester, NY 14623<EMAIL_ADDRESS>
###### Abstract

We demonstrate that the asymptotic approximant applied to the Blasius boundary
layer flow over a flat plat (Barlow et al, 2017 Q. J. Mech. Appl. Math., 70
(1): 21-48) yields accurate analytic closed-form solutions to the Falkner-Skan
boundary layer equation for flow over a wedge having angle $\beta\pi/2$ to the
horizontal. A wide range of wedge angles satisfying $\beta\in[-0.198837735,1]$
are considered, and the previously established non-unique solutions for
$\beta<0$ having positive and negative shear rates along the wedge are
accurately represented. The approximant is used to determine the singularities
in the complex plane that prescribe the radius of convergence of the power
series solution to the Falkner-Skan equation. An attractive feature of the
approximant is that it may be constructed quickly by recursion compared with
traditional Padé approximants that require a matrix inversion. The accuracy of
the approximant is verified by numerical solutions, and benchmark numerical
values are obtained that characterize the asymptotic behavior of the Falkner-
Skan solution at large distances from the wedge.

##  1 Introduction

The Falkner-Skan equation describes boundary layer flow over a wedge of angle
$\beta\pi/2$ to the horizontal that is driven by an external pressure gradient
predicted from potential flow (see Fig. 1). The equation also applies to
regimes where the pressure gradient opposes the flow when $\beta<0$ (Fig.
1c,d) for which boundary layer separation may occur. Through a similarity
transform, the governing 2D boundary-layer equations can be written as a
nonlinear ODE system for the dimensionless stream function, $f$, as a function
of a similarity variable $\eta(x,y)$ [1]:

$\displaystyle f^{\prime\prime\prime}+ff^{\prime\prime}+\beta(1-f^{\prime
2})=0$ $\displaystyle f(0)=0,f^{\prime}(0)=0,f^{\prime}({\infty})=1.$ (1)

Once (1) is solved for $f(\eta)$, the dimensional stream function,
$\psi(x,y)$, may be extracted as [1]

$\eta=y\sqrt{\frac{m+1}{2}\frac{U}{\nu x}}$ (2) $\psi(x,y)=\sqrt{\frac{2\nu
Ux}{m+1}}f(\eta)$

where $x$ and $y$ are coordinates along and perpendicular to the wedge surface
(see Fig. 1), $m=\beta/(2-\beta)$, $\nu$ is kinematic viscosity, and $U=U(x)$
is the velocity at the wedge surface determined from potential flow, as
indicated in Fig. 1. Although a power series solution can be found for (1), it
diverges at a finite value of $\eta$ for all physical values of wedge angle
$\beta$. The lack of an exact analytical solution has necessitated several
numerical [2, 3, 4, 5, 6, 7, 8] and approximate analytical [9, 10, 11]
solution approaches to the system (1). Here, we implement the recent method of
asymptotic approximants to analytically continue the power series solution,
and thereby construct a highly accurate closed form solution to (1).

Figure 1: Geometry of flow over a wedge showing potential flow and viscous
boundary layer regions. a) Domain for the potential flow solution for
$\beta>0$, from which the nonzero velocity $U(x)$ and associated Bernouli
pressure $P(x)$ along $y=0$ is extracted. b) Domain for the thin viscous
boundary layer (Falkner-Skan problem) that is driven by the potential flow
solutions for $U(x)$ and $P(x)$. There are no pressure variations in the
$y$-direction. On the length scale of figure a) the boundary layer cannot be
observed. The analogous wedge configuration for $\beta<0$ are shown in Figures
c) and d).

Asymptotic approximants, originally developed for divergent series problems of
thermodynamics [12, 13, 14], may be constructed when asymptotic behaviors are
known in two different regions of a domain; implementation details are given
in [15]. The method is a generalization of two well-known mathematical
techniques: asymptotic matching and Padé approximants [16]. Whereas asymptotic
matching leads to a single expression that combines two overlapping asymptotic
expansions, asymptotic approximants do not require that the two series
overlap; if the two series diverge before overlap, an asymptotic approximant
has the ability to analytically continue each series and accurately fill-in
the gap between them. In the same way as Padé approximants, asymptotic
approximants are constructed such that the series expansion of the approximant
about a given point is the same as that of the true expansion about that
point. However, whereas Padé approximants are restricted to rational functions
and thus have a specific asymptotic behavior about a chosen expansion point,
asymptotic approximants are tailor-made to have the correct behavior in both
regions of the domain. For example, asymptotic approximants are shown to
accurately describe the light trajectory around a Kerr black hole [17, 18] by
incorporating the correct logarithmic behavior near the black hole; a Padé is
incapable of representing such behavior efficiently. Asymptotic approximants
are used to construct accurate solutions for boundary flows over a stationary
flat plate (the Blasius problem) and for a flat plate moving through a
stationary fluid (the Sakiadis problem) [15]. The Blasius approximant uses a
power series expansion about $\eta=0$ and the leading-order behavior as
$\eta\to\infty$; the full asymptotic behavior in this latter regime is not
needed to obtain a highly accurate solution. The leading-order $\eta\to\infty$
behavior of Blasius is exactly the same as that for the Falkner-Skan problem
(1) for flow over a wedge. Thus the asymptotic approximant used in [15] for
the Blasius problem may be applied to the Falkner-Skan equation. In this work,
we demonstrate that this may be done with high accuracy.

The paper is organized as follows. In Section 2, the series solution of (1)
and numerically-obtained asymptotic properties are provided as a function of
wedge angle. The Blasius asymptotic approximant of [15] is applied to the
Falkner-Skan equation in Section 3; it is validated against numerical
solutions for key values of wedge angle. The Blasius-type approximant is found
to be accurate for positive and negative shear rates at the wedge surface over
a significant range of wedge angles – the range of $\beta$ encompasses all
physical configurations for which there is no boundary layer separation. An
alternative asymptotic approximant is introduced to handle non-monotonic
solutions to the Falkner-Skan equation that occur over a small range of
negative shear rates at the wedge surface for $\beta<0$. In Section 3.1, the
Blasius-type approximant is used to extract the radius of convergence of the
Falkner-Skan series as a function of wedge angle; to the authors’ knowledge,
this is the first time this dependence has been disclosed. Concluding remarks
are made in section 4.

##  2 Asymptotic properties and series expansion of the Falkner-Skan equation

Solutions to the Falkner-Skan equation system (1) are found for a given flow
by fixing the parameter $\beta$, which is related to the wedge angle shown in
Fig. 1. Note that this parameter incorporates the effect of wedge angle on the
potential flow that drives the fluid motion in the boundary layer. Although
(1) is a boundary value problem, a series solution may be generated if the
infinity boundary condition is relaxed, and the dimensionless velocity
gradient at the wedge surface, $\kappa(\beta)$, is applied at $\eta=0$ as

$f^{\prime\prime}(0)=\kappa(\beta).$ (3)

Note that the function $\kappa(\beta)$ is determined such that the
$\eta\to\infty$ boundary condition in (1) is applied, and is typically
determined numerically. Fig. 2a shows the dependence of $\kappa$ on the wedge
angle parameter $\beta$; the figure is constructed by implementing the
shooting method of [2]. Key values in Fig. 2a are as follows. When $\beta$=0
(point II in Fig. 2a) there is no pressure gradient in the physical system,
and the system (1) describes the classical Blasius problem for flow along a
flat plate111Note that the common form of the Blasius equation includes a
coefficient of 1/2 in front of the $ff^{\prime\prime}$ term, as a result of
omitting the factor of 2 in the denominator of (2) when deriving (1) from the
governing equations with $\beta=0$ [1]. A ramification of this is that the
values reported here are $\sqrt{2}\times$ the literature value of
$\kappa$=0.3320573362152 and $1/\sqrt{2}\times$ the literature value of
$B$=$-$1.72078765752 [19, 20].. When $\kappa$=0 (Point IV in Fig. 2a,
$\beta\approx-0.1988$) the velocity gradient along the wedge is zero, and this
condition corresponds to the onset of boundary layer separation [1]. As shown
in Fig. 2a, there are multiple solutions to (1) for $\beta<$0\. Negative
values of $\kappa$ (e.g., Points V and VI in Fig. 2a) correspond to solutions
of (1) that contain a region where $f^{\prime}$ is negative before eventually
switching sign to tend towards the asymptotic behavior $f^{\prime}\to 1$,
given in (1); this is referred to as a region of “reversed flow” by Stewartson
[21].

(a)

(a)

(b)

(b)

Figure 2: Solution properties as a function of the wedge angle parameter,
$\beta$, obtained from the Falkner-Skan system (1). (a) Dimensionless velocity
gradient, $\kappa=f^{\prime\prime}(0)$, used to construct the series expansion
(2). (b) Asymptotic constant $B$ defined according to (5). The constants
$\kappa$ and $B$ are determined numerically using the shooting method of [2]
with marching done using the 4th-order Runge-Kutta method with a step size of
$10^{-4}$. Specific values of $(\beta,\kappa,B)$ associated with the roman
numerals in the plots are as follows: I. (0.5, 0.927680039836653,
$-$0.804548615), II. (0, 0.469599988361, $-$1.21678062) III. ($-$0.12,
0.28176052424, $-$1.511343148) IV. ($-$0.198837735, 0, $-$2.359), V. ($-$0.12,
$-$0.1429351943576, $-$4.3989990662), VI. ($-$0.02, $-$0.065168585542904,
$-$9.18639214). Convergence has been established to within the digits reported
above by successively increasing the domain length ($\eta\in[0,L]$, $L$=10,
20, 30, 40) of the shooting method. A tabulated version of this data is
provided in Supplementary Material [22].

