# Quantum revivals and fractality for the Schrödinger equation

Gunwoo Cho, Jimyeong Kim, Seongyeon Kim, Yehyun Kwon and Ihyeok Seo
Department of Mathematics, Sungkyunkwan University, Suwon 16419, Republic of
Korea<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>School of Mathematics,
Korea Institute for Advanced Study, Seoul 02455, Republic of Korea
<EMAIL_ADDRESS>Department of Mathematics, Changwon National University,
Changwon 51140, Republic of Korea<EMAIL_ADDRESS>
###### Abstract.

We investigate the behavior of the Schrödinger equation under the influence of
potentials, focusing on its relationship to quantum revivals and fractality.
Our findings reveal that the solution displays fractal behavior at irrational
times, while exhibiting regularity similar to the initial data at rational
times. These extend the results of Oskolkov [16] and Rodnianski [18] on the
free Schrödinger evolution to the general case regarding potentials.

###### Key words and phrases:

Quantum revival, fractality, Schrödinger equation

###### 2020 Mathematics Subject Classification:

Primary: 35B45; Secondary: 35B10

This work was supported by a KIAS Individual Grant (MG082901) at Korea
Institute for Advanced Study and the POSCO Science Fellowship of POSCO TJ Park
Foundation (S. Kim), NRF-2020R1F1A1A0107352913 (Y. Kwon), and
NRF-2022R1A2C1011312 (I. Seo).

##  1\. Introduction

In 1836, H. F. Talbot [20] observed an intriguing phenomenon in classical
optics. When a plane wave passes through a periodic grating, the image of the
diffracted wave is refocused and recovers the initial grating pattern with a
specific periodicity. This phenomenon is now known as the Talbot effect. The
distance between the grating and the point where the image is fully
reconstructed is called the Talbot distance, denoted as $d_{\mathrm{T}}$. Lord
Rayleigh [17] later derived a formula $d_{\mathrm{T}}=a^{2}/\lambda$, where
$a$ is the spacing of the grating and $\lambda$ is the wavelength of the
incident wave.

The Talbot effect, originally an optical experiment, can be described by using
the Schrödinger evolution which leads us to mathematical studies of the
physical consequences. One of such consequence is the phenomenon of quantum
revivals, where a quantum wave packet is reconstructed at fractional multiples
of a quantum period dependent on the Planck constant and energy levels [4]. In
particular, Berry and Klein [3] used the free Schrödinger evolution to model
the Talbot effect. They found a dichotomy between “rational” and “irrational”
times, where a finite copy of the grating pattern reappears at every $t\in
d_{\mathrm{T}}\mathbb{Q}$, but the wave function exhibits a fractal, nowhere-
differentiable profile at every $t\notin d_{\mathrm{T}}\mathbb{Q}$.

As the grating is periodic in $x$, many authors have studied intensively the
periodic Schrödinger equation

$\begin{cases}i\partial_{t}u+\partial_{xx}u=0,\quad
x\in\mathbb{T}:=\mathbb{R}/2\pi\mathbb{Z},\ t\in\mathbb{R},\\\
u(0,x)=f(x).\end{cases}$ (1.1)

Assuming the initial function $f$ is of bounded variation, Oskolkov [16]
proved that $u(t,x)$, the wave function, is continuous but nowhere
differentiable in $x$ if $t\notin\pi\mathbb{Q}$, and discontinuous at every
$t\in\pi\mathbb{Q}$ where $f$ contains discontinuities. Later, Kapitanski and
Rodnianski [13] showed that $u(t,\cdot)$ has better regularity for
$t\notin\pi\mathbb{Q}$. For every $t\in\pi\mathbb{Q}$, Taylor [22] found that
if $f$ is the delta function, $u(t,\cdot)$ is a finite linear combination of
translated delta functions with coefficients being Gauss sums.

The Talbot effect has also led to the study of the Schrödinger equation,
particularly in relation to Berry’s conjectures on the fractality of the
graphs of $\mathrm{Re}u(t,\cdot)$, $\mathrm{Im}u(t,\cdot)$, and
$|u(t,\cdot)|^{2}$. Berry [2] calculated the fractal dimension of the
temporal, spatial, and diagonal restrictions of the graph of the density
function $|u(t,x)|^{2}$ for a step function $f$. Based on his calculations, he
conjectured that the graphs of $\mathrm{Re}u(t,\cdot)$,
$\mathrm{Im}u(t,\cdot)$, and $|u(t,\cdot)|^{2}$ have fractal dimension
$D=n+1/2$ at almost all irrational times $t$.