The series solution to (1) may be obtained [23] as

$f=\sum_{n=0}^{\infty}a_{n}{\eta}^{n},$ (4a)

where

$a_{n+3}=\frac{\displaystyle\sum_{j=0}^{n}\beta(j+1)(n-j+1)a(j+1)a(n-j+1)-(j+1)(j+2)a_{j+2}a_{n-j}}{(n+1)(n+2)(n+3)}.$
(4b)

The recursion above requires knowledge of the first three coefficients in
order to generate the full series. The first two coefficients $a_{0}=f(0)$=0
and $a_{1}=f^{\prime}(0)$=0 are given by the first two boundary conditions in
(1). The third coefficient $a_{2}=f^{\prime\prime}(0)/2=\kappa/2$ is taken
from the numerical solution described above and shown in Fig. 2a. As evidenced
by Figs. 3a through 7a, the utility of the series (2) is limited for all
values of beta as it has a finite radius of convergence. The dependence of the
radius of convergence on $\beta$ is investigated in Section (3.1).

The far-field condition $f^{\prime}(\infty)=1$ appearing in (1) holds true for
all wedge angles $\beta$, as does the admission of an integration constant in
this limit, such that the leading-order asymptotic behavior may be written as
[24]

$\lim_{\eta\to\infty}\left(f-\eta\right)\equiv B(\beta)$ (5)

where $B$222Note that the parameter $B$ is the negative of the displacement
thickness defined in terms of the similarity variables as
$\delta_{1}=\int_{0}^{\infty}\left(1-f^{\prime}\right)d\eta$ [21]. In this
paper, we define the quantity $B$ such that (5) is consistent with the
notation typically used for the Blasius problem ($\beta=0$) [19]. is a
function of wedge angle $\beta$; the dependence is shown in Fig. 2b. Both
Figs. 2a and 2b are constructed by implementing the numerical shooting method
of [2], replacing $\infty$ in (5) with a finite domain length $L$, calculating
$\kappa$ and $B\approx[f(\eta=L)-L]$, and recomputing the numerical solution
for successively increased $L$ until $\kappa$ and $B$ are converged to within
the tolerance reported in the figure caption and in [22]. Although presented
in Fig. 2 using newly generated numerical solutions, the functionality of both
$\kappa$ and $B$ on $\beta$ has long been known [21].

##  3 Asymptotic Approximant

The divergence of the Falkner-Skan series (2) demonstrated in Figs. 3a through
9a is overcome using the method of asymptotic approximants, which constrains
the analytic continuation of the series via an asymptotic behavior away from
the point of expansion [15]. An approximant that satisfies the $\eta\to\infty$
behavior (5) is given as

$f_{A}=\eta+B-B\left(1+\sum_{n=1}^{N}A_{n}~{}\eta^{n}\right)^{-1},$ (6a)

where the $A_{n}$ coefficients are chosen such that the expansion of (6a)
about $\eta=0$ is exactly (2) up to $N^{\mathrm{th}}$ order. Note that the
above form is not a Padé approximant, in that if one combines the terms of
(6a) through a common denominator, the coefficients of the numerator will have
an explicit dependence on those in the denominator – this is not the case for
standard Padés. The form of the approximant (6a) is used in [15] for $\beta=0$
to generate accurate solutions of the Blasius boundary layer problem for flow
over a flat plate.

A recursion for the coefficients $A_{n}$ in (6a) is obtained as follows. The
$N$-term Taylor expansion of (6a) about $\eta=0$ is set equal to the $N$-term
truncation of (2). Then, taking the limit $N\to\infty$, and re-arranging
yields

$\sum_{n=0}^{\infty}\tilde{a}_{n}\eta^{n}=-B\left(\sum_{n=0}^{\infty}A_{n}\eta^{n}\right)^{-1},$

where (noting that $a_{0}=a_{1}=0$) $\tilde{a}_{0}=-B$, $\tilde{a}_{1}=-1$,
$\tilde{a}_{2}=a_{2}=\kappa/2$, and $\tilde{a}_{j>2}=a_{j}$ (given by (4b)).
Multiplying both sides of the above by the $A_{n}$ series and applying the
well-known identity for the Cauchy product of two series [25], the expression
becomes

$\sum_{n=0}^{\infty}\left(\sum_{j=0}^{n}\tilde{a}_{j}A_{n-j}\right)\eta^{n}=-B.$

Noting that $\tilde{a}_{0}=-B$, the above can be separated as

$-BA_{0}+\sum_{n=1}^{\infty}\left(-BA_{n}+\sum_{j=1}^{n}\tilde{a}_{j}~{}A_{n-j}\right)\eta^{n}=-B.$

Equating like terms of $\eta^{0}$ and $\eta^{n\neq 0}$ in the expression
above, we arrive at the following

$A_{0}=1,$
$A_{n>0}=\frac{1}{B}\sum_{j=1}^{n}\tilde{a}_{j}~{}A_{n-j},~{}\tilde{a}_{1}=-1,~{}\tilde{a}_{j>1}=a_{j},$
(6b)

such that now all $A_{n}$ coefficients may be computed recursively. Note that
the result (6b) is identical to that provided in previous work [15] for the
Blasius solution ($\beta=0$), but here we provide the intermediate steps for
purposes of clarity.

The approximant (3) is compared with the series and numerical solutions in
Figs. 3 through 7 for the specific values of $\beta$ enumerated as I through V
in Fig. 2 (case VI is discussed separately). As is evident in the figures,
curves for $f_{A}$ approach a final curve shape as $N$ increases. In fact, a
key property of a valid approximant is that, upon increasing the order $N$, a
convergent sequence of approximants is obtained in the sense of Cauchy. The
justification for the choice of approximant form is this convergent behavior.
Since all derivatives of the approximant can be computed exactly, it is not
surprising that derivatives of the approximant are in excellent agreement with
the numerical solution, as shown in Figs. 3b through 7b. The advantage that
the analytic form provided by the approximant has over the numerical solution
is that highly-resolved domains of arbitrary length can be obtained without
(a) interpolation or extrapolation of the numerical solution or (b) re-running
the numerical solution at higher computational cost.

(a)

(a)

(b)

(b)

Figure 3: (a) The $N$-term series (2) labeled S$N$ and approximant (3) labeled
A$N$ compared with numerical solution ($\bullet$). (b) Derivatives of
approximant (3) for $N$=20. Data shown here corresponds to conditions at point
I in Fig. 2 ($\beta=0.5$, $\kappa=0.927680039836653$).

(a)

(a)

(b)

(b)

Figure 4: (a) The $N$-term series (2) labeled S$N$ and approximant (3) labeled
A$N$ compared with numerical solution ($\bullet$). (b) Derivatives of
approximant (3) for $N$=20. Data shown here corresponds to conditions at point
II in Fig. 2 ($\beta=0$, $\kappa=0.469599988361$). These results correspond to
the classical boundary layer solution of Blasius.

(a)

(a)

(b)

(b)

Figure 5: (a) The $N$-term series (2) labeled S$N$ and approximant (3) labeled
A$N$ compared with numerical solution ($\bullet$). (b) Derivatives of
approximant (3) for $N$=20. Data shown here corresponds to conditions at point
III in Fig. 2 ($\beta=-0.12$, $\kappa=0.28176052424$). This solution is not
unique – there is another solution for the same value of $\beta$ corresponding
to point V in Fig. 2, as shown in Fig. 7.

(a)

(a)

(b)

(b)

Figure 6: (a) The $N$-term series (2) labeled S$N$ and approximant (3) labeled
A$N$ compared with numerical solution ($\bullet$). (b) Derivatives of
approximant (3) for $N$=20. Data shown here corresponds to conditions at point
IV in Fig. 2 ($\beta=-0.198837735$, $\kappa=0$). These results correspond to
the conditions at the onset of boundary layer separation, for which the shear
rate at wedge surface is zero.

(a)

(a)

(b)

(b)

Figure 7: (a) The $N$-term series (2) labeled S$N$ and approximant (3) labeled
A$N$ compared with numerical solution ($\bullet$). (b) Derivatives of
approximant (3) for $N$=20. Data shown here corresponds to conditions at point
V in Fig. 2 ($\beta=-0.12$, $\kappa=-0.1429351943576$). This solution is not
unique – there is another solution for the same value of $\beta$ corresponding
to point III in Fig. 2, as shown in Fig. 5.