Rodnianski [18] rigorously justified Berry’s conjecture in one dimension for
the real and imaginary parts of the solution $e^{it\partial_{xx}}f$ with
almost all irrational times, showing that the graphs have fractal dimension
$D=3/2$ for every bounded variation $f$ not in
$\bigcup_{\epsilon>0}H^{\frac{1}{2}+\epsilon}$. Recently, Chousionis, Erdoğan,
and Tzirakis [7] extended this result to initial data in the larger class
$BV\setminus\bigcup_{\epsilon>0}H^{s_{0}+\epsilon}$, $\frac{1}{2}\leq
s_{0}<\frac{3}{4}$, and settled Berry’s conjecture in one dimension for the
density $|e^{it\partial_{xx}}f|^{2}$, showing that the dimension of the graph
is $3/2$ at almost all $t\notin\pi\mathbb{Q}$ when $f$ is a step function with
jumps only at rational points.

Drawing inspiration from previous work on the free Schrödinger evolution, we
investigate quantum revivals and fractality for the Schrödinger equation in
the presence of potentials:

$\begin{cases}iu_{t}+u_{xx}=V(x)u,\quad x\in\mathbb{T},\ t\in\mathbb{R},\\\
u(0,x)=f(x).\end{cases}$ (1.2)

To state our results clearly, we introduce some notation. The space of
functions of bounded variation on $\mathbb{T}$ is denoted by $BV$. For a
$2\pi$-periodic function $f$, we define its Fourier coefficient as
$\widehat{f}(k)=\frac{1}{2\pi}\int_{r}^{r+2\pi}e^{-ikx}f(x)dx$ for any
$r\in\mathbb{R}$. For every $s\geq 0$, we denote the Sobolev space on
$\mathbb{T}$ equipped with the norm

$\|f\|_{H^{s}}:=\Big{(}\sum_{k\in\mathbb{Z}}\langle
k\rangle^{2s}|\widehat{f}(k)|^{2}\Big{)}^{1/2}$ (1.3)

by $H^{s}$, where $\langle k\rangle=(1+|k|^{2})^{1/2}$. We also use the
notation $H^{s+}:=\bigcup_{\epsilon>0}H^{s+\epsilon}$ and $H^{+}:=H^{0+}$.
Moreover, throughout this paper, if a Banach space of functions
$\mathcal{B}^{s}$ is decreasing (in the sense of the set inclusion) with
respect to a regularity index $s\in\mathbb{R}$, we use the following
convenient notation:

$\mathcal{B}^{s+}:=\bigcup_{\epsilon>0}\mathcal{B}^{s+\epsilon},\quad\mathcal{B}^{s-}:=\bigcap_{\epsilon>0}\mathcal{B}^{s-\epsilon}.$
(1.4)

Examples of $\mathcal{B}^{s}$ include the Sobolev space $H^{s}$, the Besov
space $B_{p,q}^{s}$, and the Hölder space $C^{s}$.

Our first result concerns the dichotomy in regularity at rational and
irrational times.

######  Theorem 1.1.

Let $u$ be a solution to (1.2) with $f\in BV$ and $V\in H^{+}$. For
$t\notin\pi\mathbb{Q}$, the function $u(t,x)$ is continuous with respect to
$x$. If $t\in\pi\mathbb{Q}$ and $f$ has at least one discontinuity on
$\mathbb{T}$, then $u(t,x)$ is bounded and necessarily contains (at most
countable) discontinuities. Moreover, if $f$ is continuous, then $u(t,x)$ is
jointly continuous on $\mathbb{R}\times\mathbb{T}$.

We also determine the fractal dimension of the solution’s graph at irrational
time slices, in relation to the regularity of the potentials and initial data.

######  Theorem 1.2.

Let $u$ be a solution to (1.2) with $f\in BV$ and $V\in H^{+}$. Suppose that

$\sigma_{0}:=\sup\\{\sigma\in\mathbb{R}\colon f\in H^{\sigma}\\}<3/4$

(since $f\in BV$, we have $\sigma_{0}\geq 1/2$). Let
$r_{0}:=\sup\\{r\in\mathbb{R}\colon V\in H^{r}\\}$. Then, for almost all
$t\in\mathbb{R}\setminus\pi\mathbb{Q}$:

  1. $(1)$

The upper Minkowski dimension of the graphs of $\mathrm{Re}u(t,\cdot)$,
$\mathrm{Im}u(t,\cdot)$, and $|u(t,\cdot)|^{2}$ is less than or equal to
$\max\big{\\{}\frac{3}{2},\frac{5}{2}-\sigma_{0}-r_{0}\big{\\}}$.

  2. $(2)$

The upper Minkowski dimension of the graphs of $\mathrm{Re}u(t,\cdot)$ and
$\mathrm{Im}u(t,\cdot)$ is greater than or equal to $\frac{5}{2}-2\sigma_{0}$,
provided that $\sigma_{0}-\frac{1}{2}<r_{0}$.

######  Remark 1.3.

In the above theorems, the potential space $H^{+}$ contains several
interesting functions. A classical result by Haslam-Jones [12] states that the
Fourier coefficients of the unbounded function

$V(x)=\frac{\phi(x)}{|x|^{\nu}(\log|\kappa/x|)^{a}},\quad-\pi\leq x<\pi,$
(1.5)

where $\phi\in BV$, $0<\nu<1$, $\kappa>\pi$ and $a\in\mathbb{R}$, satisfy

$|\widehat{V}(0)|\lesssim
1\quad\text{and}\quad\widehat{V}(k)=O\big{(}|k|^{\nu-1}(\log|k|)^{-a}\big{)},\
\ k\neq 0.$

Therefore, for every $a\in\mathbb{R}$ and $\nu<\frac{1}{2}$, the unbounded
potential (1.5) belongs to $H^{+}$. Moreover, since $C^{\alpha}\hookrightarrow
H^{s}$ for $0<s<\alpha\leq 1$ ([14, Theorem 1.13]), we conclude that the space
$H^{+}$ includes all classes of Hölder continuous periodic functions.

#### Organization

In Section 2, we present the crucial smoothing estimate (Proposition 2.1) for
the Duhamel part of the initial value problem (1.2), as well as the relevant
results on the free Schr”odinger evolution that we require. We then prove
Theorems 1.1 and 1.2. The remaining sections are devoted to proving the
smoothing estimate. In Section 3, we first establish bilinear estimates in the
Bourgain space for the potential term $Vu$ in (1.2). Building on this, we
establish the well-posedness of (1.2) in Section 4, and subsequently prove the
smoothing estimate.

#### Notation

In our use of inequalities, we adopt the symbol $C$ to represent a positive
constant that may vary in each occurrence. For $A,B>0$, we express $A\lesssim
B$ when $A\leq CB$ with some constant $C>0$. We also use the notation
$A\approx B$ to signify that $A\lesssim B$ and $B\lesssim A$.

##  2\. Proofs of the theorems

In this section, we prove Theorems 1.1 and 1.2. Let us first rearrange the
equation in (1.2) as follows:

$iu_{t}+\big{(}\partial_{xx}-\widehat{V}(0)\big{)}u=R(V,u),$

where $R(V,u)$ is the function on $\mathbb{R}\times\mathbb{T}$ defined by

$R(V,u)(t,x)=\big{(}V(x)-\widehat{V}(0)\big{)}u(t,x).$

Then, by Duhamel’s formula, the solution to the initial value problem (1.2)
can be equivalently written as

$u(t,x)=e^{it(\partial_{xx}-\widehat{V}(0))}f(x)-i\int_{0}^{t}e^{i(t-t^{\prime})(\partial_{xx}-\widehat{V}(0))}R(V,u)(t^{\prime},x)dt^{\prime}.$

We shall prove that the Duhamel term

$\mathcal{P}(t,x):=-i\int_{0}^{t}e^{i(t-t^{\prime})(\partial_{xx}-\widehat{V}(0))}R(V,u)(t^{\prime},x)dt^{\prime}$
(2.1)

is in fact continuous on $\mathbb{R}\times\mathbb{T}$. For the purpose we
obtain the following smoothing property for $\mathcal{P}(t,\cdot)$, which is
the key ingredient in this paper. We provide the proof of the property in
Section 4.2.