Convergence of approximant (3) is shown on a more sensitive scale in Fig. 8a
for $\beta=0.5$, where the absolute error between the approximant and the
numerical solution is shown for different $N$ values. The convergence behavior
shown in the figure is representative of other $\beta$ values within the range
examined here. The infinity norm of the error for $\eta\in[0,14]$ is shown in
Fig. 8b over the full range of $\beta$. For the cases of positive $\kappa$
(filled symbols of Fig. 8b), convergence of approximant (3) is established for
all $\beta$, such that accurate solutions are obtained. For the cases of
negative $\kappa$ (open symbols of Fig. 8b), convergence is apparent but is
increasingly limited as $\beta\to 0^{-}$. Note from Fig. 8b that the overall
error increases in this limit, and approximant (3) becomes worse at
representing the solution. This limited convergence is demonstrated in Fig. 9
for $\beta=-0.02$ ($\kappa<0$, point VI in Fig. 2), where approximant (3) is
shown to poorly represent the solution for intermediate values of $\eta$; an
improvement is discussed below.

(a)

(a)

(b)

(b)

Figure 8: Error associated with approximant (3), measured against numerical
solutions with truncation error of O(10-16). (a) Absolute error in approximant
(3) for $\beta$=0.5. (b) Infinity norm (taken over $0\leq\eta\leq 14$) of the
absolute error associated with the approximant for $-0.198837735\leq\beta\leq
2$. The multivalued solutions for $\beta<0$ are represented here; the filled
and open symbols correspond to $\kappa>0$ and $\kappa<0$ solutions,
respectively.

(a)

(a)

(b)

(b)

Figure 9: (a) The $N$-term series (2) labeled S$N$ and approximant (3) labeled
A$N$ compared with numerical solution ($\bullet$). (b) Derivatives of
approximant (3) for $N$=30. Data shown here corresponds to conditions at point
VI in Fig. 2 ($\beta=-0.02$, $\kappa=-0.065168585542904$).

The monotonic or nearly-monotonic solutions in $f(\eta)$ shown in Figs. 3
through 7 (representing cases I through V in Fig. 2) enable the simple form of
approximant (3) to provide accurate solutions over the physical domain, from
$\eta=0$ towards “large” $\eta$ such that $f^{\prime}\to 1$. As $\beta$
approaches 0 from the left for $\kappa<0$, $f(\eta)$ becomes less monotonic
and a minima manifests prior to reaching the $f^{\prime}\to 1$ large $\eta$
behavior, as shown in Fig. 9 for $\beta=-0.02$ (point VI in Fig. 2). As
mentioned above, approximant (3) is incapable of correctly resolving this
minima in $f(\eta)$. To allow for more flexible curve shapes, the approximant
is adjusted to have a cubic form in its numerator as follows:

$f_{A}={\eta}+B-B\left[\frac{A_{0}+A_{1}{\eta}+A_{2}{\eta}^{2}+A_{3}{\eta}^{3}}{1+\displaystyle{\sum_{n=1}^{N-3}}d_{n}{\eta}^{n}}\right],~{}~{}N>3.$
(7)

Like (3), approximant (7) satisfies the large $\eta$ behavior (5) by
construction. The unknown coefficients $A_{0}\dots A_{3}$ and $d_{n}$ in (7)
are chosen such that the expansion of (7) about $\eta$=0 is equal to the
series given by (2) to order $N$. Note that the term in square brackets in (7)
is a [3/N-3] Padé approximant, for which solvers are readily available [26] to
handle the required matrix inversion. One may compute the $A_{0}\dots A_{3}$
and $d_{n}$ coefficients by isolating the bracketed term of (7) onto one side
of the equation and finding the [3/N-3] Padé for the power series
$1+\left(\eta-\sum_{n=0}^{N}a_{n}\eta^{n}\right)/B$. As before, the constants
$\kappa$ and $B$ are taken from the numerical results shown in Fig. 2 and are
inputs to the approximant. Fig. 10 shows the convergence of approximant (7)
applied to $\beta=-0.02$ ($\kappa<0$, point VI in Fig. 2), and demonstrates a
significant improvement over approximant (3) (see Fig. 9 for comparison). Note
that, although a Padé solver can be used, (7) is not a standard Padé as they
are defined today – it is a Padé as it was originally intended by Baker &
Gammel [27], constructed such that it is consistent with the correct
$\eta\to\infty$ limit.

(a)

(a)

(b)

(b)

Figure 10: (a) The $N$-term series (2) labeled S$N$ and approximant (7)
labeled A$N$ compared with numerical solution ($\bullet$). (b) Derivatives of
approximant (7) for $N$=25. Data shown here corresponds to conditions at point
VI in Fig. 2 ($\beta=-0.02$, $\kappa=-0.065168585542904$).

Although an improvement is found using approximant (7) over the simple (3) for
the cases of negative shear and small $\beta$, for all other $\beta$ values
(including those that are physically relevant), both approximants perform
equally well. Approximant (3), however, is simpler to compute, as it is based
solely on recursion.

###  3.1 Complex singularities that limit convergence of power series solution
(2)

The power series result (2) is an exact representation of the solution to (1)
within its radius of convergence. As there is no exact solution to (1), the
radius of convergence must be extracted by examining successive terms in the
power series using standard methods such as the ratio test [28].
Alternatively, as has been done using Padé approximants [29, 30], the
asymptotic approximant (3) may be used to determine the location of these
singularities and the radius of convergence of the series. The radius itself
is set from the singularities closest to $\eta=0$ in the solution of (1) when
viewed in the complex plane; it follows, then, that the roots of the
denominator of the approximant (6a) should be examined. Here, we increase the
number of terms in the $A_{n}$ series in the denominator of (6a) and obtain
convergent solutions for the roots nearest to $\eta=0$. Fig. 11a provides
results corresponding to three different values of $\beta$. Roots for
$\beta=0$ lie on the negative real axis, the first quadrant, and the fourth
quadrant of the complex plane, and yield a radius of convergence of
approximately 4.024 in agreement with Boyd [31, 19]333As a result of using a
different similarity variable, the value reported in [31, 19] is
$\sqrt{2}\times$ our value (see footnote $\dagger$ in Section 2). For
$0<\beta<0.6$, the root structure changes, as instead only one singularity
appears that sets the circle convergence, lying on the negative real
axis444For $0.5<\beta<0.6$, the singularity location (orientation and radius)
does not converge using (3). However, the series terms eventually alternate
sign at higher-order, indicating that the closest singularity lies on the
negative real axis., as shown in Fig. 11a for $\beta=0.5$. For $\beta<0$
($\kappa>0$ branch) there are a pair of singularities that set circle of
convergence, shown in Fig. 11a for $\beta=-0.12$ ($\kappa>0$); this
singularity orientation persists up until separation ($\beta=-0.199$). Outside
of these ranges, for $\beta>-0.199$ ($\kappa<0$) and $\beta>0.6$, estimates
for the roots using the approximant do not converge with increasing $N$.
However, even when the roots do not converge, the singularities move in such a
way that their distance from $\eta=0$ does converge for all $\beta$ excluding
$0.5<\beta<0.6$ (over this limited range of $\beta$, the ratio-test provides
an accurate radius of convergence) – thus, the radius of convergence of (2)
may be determined. Fig. 11b shows the dependence of radius of convergence on
$\beta$.

(a)

(a)

(b)

(b)

Figure 11: (a) Location of singularities that limit convergence of the
Falkner-Skan series (2), shown for three values of $\beta$. (b) Radius of
convergence of (2); a tabulated version of this curve is provided in
Supplementary Material [22].

Note that, although Fig. 11b indicates an apparent infinite radius of
convergence for $\beta\to 0^{-}$ ($\kappa<0$), Figs. 7 and 9 (typical of other
cases) show that the local minima in the $f(\eta)$ curve roughly tracks along
with the radius of convergence and is pushed farther out as $\beta$ becomes
small. Thus, although the radius of convergence increases for smaller $\beta$
past separation ($\kappa<0$), the asymptotic behavior ($f^{\prime}\to 1$) is
never captured by the series.

##  4 Conclusions

Asymptotic approximants provide nearly exact closed-form solutions to the
Falkner-Skan boundary layer equation for varying wedge angle. This adds to the
increasing number of problems in disparate areas of mathematical physics to
which asymptotic approximants have been applied successfully [12, 13, 14, 15,
18, 17]. Advantages of asymptotic approximants, specifically for the Falkner-
Skan problem and in general for other problems, are their simple form, ability
to yield highly accurate solutions, accuracy in solution derivatives, and low
computational load. The approximant is used to determine the singularities in
the complex plane that prescribe the radius of convergence of the power series
solution to the Falkner-Skan equation for a large range of wedge angles. One
limitation of the methodology provided here is that accurate values of the
wall velocity gradient, $\kappa$, and asymptotic parameter, $B$, are needed to
construct the approximant. As these are determined numerically in the present
work, the approximants can only be utilized in post-processing once numerics
are implemented. In an approximant generated for the Sakiadis boundary layer
in a previous paper [15], however, it is possible to predict with high
accuracy these parameters by a judicious choice in approximant coefficients.
In the same paper, the approximant used here is applied to the Blasius problem
to predict $\kappa$ and $B$, but the precision does not approach that of
existing benchmarks [19] since, unlike the Sakiadis problem, higher-order
asymptotic behavior is not incorporated in the approximant. Although not
reported here, reasonable estimates for $\kappa$ and $B$ can be predicted
applying the approximant for $0\leq\beta\leq 1$, but the prediction method
fails for $\beta<0$. That said, one may interpolate between the numerically-
obtained tabulated results given in the supplementary material [22] to obtain
fairly accurate values for $\kappa$ and $B$ for values of $\beta$ not
tabulated. Implementation of approximants in this manner makes them fully
independent of further numerical predictions.