######  Proposition 2.1.

Suppose that $V\in H^{r}$ and $f\in H^{s}$ for $r\geq 0$ and $0<s<r+1$. If

$a\leq r\quad\text{and}\quad a<\min\\{1+r-s,1/2\\},$

then $\mathcal{P}(t,\cdot)\in H^{s+a}$ for every $t\in\mathbb{R}$, and is
continuous in the time variable $t$.

For the evolution $e^{it(\partial_{xx}-\widehat{V}(0))}$, we make use of the
following known result due to Oskolkov [16, Proposition 14 and page 390]:

######  Proposition 2.2.

Suppose that $g\in BV$.

  1. $(\romannum{1})$

If $t\notin\pi\mathbb{Q}$, then $e^{it\partial_{xx}}g$ is a continuous
function of $x$. If $g$ has at least one discontinuity on $\mathbb{T}$ and
$t\in\pi\mathbb{Q}$, then $e^{it\partial_{xx}}g$ necessarily contains
discontinuities.

  2. $(\romannum{2})$

If $g$ is continuous, then $e^{it\partial_{xx}}g$ is jointly continuous in
temporal and spatial variables.

######  Remark 2.3.

The quantization results of Berry and Klein [3] and Taylor [22] state that if
$t\in\pi\mathbb{Q}$ then $e^{it\partial_{xx}}g$ is a linear sum of finitely
many translates of $g$ (see [10, Theorem 2.14]). Hence, in Proposition 2.2
$(\romannum{1})$, $e^{it\partial_{xx}}g\in BV$ whenever $t\in\pi\mathbb{Q}$,
so the possible discontinuities in this case are at most countable.

###### Proof of Theorem 1.1.

By the assumption $V\in H^{+}$ it is possible to pick a sufficiently small
number $\alpha\in(0,\frac{1}{8})$ such that $V\in H^{2\alpha}$. Since $f\in
BV$ it follows that $f\in H^{\frac{1}{2}-}$, and in particular, $f\in
H^{\frac{1}{2}-\alpha}$. Thus, making use of Proposition 2.1 (with
$r=2\alpha$, $s=\frac{1}{2}-\alpha$ and $a=2\alpha$), we conclude that
$\mathcal{P}(t,x)\in C_{t}H_{x}^{\frac{1}{2}+\alpha}$. Moreover, from the
Sobolev embedding

$H^{s}\hookrightarrow C^{s-\frac{1}{2}}\quad\text{for}\quad s>1/2,$ (2.2)

it follows that

$\mathcal{P}(t,x)\in C_{t}C_{x}^{\alpha}.$ (2.3)

For $t\notin\pi\mathbb{Q}$, Proposition 2.2 $(\romannum{1})$ shows that

$e^{-it(\partial_{xx}-\widehat{V}(0))}f=e^{it\widehat{V}(0)}e^{-it\partial_{xx}}f$
(2.4)

is continuous. Hence, combined with (2.3), it follows from (2.1) that

$u(t,\cdot)=e^{-it(\partial_{xx}-\widehat{V}(0))}f(\cdot)+\mathcal{P}(t,\cdot)$
(2.5)

is also continuous.

If $t\in\pi\mathbb{Q}$ and $f$ is discontinuous, then it follows from
Proposition 2.2 $(\romannum{1})$ and Remark 2.3 that the evolution (2.4) is of
bounded variation and contains (at most countable) discontinuity. Thus, (2.3)
tells us that the solution (2.5) is a discontinuous bounded function with at
most countable discontinuity.