## References

  * [1] H. Schlichting.  Boundary Layer Theory.  McGraw-Hill, seventh edition, 1979.
  * [2] T. Cebeci and H. B. Keller.  Shooting and parallel shooting methods for solving the Falkner-Skan boundary-layer equation.  J. Comp. Phys., 7:289–300, 1971.
  * [3] C. Laine and L. Reinhart.  Further numerical methods for the Falkner-Skan equaitons: shooting and continuation techniques.  I. J. Num. Meth. Fluids, 4:833–852, 1984.
  * [4] R. Fazio.  A novel approach to the numerical solution of boundary value problems on infinite intervals.  SIAM J. Num. Anal., 33(4):1473–1483, 1996.
  * [5] A. Asaithambi.  A finite-difference method for the Falkner-Skan equation.  Appl. Math. Comp., 92:135–141, 1998.
  * [6] S. S. Motsa and P. Sibanda.  An efficient numerical method for solving Falkner-Skan boundary layer flows.  I. J. Num. Meth. Fluids, 69:499–508, 2012.
  * [7] R. Fazio.  Blasius problem and Falkner-Skan model: Töpfer’s algorithm and its extension.  Comp. Fluids, 73(15):202–209, 2013.
  * [8] C-S. Liu.  An iterative method based-on eigenfunctions and adjoint eigenfunctions for solving the Falkner-Skan equation.  Appl. Math. Lett., 67:33–39, 2017.
  * [9] H. Bararnia, N. Haghparast, M. Miansari, and A. Barari.  Flow analysis for the Falkner-Skan wedge flow.  Curr. Sci., 103(2):169–177, 2012.
  * [10] B. I. Yun.  New approximate analytical solutions of the Falkner-Skan equation.  J. Appl. Math., 2012:170802: 1–12, 2012.
  * [11] A. Khidir.  A note on the solution of general Falkner-Skan problem by two novel semi-analytical techniques.  Propul. Power Res., 4(4):212–220, 2015.
  * [12] N. S. Barlow, A. J. Schultz, S. J. Weinstein, and D. A. Kofke.  An asymptotically consistent approximant method with application to soft- and hard-sphere fluids.  J. Chem. Phys., 137:204102, 2012.
  * [13] N. S. Barlow, A. J. Schultz, S. J. Weinstein, and D. A. Kofke.  Critical isotherms from virial series using asymptotically consistent approximants.  AIChE J., 60(9):3336–3349, 2014.
  * [14] N. S. Barlow, A. J. Schultz, S. J. Weinstein, and D. A. Kofke.  Communication: Analytic continutation of the virial series through the critical point using parametric approximants.  J. Chem. Phys., 143:071103:1–5, 2015.
  * [15] N. S. Barlow, C. R. Stanton, N Hill, S. J. Weinstein, and A. G. Cio.  On the summation of divergent, truncated, and underspecified power series via asymptotic approximants.  Q. J. Mech. Appl. Math., 70(1):21–48, 2017.
  * [16] C. M. Bender and S. A. Orszag.  Advanced Mathematical Methods for Scientists and Engineers I: Asymptotic Methods and Perturbation Theory.  McGraw-Hill, 1978.
  * [17] R. J. Beachley, M. Mistysyn, J. A. Faber, S. J. Weinstein, and N. S. Barlow.  Accurate closed-form trajectories of light around a kerr black hole using asymptotic approximants.  Class. Quant. Grav., 35(20):1–28, 2018.
  * [18] N. S. Barlow, S. J. Weinstein, and J. A. Faber.  An asymptotically consistent approximant for the equatorial bending angle of light due to kerr black holes.  Class. Quant. Grav., 34(135017):1–16, 2017.
  * [19] J. P. Boyd.  The Blasius function: Computations before computers, the value of tricks, undergraduate projects, and open research problems.  SIAM Rev., 50(4):791–804, 2008.
  * [20] S. Anil Lal and Neeraj Paul M.  An accurate Taylor series solution with high radius of convergence for the Blasius function and parameters of asymptotic variation.  JAFM, 7(4):557–564, 2014.
  * [21] K. Stewartson.  Further solutions of the Falkner-Skan equation.  Proc. Roy. Phil. Soc., 50:454–465, 1954.
  * [22] See (insert url provided by journal) for a spreadsheet of values that were computed in this work.
  * [23] P. L. Sachdev, R. B. Kudenatti, and N. M. Bujurke.  Exact analytic solution of a boundary value problem for the Falkner-Skan equation.  Stud. Appl. Math, 120:1–16, 2008.
  * [24] W. A. Coppel.  On a differential equation of boundary layer theory.  Phil. Trans. R. Soc. Lond. A, 253(1023):101–136, 1960.
  * [25] R. V. Churchill.  Complex Variables.  McGraw-Hill, 1948.
  * [26] P. Gonnet, S. Güttel, and L. N. Trefethen.  Robust Padé approximation via svd.  SIAM Rev., 55(1):101–117, 2013.
  * [27] G. A. Baker Jr. and J. L. Gammel.  The Padé approximant.  J. Math. Anal. Appl., 2:21–30, 1961.
  * [28] E. C. Titchmarsh.  The Theory of Functions.  Oxford University Press, 2 edition, 1968.
  * [29] M. Van Dyke.  Perturbation Methods in Fluid Mechanics.  Parabolic, 1975.
  * [30] A. J. Guttmann and I. Jensen.  Series analysis.  In Polygons, Polyominoes, and Polycubes, pages 181–202. Springer, 2009.
  * [31] J. P. Boyd.  The Blasius function in the complex plane.  Exper. Math., 8(4):381–394, 1999.


# Learning from Perturbations: Diverse and Informative
Dialogue Generation with Inverse Adversarial Training

Wangchunshu Zhou Qifei Li∗ Chenle Li
Beihang University, Beijing, China
<EMAIL_ADDRESS>Equal contribution. Corresponding author

###### Abstract

In this paper, we propose Inverse Adversarial Training (IAT) algorithm for
training neural dialogue systems to avoid generic responses and model dialogue
history better. In contrast to standard adversarial training algorithms, IAT
encourages the model to be sensitive to the perturbation in the dialogue
history and therefore learning from perturbations. By giving higher rewards
for responses whose output probability reduces more significantly when
dialogue history is perturbed, the model is encouraged to generate more
diverse and consistent responses. By penalizing the model when generating the
same response given perturbed dialogue history, the model is forced to better
capture dialogue history and generate more informative responses. Experimental
results on two benchmark datasets show that our approach can better model
dialogue history and generate more diverse and consistent responses. In
addition, we point out a problem of the widely used maximum mutual information
(MMI) based methods for improving the diversity of dialogue response
generation models and demonstrate it empirically.

##  1 Introduction

In recent years, neural end-to-end dialogue response generation models Sordoni
et al. (2015); Serban et al. (2016); Bordes et al. (2016) has gained
increasing popularity with the recent advancements of neural sequence-to-
sequence (seq2seq) learning models Sutskever et al. (2014); Vaswani et al.
(2017). While neural dialogue models can generate seemingly fluent responses,
due to the over-simplified maximum likelihood estimation (MLE) training
objective and the high frequency of generic responses in training corpora,
they tend to produce dull and generic responses such as “I don’t know” much
more often than that humans generally do Li et al. (2015), which makes
dialogue agents less engaging and ineffective.

In addition, recent research on whether neural dialogue systems use dialogue
history effectively Sankar et al. (2019) shows that most neural dialogue
agents fail to take the dialogue history into account when generating
responses. This problem makes neural dialogue systems tend to generate
responses irrelevant to the current topic of the conversation and are not
consistent with the dialogue history. This problem may also intensify the
generic response problem, as dull responses are generally off-topic and
irrelevant to the dialogue history.

To address the above issues, in this paper, we propose Inverse Adversarial
Training (IAT) algorithm for training neural dialogue systems to avoid generic
responses and model dialogue history better, thus generating diverse and
informative responses. Conventional adversarial training methods generally
generate label-preserving adversarial inputs with carefully designed methods
and train the model to generate the same output to enhance the model’s
robustness. In contrast, our approach perturbs in input dialogue history such
that a good dialogue model should not generate the same output if the output
is non-generic and relevant to the dialogue history. We name our proposed
method as inverse adversarial training because it is related to conventional
adversarial training methods which aim to improve the model’s adversarial
robustness but our proposed objective is motivated in the opposite direction.
Note that our work is not directly related to TextGANs as well as their
applications on dialogue response generation.

Specifically, the proposed inverse adversarial training assigns higher rewards
to generated responses or ground-truth responses if their likelihood decreases
more when the dialogue history is perturbed, and penalize the model when it
generates responses whose likelihood is almost unchanged given either original
or perturbed dialogue history as input. This encourages the model to generate
more relevant and informative responses and capture dialogue history better.
The proposed IAT algorithm can be used in both supervised and self-supervised
fashion (with/without reference response), which can be viewed as a form of
reward-augmented maximum likelihood (RAML) method Norouzi et al. (2016) that
improves the original MLE objective or a rewarding scheme for RL-based text
generation algorithms. The inverse adversarial learning framework is also
conceptually related to self-adversarial learning (Zhou et al., 2020) where
the the comparison is made between different checkpoints of the same model to
provide reward for RL training of the NLG model.

In addition, we identify a limitation of the widely-used maximum mutual
information (MMI) based methods for improving the diversity of dialogue
response generation models. This will be discussed in detail in section 2.1
and empirically demonstrated in section 4.2.

We conduct experiments on two dialogue datasets, OpenSubtitiles and
DailyDialog, to demonstrate the effectiveness of the proposed approach.
Experimental results show IAT helps neural dialogue systems model dialogue
history better and generate more diverse and informative responses.

##  2 Related Work

###  2.1 Dull Response Problem

Neural dialogue models tend to generate generic or dull responses such as I
don’t know which are not engaging for the users Sordoni et al. (2015). This
behavior can be ascribed to the high frequency of generic responses in the
training corpus and the over-simplified MLE training objective. How to avoid
generic responses and to make the dialogue agent more engaging has been a
long-standing problem. Previous work attempts to address this problem with
different approaches: 1) Li et al. (2015) propose a diversity-promoting
objective based on Maximum Mutual Information (MMI). Given source $S$ and
target $T$, their approach first generates N-best lists based on $P(T|S)$ and
then rerank the list by combining $p(T|S)$ and $\lambda p(S|T)$; 2) Zhang et
al. (2018b) propose to directly optimize $p(S|T)$ together with $p(T|S)$ with
an Adversarial Information Maximization objective; and 3) adversarial learning
Li et al. (2017a) and dual adversarial learning Cui et al. (2019) based on the
intuition that real responses are of high diversity, thus can be distinguished
from generated responses which are often dull and generic. There are also
other methods using distributional constraints of the target responses Baheti
et al. (2018); Csáky et al. (2019) or commonsense knowledge (Wu et al., 2020).

While shown to be effective in several datasets, these approaches suffer from
several drawbacks. For the first two approaches, while the MMI objective may
lead to larger mutual information, it often does not actually result in more
informative and engaging responses according to our observations. For example,
given a dialog context: “What have you done with him in the bar last night?”
The top response re-ranked by the MMI objective is “I have done nothing with
him in the bar last night.”, which is non-informative and less natural
compared with the response “Nothing at all.” generated by a standard seq2seq
dialogue model. This is also confirmed in the experiment section. We suspect
this phenomenon is caused by the term $p(S|T)$ in the MMI objective. It
encourages generating responses that make the last utterance in the dialogue
history have a high likelihood given the generated responses. While a truly
informative response may yield a high $p(S|T)$, the model can easily find a
“shortcut” to cheat this objective by simply copying a portion of tokens in
the last utterance, which is likely to have high $p(S|T)$ as well as $p(T|S)$.
The adversarial learning based dialogue model is notoriously hard to train and
may suffer from the problem of mode collapse, which decreases the diversity of
generated responses.

In contrast, our proposed IAT approach is based on the intuition that a
diverse, relevant, and consistent response should be sensitive to the
perturbation in the dialogue history, which is from a different perspective
and may be complementary with the aforementioned approaches.

###  2.2 Dialogue History Modeling

Recently, Sankar et al. (2019) evaluated whether existing neural dialogue
systems use dialogue history effectively by perturbing dialogue history and
observing the variation of model output. They corrupted the dialogue history
with both utterance-level and word-level perturbation and see whether and how
much the output perplexity decreases. Their experimental results show that
end-to-end neural dialogue systems are generally non-sensitive to the
perturbation of dialogue history, suggesting that they may perform poorly in
modeling dialogue history. Previous work Serban et al. (2016); Zhao et al.
(2017) improves the context modeling ability with modification in model
architectures. In contrast, our approach employs a novel training objective to
enhance the dialogue history modeling ability, which is orthogonal and may be
complementary with them.

##  3 Inverse Adversarial Training

Figure 1: Illustration of IAT. Our algorithm assigns high reward and low
penalty when the dialogue model generates relevant and engaging responses
given original and perturbed dialogue history respectively. The reward and
penalty are respectively decreased and increased when the dialogue model
generates dull responses. Note that both dull responses and engaging responses
are gold human-written reference responses. They are not labeled in the
dataset but automatically detected by the difference of their generation
likelihood when given original and perturbed dialogue history. (Best view in
color.)

In this section, we describe the proposed inverse adversarial training
algorithm in detail. We first describe how we perturb the dialogue history and
then formally introduce the inverse adversarial training algorithm.

###  3.1 Perturbation Approaches

Following previous study Sankar et al. (2019), we perturb the dialogue history
in both utterance and word level and apply them jointly during training.

#### Utterance-level Perturbations

We consider the following operations 1) Shuf that shuffles the sequence of
utterances in the dialog history, 2) Rev that reverses the order of utterances
in the history (but maintains word order within each utterance) 3) Drop that
completely drops certain utterances, 4) Truncate that truncates the dialog
history to contain only the k most recent utterances where k $\leq$ n, where n
is the length of dialog history, and 5) Repl that randomly replaces each
utterance in the dialogue history by another utterance in the dataset with a
probability of 30%, which resembles the negative sampling Mikolov et al.
(2013) approach111The first four kinds of perturbation is originally proposed
in Sankar et al. (2019) and the last is proposed in this paper..

#### Word-level perturbations

We consider similar operations but at the word level within every utterance 1)
word-shuffle that randomly shuffles the words within an utterance 2) reverse
that reverses the ordering of words, 3) word-drop that drops 30% of the words
uniformly 4) noun-drop that drops all nouns, 5) verb-drop that drops all
verbs, and 6) word-repl that replace 30% of words with a random word in the
vocabulary uniformly.

We explain the role of different perturbations and their potential effects
briefly. The Shuf and Rev perturbations change the chronological order of
utterances. Inverse adversarial training with these kinds of perturbation may
help the model to capture some common-senses about the chronological order of
utterances. The Drop and Repl perturbations may help the model to capture some
kinds of casual effects. Finally, the Truncate perturbation may help the model
capture long-term and multi-turns dialogue history better.

###  3.2 Inverse Adversarial Training

In contrast to the adversarial training objective which maximize the
likelihood of generating the same output given perturbed input, the inverse
adversarial training objective maximizes the reduction of the likelihood of
generating the same output when the input is perturbed, which is opposite to
the conventional adversarial training.

A straightforward approach is to maximize the likelihood of generating ground-
truth responses given original dialogue history while minimizing this
likelihood when given perturbed dialogue history. However, this approach
suffers from several problems: First, as a previous study Sankar et al. (2019)
has shown, neural dialogue models generally capture the perturbation in the
dialogue history poorly, which is suggested by the fact that the output
embeddings of the encoder are very similar when given original and perturbed
input dialogue histories. This results in training the decoder to
simultaneously maximize and minimize the likelihood of the same output given
very similar input, which is undesirable and makes the training ineffective.
The second problem is that this training objective does not capture the
variation of likelihood and thus treats relevant and engaging responses
equally with dull and generic responses. This is undesirable as we only want
to maximize/minimize the likelihood for relevant and engaging responses when
conditioning on original/perturbed dialogue history and dull responses should
be avoided in both cases.

In this paper, we propose a sequence-level objective which is able to capture
the variation of the likelihood of responses given original or perturbed
input. This makes it possible to model dialogue history better and avoid
generic response problem at the same time. The idea is to evaluate generated
sentences based on the variation of the likelihood of responses given original
or perturbed dialogue history and use this variation as rewards for training
the dialogue model.

Given original dialogue history $X$ and perturbed dialogue history
$X^{\prime}$, the reward $\mathcal{R}(Y|X,X^{\prime})$ of generating response
$Y$, which is a sequence of $n$ tokens $y_{i},i\in{1,2,...,n}$, is measured by
how much $Y$ is more likely to be generated by the dialogue model given $X$
compared with that given $X^{\prime}$, which is computed by the difference of
negative log-likelihood losses (NLL) in two cases, as described below.

$\text{NLL}_{orig}=-\sum_{i=1}^{n}{\log{P(y_{i}|y_{<i},X)}}$ (1)
$\text{NLL}_{adv}=-\sum_{i=1}^{n}{\log{P(y_{i}|y_{<i},X^{\prime})}}$ (2)
$\mathcal{R}(Y|X,X^{\prime})=\text{NLL}_{adv}-\text{NLL}_{orig}$ (3)

Intuitively, the reward $\mathcal{R}$ would be high when the response $Y$ is
engaging and relevant to the dialogue history. A generic response should be
assigned with a low or even negative reward as it is irrelevant to the
dialogue history. The inverse adversarial training objective is to generate
responses to maximize its reward. With likelihood ratio Sutton et al. (2000),
we can formulate the gradient of the objective function for dialogue response
generator $G_{\theta}$ as:

$\nabla_{\theta}J(\theta)=\sum_{Y}{\sum_{i=1}^{n}\nabla_{\theta}\log
G_{\theta}(y_{i}|y_{<i},X)\cdot\mathcal{R}(Y|X,X^{\prime})}$ (4)

The above training objective encourages the dialogue model to generate non-
generic responses and model dialogue history better by giving higher rewards
when generating good responses based on original dialogue history.