If $f$ is continuous, then the function (2.4) is also continuous by
Proposition 2.2 $(\romannum{2})$. Hence, combining with (2.3) we see that the
solution (2.5) is jointly continuous on $\mathbb{R}\times\mathbb{T}$. ∎

Now we prove Theorem 1.2. Let us first recall the Besov space and its
properties that we need. Let $\phi\in C^{\infty}_{0}([-2,-1/2]\cup[1/2,2])$ be
such that $\sum_{j\in\mathbb{Z}}\phi(2^{-j}t)=1$ for
$t\in\mathbb{R}\setminus\\{0\\}$, and let $\phi_{0}(t):=1-\sum_{j\geq
1}\phi(2^{-j}t)$. We denote by $P_{j}$ the projections defined by

$P_{0}f(x):=\sum_{k\in\mathbb{Z}}\phi_{0}(k)\widehat{f}(k)e^{ikx},\quad
P_{j}f(x):=\sum_{k\in\mathbb{Z}}\phi(2^{-j}k)\widehat{f}(k)e^{ikx},\ \ j\geq
1.$

For $1\leq p,q\leq\infty$ and $s\geq 0$, the inhomogeneous Besov space
$B^{s}_{p,q}$ on the periodic domain $\mathbb{T}$ is a Banach space of
functions equipped with the norm

$\|f\|_{B_{p,q}^{s}}:=\begin{cases}\|P_{0}f\|_{L^{p}(\mathbb{T})}+\big{(}\sum_{j\geq
1}(2^{sj}\|P_{j}f\|_{L^{p}(\mathbb{T})})^{q}\big{)}^{1/q}&\text{if}\quad
q<\infty,\\\ \sup_{j\geq 0}2^{sj}\|P_{j}f\|_{L^{p}(\mathbb{T})}&\text{if}\quad
q=\infty.\end{cases}$ (2.6)

It is well-known111See, for example, [19] (Remark 4 on p. 164 and Theorem (i),
(v) on pp. 168–169). that $B_{2,2}^{s}=H^{s}$ for every $s$ and
$B^{\alpha}_{\infty,\infty}=C^{\alpha}$ for
$\alpha\in(0,\infty)\setminus\mathbb{N}$. By complex interpolation between
Besov spaces (see [1, Theorem 6.4.5]) and Hölder’s inequality, we have, for
$s_{1}\neq s_{2}$,

$\big{(}B_{1,\infty}^{s_{1}},B_{\infty,\infty}^{s_{2}}\big{)}_{[\frac{1}{2}]}=B_{2,\infty}^{\frac{s_{1}+s_{2}}{2}}\hookrightarrow
B_{2,2}^{\frac{s_{1}+s_{2}}{2}-}=H^{\frac{s_{1}+s_{2}}{2}-}.$ (2.7)

We also make use of the following lemmas due to Chousionis–Erdoğan–Tzirakis
[7] and Deliu–Jawerth [8].

######  Lemma 2.4 ([7]).

Let $\frac{1}{2}\leq\sigma_{0}<\frac{3}{4}$ and suppose $g\in BV\setminus
H^{\sigma_{0}+}$. Then, for almost all $t\in\mathbb{R}\setminus\pi\mathbb{Q}$,
we have $e^{it\partial_{xx}}g\in C^{\frac{1}{2}-}\setminus
B_{1,\infty}^{2\sigma_{0}-\frac{1}{2}+}$.

######  Lemma 2.5 ([8]).

Let $0<s<1$. Assume that $f\colon\mathbb{T}\to\mathbb{R}$ is continuous and
$f\notin B_{1,\infty}^{s+}$. Then the upper Minkowski dimension of the graph
of $f$ is at least $2-s$.

###### Proof of Theorem 1.2.

Let us first prove the part (1). By the definition of $\sigma_{0}$ and $r_{0}$
it is clear that

$f\in H^{\sigma_{0}-}\quad\text{and}\quad V\in H^{r_{0}-}.$

By applying Proposition 2.1 (with $s=\sigma_{0}-\epsilon$ and
$r=r_{0}-2\epsilon$ for an infinitesimal $0<\epsilon\ll 1$) we have

$\mathcal{P}(t,\cdot)\in H^{\sigma_{0}+a},\quad\forall t\in\mathbb{R},$ (2.8)

whenever

$a<\min\\{r_{0},1+r_{0}-\sigma_{0},1/2\\}=\min\\{r_{0},1/2\\}=:a_{0}.$

Hence, by the Sobolev embedding (2.2) we have

$\mathcal{P}(t,\cdot)\in C^{\sigma_{0}+a-\frac{1}{2}},\quad\forall
t\in\mathbb{R},$ (2.9)

for every $a<a_{0}$. On the other hand, by Lemma 2.4, we have

$e^{it\partial_{xx}}f\in C^{\frac{1}{2}-}\quad\text{for almost all}\ \
t\in\mathbb{R}\setminus\pi\mathbb{Q}.$ (2.10)