Similarly, we would also want to penalize the dialogue model when it generates
the same response given perturbed dialogue history to explicitly force the
dialogue system to effectively model the dialogue history. We propose to model
this penalty with a max-margin reward scheme. Given margin $\mathcal{M}$, the
penalty $\mathcal{P}(Y|X,X^{\prime})$ of generating $Y$ is computed by

$\mathcal{P}(Y|X,X^{\prime})=\min(0,\text{NLL}_{adv}-\text{NLL}_{orig}-\mathcal{M})$
(5)

The insight behind equation 5 is that when the variation of likelihood of
generating $Y$ given $X$ and $X^{\prime}$ is large enough (i.e.
$\text{NLL}_{orig}-\text{NLL}_{adv}-\mathcal{M}>0$), the model should be
considered successfully captured the perturbation in the dialogue history and
should not be penalized. In contrast, when the variation is not large enough,
we penalize the dialogue agent for generating $Y$ giving $X^{\prime}$ because
a small variation of likelihood implicates: (1) the dialogue agent models
dialogue history poorly and (2) the generated responses $Y$ may be irrelevant
to the dialogue history $X$ and thus be generic and non-informative. The
corresponding gradient can be formulated as:

$\nabla_{\theta}J^{\prime}(\theta)=\sum_{Y}{\sum_{i=1}^{n}\nabla_{\theta}\log
G_{\theta}(y_{i}|y_{<i},X^{\prime})\cdot\mathcal{P}(Y|X,X^{\prime})}$ (6)

The penalty and reward are combined by directly summing up the gradient in Eq
(4) and Eq (6). The proposed inverse adversarial training algorithm can be
applied in both supervised fashion where responses $Y$ are ground-truth
responses in the dataset and self-supervised fashion where $Y$ is generated by
the dialogue model itself. The only difference between the self-supervised and
supervised version is whether the reference responses are generated (self-
supervised) or ground-truth responses (supervised). The supervised inverse
adversarial training can be viewed as a reward function algorithm for RAML
Norouzi et al. (2016) training that assigns higher rewards for “good” training
examples that help our model to generate relevant responses and learn to model
dialogue history better. The self-supervised inverse adversarial training, in
contrast, allows the model to explore freely and train the model with policy
gradient Sutton et al. (2000), a reinforcement learning approach.

##  4 Experiments

To validate the effectiveness of the proposed inverse adversarial training
algorithm, we conduct experiments in order to answer the following two
research questions:

(1) Do inverse adversarial training help neural dialogue systems model
dialogue history better?

(2) Do inverse adversarial training help neural dialogue models generate more
diverse, engaging, and informative dialogue responses?

###  4.1 Experimental Settings

Datasets We employ two datasets in our experiments. The first dataset is the
OpenSubtitles corpus Lison and Tiedemann (2016) which is a large, open-domain
dataset containing scripts of movie characters. Following previous work, we
consider each turn in the dataset as the target response and the two previous
sentences as the dialogue history. We remove the pairs whose response is
shorter than 5 words and randomly sample 1,800K, 500K, and 12K dialogue turns
for training, validation, and testing, respectively.

We employ the DailyDialog dataset Li et al. (2017b) as the second dataset
which consists of dialogues that resemble daily conversations across multiple
topics. It comprises of 13k dialogues, which is much smaller compared with the
OpenSubtitles dataset. However, it has an average of 7.9 turns per dialog,
which is more suitable for evaluating whether the proposed approach is able to
improve the model’s ability of modeling long-term dialogue history.

Compared Models We build dialogue systems with seq2seq Sutskever et al. (2014)
models. Following previous work Li et al. (2017a, 2015), we employ LSTM-based
seq2seq model for the OpenSubtitles dataset. For the DailyDialog dataset, we
employ the transformer Vaswani et al. (2017) model which yields superior
results in preliminary experiments while shown to perform poorly in modeling
dialogue history Sankar et al. (2019). Specifically, following previous work
Xu et al. (2018), we set the hidden size to 256, embedding size to 128,
vocabulary size to 50K, and batch size to 64 for the proposed models and the
baselines. We use the Adam optimizer with the initial learning rate 0.1 for
model training.

We compare the dialogue model trained with the proposed inverse adversarial
learning algorithm with the following baseline methods (all compared models
are using the same backbone architecture):

  * •

Seq2Seq: The vanilla seq2seq dialogue model trained with MLE objective.

  * •

Seq2Seq + MMI: The dialogue model using mutual information method Li et al.
(2015), which substracts the score of the target sequence $\log p(T|S)$ by its
language model score $\log p(T)$ (MMI-anti) or by a backward generation score
$\log p(S|T)$ (MMI-bidi) for decoding.

  * •

Seq2Seq + Adversarial Learning: A dialogue model trained with adversarial
learning objective Li et al. (2017a). The model is pretrained with MLE
objective and then fine-tuned with adversarial learning.

  * •

Seq2Seq + DS: A strong baseline using distributional constraints over the
generated responses Baheti et al. (2018).

  * •

CVAE: A dialogue response generation model using conditional VAE (Zhao et al.,
2017) to improve the discourse-level diversity of generated responses.

Our models are pretrained with the MLE objective until the validation
perplexity stops decreasing. We then apply the inverse adversarial training
algorithm for continual training. During training, reference responses are
either generated responses or ground-truth responses in self-supervised and
supervised inverse-adversarial training respectively. We combine both
supervised and self-supervised inverse adversarial training by alternatively
switching between these two objectives for each training iteration.

Evaluation Metrics We employ different automated evaluation metrics to
respectively answer the three research questions introduced at the beginning
of this section. To evaluate how well dialogue systems are able to model
dialogue history, we adopt the approach proposed by Sankar et al. (2019),
which measures the increases in perplexity when the model is fed with
perturbed dialogue history instead of original dialogue history. We report the
result in both utterance-level and word-level perturbation.

To evaluate if inverse adversarial learning can effectively reduce the generic
response problem, following Li et al. (2015), we evaluate the diversity of
generated responses by calculating the number of distinct unigrams, bigrams,
and trigrams in generated responses. The value is scaled by the total number
of generated tokens to avoid favoring long sentences, which are shown as
distinct-1, distinct-2, and distinct-3 in Table 2. Lastly, we compare the
percentage of stop-words222Stopword List from https://www.ranks. nl/stopwords.
We appended punctuations to this list. of the responses generated by each
model (smaller values that are closer to the distribution of human
conversations are preferred). We also report the token-level overlap between
the generated response and the last utterance in the dialog history to
demonstrate the “shortcut”problem of MMI-based methods decribed in Section
2.1.

As our approach is training in an “opposite” direction compared to
conventional adversarial training employed to enhance the robustness of
trained models, we also conduct experiments to evaluate the robustness of the
dialogue response generation models with respect to non label-changing
adversarial dialogue history. Similar to the method of evaluating the dialogue
history modeling ability, we measure the perplexity changes when the model is
given a different but meaning-preserving dialogue history, which is
constructed by performing word substitution with a BERT-based lexical
substitution method Zhou et al. (2019) and paraphrase generation Kumar et al.
(2020) as word-level and utterance-level perturbation respectively on the
original dialogue history, as the input.

In addition, as demonstrated by Liu et al. (2016); Zhou and Xu (2020),
automated metrics are notoriously poor for evaluating dialogue systems. We
thus conduct a human evaluation to better evaluate the effectiveness of the
proposed algorithm. For human evaluation, we invite 20 human annotators which
are all graduate students with good English proficiency to evaluate the
quality of the model. Following Zhang et al. (2018a), we ask human annotators
to interact with compared models for 50 utterances with each compared dialogue
system and evaluate the fluency, consistency, and diversity of the model
(scored between 1- 5). Fluency measures how likely the generated text is
produced by human. Consistency measures how likely the generated text is
related to the input dialogue history, which corresponds to the first research
question. Diversity measures how much the generated text provides specific
information, rather than “dull” and repeated information, which corresponds to
the second research question.

Method | DailyDialog | OpenSubtitles
---|---|---
Seq2Seq |  |
\- base model | 2.71 (1.18) | 1.94 (0.33)
\- + AL | 2.76 (1.22) | 1.67 (0.41)
\- + DS | 2.79 (1.24) | 1.87 (0.44)
\- + CVAE | 3.25 (1.41) | 2.11 (0.49)
\- + IAT | 3.69 (1.65) | 2.37 (0.42)

Table 1:  Results on the dialogue history modeling ability of compared models,
which is measured by the difference between perplexity of gold responses when
receiving original dialogue history and receiving perturbed dialogue history.
Mean and standard deviation of 5 runs are reported.

Method | DailyDialog | OpenSubtitles
---|---|---
Dist-1 | Dist-2 | Dist-3 | overlap | stop-word | Dist-1 | Dist-2 | Dist-3 | overlap | stop-word
Seq2Seq |  |  |  |  |  |  |  |  |  |
\- base model | 2.32 | 6.28 | 9.43 | 15.6 | 67.4 | 1.72 | 5.37 | 7.64 | 22.5 | 77.8
\- + MMI-anti | 4.15∗ | 11.27∗ | 19.61∗ | 26.7 | 62.4 | 3.45 | 11.35 | 18.12 | 30.1 | 74.2
\- + MMI-bidi | 3.52 | 9.29 | 17.43 | 31.5 | 63.1 | 3.52∗ | 12.11∗ | 18.56∗ | 37.8 | 74.7
\- + AL | 2.25 | 6.01 | 9.39 | 16.1 | 66.8 | 2.97 | 5.44 | 7.46 | 23.5 | 76.4
\- + DS | 3.19 | 7.84 | 11.61 | 18.4 | 61.5 | 3.05 | 6.30 | 11.59 | 21.3 | 71.2
\- + CVAE | 3.59 | 9.41 | 12.93 | 17.7 | 61.1 | 3.35 | 10.13 | 17.02 | 22.5 | 71.4
\- + IAT | 3.72 | 9.81 | 14.93 | 15.4∗ | 60.9 | 3.29 | 10.16 | 17.30 | 20.8∗ | 70.9∗

Table 2:  Results of the diversity of generated responses of compared models.
We report the average value of 5 runs on both datasets. ∗ denotes
statistically significant with p-value $<$ 0.01.

Method | DailyDialog | OpenSubtitles
---|---|---
Seq2Seq |  |
\- base model | 0.75(0.41) | 0.42(0.29)
\- + MMI | - | -
\- + AL | 0.83(0.47) | 0.49(0.34)
\- + DS | 0.78(0.44) | 0.46(0.33)
\- + IAT | 0.77(0.45) | 0.44(0.31)

Table 3:  Results on the adversarial robustness of compared models, which is
measured by the difference between perplexity of gold responses when receiving
original dialogue history and receiving non-label changing adversarial
dialogue history. AL denotes adversarial learning and IAT denotes inverse
adversarial training.

###  4.2 Experimental Results

Results on dialogue history modeling We first present the results on dialogue
history modeling ability. The results are shown in Table 1. We can see that
the dialogue model trained with the proposed inverse adversarial training
algorithm performs significantly better than the compared baselines as the
perplexity dramatically increases when the input dialogue history is
perturbed. This is not surprising as our approach is the first learning
objective which explicitly forces the dialogue system to better model dialogue
history. In contrast, the MMI criterion and the adversarial learning objective
do not significantly influence the dialogue history modeling ability of
dialogue systems. The dialogue model based on CVAE models dialogue history
better than other baselines while still under-performs our approach.

Method | Fluency | Consistency | Diversity
---|---|---|---
Seq2Seq |  |  |
\- base model | 2.83 | 2.69 | 3.05
\- + MMI-anti | 2.73 | 2.78 | 3.10
\- + MMI-bidi | 2.80 | 2.82 | 3.02
\- + AL | 2.77 | 2.69 | 2.91
\- + DS | 2.85 | 2.88 | 3.12
\- + CVAE | 2.93 | 2.91 | 3.19
\- + IAT | 3.02∗ | 3.05∗ | 3.34∗

Table 4:  Human evaluation results of compared model on the DailyDialog
dataset. The Kappa score between human annotators is 0.73. ∗ denotes
statistically significant with p-value $<$ 0.05.

Reults on diversity The results of the diversity of responses generated by
compared models are shown in Table 2. We can see that both the Maximum Mutual
Information objective and the proposed inverse adversarial learning succeed in
improving the diversity of generated responses. In contrast, the adversarial
learning objective hardly improves the diversity, which may be due to the
instability of adversarial learning on text generation. While the MMI
objective yields slightly larger improvements on distinct n-gram based
metrics, their approach is used only for re-ranking during inference, which is
orthogonal and may be complementary to the proposed approach.

Source | how long will it take us to drive to London ? … ok . You haven’t seen my company car , have you ?
---|---
Seq2Seq | I don’t know about what you are talking about.
+MMI-anti | I have seen a car in London.
+MMI-bidi | I haven’t seen your company car, have you ?
+DS | No, I haven’t seen a car.
+CVAE | No, I think no.
+IAT | No, I don’t think we can arrive on time.
Source | Your throat is inflamed . And your tongue is heavily coated . You have all the symptoms of influenza .
Seq2Seq | I agree with that you have just said.
+MMI-anti | I want to have all the influenza.
+MMI-bidi | I do not want to have the influenza and the throat.
+DS | Oh That’s too bad.
+CVAE | That’s a sad thing.
+IAT | oh what should I do for my tougue?
Source | Good evening , madam . Can I help you ? Yes . I bought this sheet here yesterday .
Seq2Seq | I want to buy a sheet too.
+MMI-anti | What is the color of the shirt you bought yesterday?
+MMI-bidi | What is wrong with this sheet bought here yesterday?
+DS | What problem about the sheet?
+DS | What do you think of the sheet?
+IAT | Anything wrong with the sheet?

Table 5:  Case study of dialogue responses generated by different compared
models.

In addition, as described in section 2.1, the MMI objective may favor non-
engaging responses that simply repeats the last utterance in the dialogue
history. This is empirically demonstrated by their high overlap with the last
utterance in the dialog history, as measured by the “overlap” metric. In
contrast, our approach does not suffer from this problem and also generate
fewer stop-words compared to the MMI-based methods. In addition, our approach
also outperforms the strong baselines including that using distributional
constraint and CVAE, demonstrating its effectiveness in improving the
diversity of generated responses.

Results on adversarial robustness We also conduct experiments to test the
robustness of the dialogue model trained with the proposed inverse adversarial
training objective. The results are shown in Table 3. We see that the increase
in the perplexity of ground-truth responses under our model is roughly the
same with the baseline transformer model and the other compared models. This
suggests that our proposed IAT objective does not harm the adversarial
robustness.

Human evaluation We conduct a human evaluation of compared models on the
DailyDialog dataset. The results are shown in Table 4. We can see that the
proposed inverse adversarial training objective substantially improves the
consistency of the dialogue model over all compared baselines, which confirms
its ability to train dialogue agents to model dialogue history better. As for
the diversity of generated responses, we find that human annotators do not
prefer the responses selected by the MMI objective over that generated by the
baseline model with a large margin. We find that this is mainly because the
MMI objective prefers repeating tokens which appear in the last utterance and
human annotators find it non-informativeness. In contrast, our approach yields
even larger improvements in the diversity of the generated responses. We do
not find the adversarial learning method improves the diversity of dialogue
models, which may be due to the problem of mode collapse in adversarial
learning. The over-all fluency of compared models is roughly the same, which
may be because they are all trained or pretrained with MLE objective.

Method | Fluency | Consistency | Diversity
---|---|---|---
Ours | 2.86 | 3.02 | 3.36
\- w/o supervised | 2.68 | 2.91 | 3.34
\- w/o self-supervised | 2.91 | 2.98 | 3.31
\- w/o reward | 2.74 | 2.88 | 3.12
\- w/o penalty | 2.88 | 2.82 | 3.26
\- w/o utter_pertub | 2.82 | 2.71 | 3.23
\- w/o token_pertub | 2.85 | 2.84 | 3.28

Table 6: Ablation study results of compared model on the DailyDialog dataset.
AL denotes adversarial learning and IAT denotes inverse adversarial training.

###  4.3 Qualitative Analysis

To better compare and analyze the inverse adversarial training objective, we
conduct a qualitative analysis of dialogue responses generated by different
compared models. The samples are presented in Table 5. We can see that the
vanilla transformer-based dialogue response generation model tends to generate
irrelevant and generic responses. Applying the MMI objective for re-ranking
successfully avoids those generic responses. However, it leads to another kind
of non-informative response that repeats the majority of tokens in the latest
utterance, which is also quite unnatural. In contrast, dialogue models trained
with the proposed inverse adversarial training objective tend to generate more
diverse responses which are also more relevant to the dialogue history.

###  4.4 Ablation Study

To better understand the relative importance of different components in the
proposed inverse adversarial training objective, we conduct an ablation study
with human evaluation to compare different model variants against the full
model. The results are shown in Table 6. We can find that both supervised-only
and self-supervised-only variant of the proposed inverse adversarial training
algorithm can improve the consistency and the diversity of dialogue models.
However, self-supervised inverse adversarial training seems to sacrifice the
fluency of generated responses for better diversity and consistency as the
model trained without the self-supervised objective are considered to be more
fluent by human annotators. The usefulness of the reward and the penalty
objectives is also demonstrated by human evaluation. Concretely, we find that
the reward described in Eq.(3) contributes more to the diversity of generated
responses. This may be because it assigns high rewards for relevant and
specific responses and negative rewards for generic responses. In contrast,
the penalty in Eq.(5) helps the dialogue system model dialogue history better
and leads to more consistent responses by punishing the dialogue model when
generating the same responses given perturbed dialogue history. As for
different perturbation approaches, we find that both utterance-level and
token-level contributes to the performance improvements. Also, we find that
utterance-level perturbation may be more effective for improving the
consistency of generated responses. We suspect this may be because the ability
of the dialogue model to distinguish utterance-level perturbation is more
important for better dialogue history modeling.