Hence, it follows from (2.5), (2.9) and (2.10) that

$u(t,\cdot)\in
C^{\min\\{\frac{1}{2},\sigma_{0}+a-\frac{1}{2}\\}-}\quad\text{for almost all}\
\ t\in\mathbb{R}\setminus\pi\mathbb{Q}$ (2.11)

for every $a<a_{0}$. It is obvious that the same statement is still valid for
$\mathrm{Re}u(t,\cdot)$, $\mathrm{Im}u(t,\cdot)$ and $|u(t,\cdot)|^{2}$ in
place of $u(t,\cdot)$ in (2.11). We now use the following classical result on
the upper Minkowski dimension for Hölder continuous functions (for a proof we
refer the reader to [11, Corollary 11.2]):

######  Lemma 2.6.

Let $0\leq\alpha\leq 1$. If a function $f\colon\mathbb{T}\to\mathbb{R}$ is
$C^{\alpha}$, then the upper Minkowski dimension of the graph of $f$ is at
most $2-\alpha$.

Indeed, by Lemma 2.6, the upper Minkowski dimension of the graphs of
$\mathrm{Re}u(t,\cdot)$, $\mathrm{Im}u(t,\cdot)$ and $|u(t,\cdot)|^{2}$ is at
most

$2-\min\\{1/2,\sigma_{0}+a_{0}-1/2\\}=\max\\{3/2,5/2-\sigma_{0}-a_{0}\\}$

for almost all $t\in\mathbb{R}\setminus\pi\mathbb{Q}$. We notice further that
if $a_{0}=1/2$, then $5/2-\sigma_{0}-a_{0}=2-\sigma_{0}\in(5/4,3/2]$, so the
maximum is equal to $3/2$. Therefore, we obtain (1).

Let us now prove the part (2). Since $f\notin H^{\sigma_{0}+}$, we see that
for almost every $t$, neither $\mathrm{Re\,}e^{it\partial_{xx}}f$ nor
$\mathrm{Im\,}e^{it\partial_{xx}}f$ belong to $H^{\sigma_{0}+}$ (see [7, Lemma
3.2]). It follows from (2.10) and the embedding $B_{1,\infty}^{s_{1}}\cap
B_{\infty,\infty}^{s_{2}}\subset H^{\frac{s_{1}+s_{2}}{2}-}$ (see (2.7)) that,
for almost all $t\in\mathbb{R}\setminus 2\pi\mathbb{Q}$, both the real and the
imaginary parts of $e^{it\partial_{xx}}f$ do not belong to
$B_{1,\infty}^{2\sigma_{0}-\frac{1}{2}+}$. On the other hand, by (2.8) we see
$\mathcal{P}(t,\cdot)\in
B_{1,\infty}^{\sigma_{0}+\min\\{r_{0},\frac{1}{2}\\}-}$ for all
$t\in\mathbb{R}$. Combining these we have

$u(t,\cdot)=\underbrace{\mathrm{Re}(e^{it(\partial_{xx}-\widehat{V}(0))}f)}_{\notin
B^{2\sigma_{0}-\frac{1}{2}+}_{1,\infty}\text{ for a.e.
}t}+i\underbrace{\mathrm{Im}(e^{it(\partial_{xx}-\widehat{V}(0))}f)}_{\notin
B^{2\sigma_{0}-\frac{1}{2}+}_{1,\infty}\text{ for a.e.
}t}+\underbrace{\mathcal{P}(t,\cdot)}_{\in
B^{\sigma_{0}+\min\\{r_{0},\frac{1}{2}\\}-}_{1,\infty}\text{ for all }t}.$

From this we conclude that if
$2\sigma_{0}-\frac{1}{2}<\sigma_{0}+\min\\{r_{0},\frac{1}{2}\\}$, that is,
either $r_{0}\geq\frac{1}{2}$ or $\sigma_{0}-\frac{1}{2}<r_{0}<\frac{1}{2}$,
then for almost all $t\notin 2\pi\mathbb{Q}$ neither the real nor the
imaginary parts of $u(t,\cdot)$ belong to
$B^{2\sigma_{0}-\frac{1}{2}+}_{1,\infty}$. By Lemma 2.5, we conclude that the
graphs of real and imaginary parts of $u(t,\cdot)$ have Minkowski dimension
$\geq 2-(2\sigma_{0}-\frac{1}{2})=\frac{5}{2}-2\sigma_{0}$. ∎

##  3\. Bilinear estimate in the Bourgain space

In this section, we obtain bilinear estimates for $R(V,u)$ in the Bourgain
space, which are essential in proving Proposition 2.1.