##  5 Conclusion

In this work, we introduce inverse adversarial training (IAT) algorithm that
is able to simultaneously reduce the dull response problem and help neural
dialogue systems model dialogue history better. IAT measures the relevance and
consistency of responses by the difference of their likelihood conditioning on
either original and perturbed dialogue history. In this way, it is able to
prevent the dialogue system from preferring generic responses, even they are
often of high frequency in the training corpora. Our method also encourages
the dialogue agent to model dialogue history better by penalizing the model
when generating the same responses given perturbed dialogue history.
Experimental results on two benchmark datasets show that the proposed inverse
adversarial training algorithm helps dialogue models capture dialogue history
better and generate more diverse and consistent responses. We also identify a
limitation of the widely-used MMI based methods for improving the diversity of
dialogue response generation models and empirically demonstrate the existence
of this problem through our experimetns.

## Boarder Impact

This work does not involve collection and release of data, nor inference of
information or judgments about individuals. However, dialogue systems may have
a social impact and we believe that making dialogue agent able to generate
more meaningful and consistent responses are beneficial. We also agree that
general control on the bias or unfairness of neural dialogue agents is
important. We believe this can be done from both the perspective of data
collection and training algorithms. We believe our proposed training algorithm
will likely not contribute to any ethical concern of chat robots.

## Acknowledgments

We thank the anonymous reviewers for their valuable comments.

## References

  * Baheti et al. (2018) Ashutosh Baheti, Alan Ritter, Jiwei Li, and Bill Dolan. 2018.  Generating more interesting responses in neural conversation models with distributional constraints.  In _Proceedings of the 2018 Conference on Empirical Methods in Natural Language Processing_ , pages 3970–3980, Brussels, Belgium. Association for Computational Linguistics.
  * Bordes et al. (2016) Antoine Bordes, Y-Lan Boureau, and Jason Weston. 2016.  Learning end-to-end goal-oriented dialog.  _arXiv preprint arXiv:1605.07683_.
  * Csáky et al. (2019) Richárd Csáky, Patrik Purgai, and Gábor Recski. 2019.  Improving neural conversational models with entropy-based data filtering.  In _Proceedings of the 57th Annual Meeting of the Association for Computational Linguistics_ , pages 5650–5669, Florence, Italy. Association for Computational Linguistics.
  * Cui et al. (2019) Shaobo Cui, Rongzhong Lian, Di Jiang, Yuanfeng Song, Siqi Bao, and Yong Jiang. 2019\.  Dal: Dual adversarial learning for dialogue generation.
  * Kumar et al. (2020) Ashutosh Kumar, Kabir Ahuja, Raghuram Vadapalli, and Partha Talukdar. 2020.  Syntax-guided controlled generation of paraphrases.  _Transactions of the Association for Computational Linguistics_ , 8:330–345.
  * Li et al. (2015) Jiwei Li, Michel Galley, Chris Brockett, Jianfeng Gao, and Bill Dolan. 2015.  A diversity-promoting objective function for neural conversation models.  _arXiv preprint arXiv:1510.03055_.
  * Li et al. (2017a) Jiwei Li, Will Monroe, Tianlin Shi, Sébastien Jean, Alan Ritter, and Dan Jurafsky. 2017a.  Adversarial learning for neural dialogue generation.  _arXiv preprint arXiv:1701.06547_.
  * Li et al. (2017b) Yanran Li, Hui Su, Xiaoyu Shen, Wenjie Li, Ziqiang Cao, and Shuzi Niu. 2017b.  Dailydialog: A manually labelled multi-turn dialogue dataset.  _arXiv preprint arXiv:1710.03957_.
  * Lison and Tiedemann (2016) Pierre Lison and Jörg Tiedemann. 2016.  Opensubtitles2016: Extracting large parallel corpora from movie and tv subtitles.
  * Liu et al. (2016) Chia-Wei Liu, Ryan Lowe, Iulian V Serban, Michael Noseworthy, Laurent Charlin, and Joelle Pineau. 2016.  How not to evaluate your dialogue system: An empirical study of unsupervised evaluation metrics for dialogue response generation.  _arXiv preprint arXiv:1603.08023_.
  * Mikolov et al. (2013) Tomas Mikolov, Ilya Sutskever, Kai Chen, Greg S Corrado, and Jeff Dean. 2013.  Distributed representations of words and phrases and their compositionality.  In _Advances in neural information processing systems_ , pages 3111–3119.
  * Norouzi et al. (2016) Mohammad Norouzi, Samy Bengio, Navdeep Jaitly, Mike Schuster, Yonghui Wu, Dale Schuurmans, et al. 2016.  Reward augmented maximum likelihood for neural structured prediction.  In _Advances In Neural Information Processing Systems_ , pages 1723–1731.
  * Sankar et al. (2019) Chinnadhurai Sankar, Sandeep Subramanian, Christopher Pal, Sarath Chandar, and Yoshua Bengio. 2019.  Do neural dialog systems use the conversation history effectively? an empirical study.  _arXiv preprint arXiv:1906.01603_.
  * Serban et al. (2016) Iulian V Serban, Alessandro Sordoni, Yoshua Bengio, Aaron Courville, and Joelle Pineau. 2016.  Building end-to-end dialogue systems using generative hierarchical neural network models.  In _Thirtieth AAAI Conference on Artificial Intelligence_.
  * Sordoni et al. (2015) Alessandro Sordoni, Michel Galley, Michael Auli, Chris Brockett, Yangfeng Ji, Margaret Mitchell, Jian-Yun Nie, Jianfeng Gao, and Bill Dolan. 2015.  A neural network approach to context-sensitive generation of conversational responses.  _arXiv preprint arXiv:1506.06714_.
  * Sutskever et al. (2014) Ilya Sutskever, Oriol Vinyals, and Quoc V Le. 2014.  Sequence to sequence learning with neural networks.  In _Advances in neural information processing systems_ , pages 3104–3112.
  * Sutton et al. (2000) Richard S Sutton, David A McAllester, Satinder P Singh, and Yishay Mansour. 2000\.  Policy gradient methods for reinforcement learning with function approximation.  In _Advances in neural information processing systems_ , pages 1057–1063.
  * Vaswani et al. (2017) Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Łukasz Kaiser, and Illia Polosukhin. 2017.  Attention is all you need.  In _Advances in neural information processing systems_ , pages 5998–6008.
  * Wu et al. (2020) Sixing Wu, Ying Li, Dawei Zhang, Yang Zhou, and Zhonghai Wu. 2020.  Diverse and informative dialogue generation with context-specific commonsense knowledge awareness.  In _ACL_ , pages 5811–5820. Association for Computational Linguistics.
  * Xu et al. (2018) Jingjing Xu, Xuancheng Ren, Junyang Lin, and Xu Sun. 2018.  Diversity-promoting gan: A cross-entropy based generative adversarial network for diversified text generation.  In _Proceedings of the 2018 Conference on Empirical Methods in Natural Language Processing_ , pages 3940–3949.
  * Zhang et al. (2018a) Saizheng Zhang, Emily Dinan, Jack Urbanek, Arthur Szlam, Douwe Kiela, and Jason Weston. 2018a.  Personalizing dialogue agents: I have a dog, do you have pets too?  _arXiv preprint arXiv:1801.07243_.
  * Zhang et al. (2018b) Yizhe Zhang, Michel Galley, Jianfeng Gao, Zhe Gan, Xiujun Li, Chris Brockett, and Bill Dolan. 2018b.  Generating informative and diverse conversational responses via adversarial information maximization.  In S. Bengio, H. Wallach, H. Larochelle, K. Grauman, N. Cesa-Bianchi, and R. Garnett, editors, _Advances in Neural Information Processing Systems 31_ , pages 1810–1820. Curran Associates, Inc.
  * Zhao et al. (2017) Tiancheng Zhao, Ran Zhao, and Maxine Eskenazi. 2017.  Learning discourse-level diversity for neural dialog models using conditional variational autoencoders.  In _Proceedings of the 55th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers)_ , pages 654–664, Vancouver, Canada. Association for Computational Linguistics.
  * Zhou et al. (2019) Wangchunshu Zhou, Tao Ge, Ke Xu, Furu Wei, and Ming Zhou. 2019.  Bert-based lexical substitution.  In _Proceedings of the 57th Annual Meeting of the Association for Computational Linguistics_ , pages 3368–3373.
  * Zhou et al. (2020) Wangchunshu Zhou, Tao Ge, Ke Xu, Furu Wei, and Ming Zhou. 2020.  Self-adversarial learning with comparative discrimination for text generation.  In _8th International Conference on Learning Representations, ICLR 2020, Addis Ababa, Ethiopia, April 26-30, 2020_. OpenReview.net.
  * Zhou and Xu (2020) Wangchunshu Zhou and Ke Xu. 2020.  Learning to compare for better training and evaluation of open domain natural language generation models.  In _AAAI_ , pages 9717–9724. AAAI Press.