###  3.1. The Bourgain space

For $s,b\in\mathbb{R}$, we denote by $X^{s,b}$ the closure of the set of
Schwartz functions $\mathcal{S}(\mathbb{R};C^{\infty}(\mathbb{T}))$ under the
norm

$\|u\|_{X^{s,b}}:=\|\langle
k\rangle^{s}\langle\tau+k^{2}\rangle^{b}\widetilde{u}(\tau,k)\|_{L^{2}_{\tau}l^{2}_{k}(\mathbb{R}\times\mathbb{Z})},$

where $\widetilde{u}$ denotes the space-time Fourier transform of $u$ defined
by

$\widetilde{u}(\tau,k)=\int_{\mathbb{R}}e^{-i\tau
t}\widehat{u}(t,k)dt=\frac{1}{2\pi}\int_{\mathbb{R}}\int^{2\pi}_{0}e^{-i(\tau
t+kx)}u(t,x)dxdt,\quad(\tau,k)\in\mathbb{R}\times\mathbb{Z}.$

The $X^{s,b}$-space is also called the _Bourgain space_ or _dispersive Sobolev
space_. We also define, for a closed interval $I\subset\mathbb{R}$, the
restricted space $X^{s,b}_{I}$ as the Banach space of functions on
$I\times\mathbb{T}$ equipped with the norm

$\|u\|_{X^{s,b}_{I}}=\inf\big{\\{}\|w\|_{X^{s,b}}\colon
w|_{I\times\mathbb{T}}=u\big{\\}}.$

We notice that $X^{s,b}_{\mathbb{R}}=X^{s,b}$.

The followings are some basic properties of the Bourgain space that we need to
prove the smoothing estimate (Proposition 2.1). The proofs of the properties,
which we omit, can be obtained by routine adaptation of those of [21,
Corollary 2.10, Lemma 2.11 and Proposition 2.12] in the spatially periodic
setting and time translation argument.

######  Lemma 3.1.

Let $s\in\mathbb{R}$, $b>\frac{1}{2}$ and $I\subset\mathbb{R}$ a closed
interval. Then $X_{I}^{s,b}\hookrightarrow C(I;H^{s})$ and

$\sup_{t\in I}\|u(t,\cdot)\|_{H^{s}}\leq C\|u\|_{X_{I}^{s,b}},$

where $C$ is a constant depending only on $b$.

######  Lemma 3.2.

Let $s\in\mathbb{R}$, $-\frac{1}{2}<b<b^{\prime}<\frac{1}{2}$ and $I$ a closed
interval of length $\delta$. Then

$\|u\|_{X^{s,b}_{I}}\leq C\delta^{b^{\prime}-b}\|u\|_{X^{s,b^{\prime}}_{I}},$

where the constant $C$ depends only on $b$ and $b^{\prime}$.

######  Lemma 3.3.

Let $s\in\mathbb{R}$, $\frac{1}{2}<b\leq 1$ and $I=[t_{0},t_{0}+\delta]$ for
$t_{0}\in\mathbb{R}$ and $0<\delta\leq 1$. Then, for $t\in I$, we have

$\displaystyle\|e^{i(t-t_{0})\partial_{xx}}f\|_{X^{s,b}_{I}}\leq
C\|f\|_{H^{s}},$
$\displaystyle\Big{\|}\int^{t}_{t_{0}}e^{i(t-t^{\prime})\partial_{xx}}F(t^{\prime},\cdot)dt^{\prime}\Big{\|}_{X^{s,b}_{I}}\leq
C\|F\|_{X^{s,b-1}_{I}},$

where $C$ depends only on $b$.

###  3.2. Bilinear estimate for $R(V,u)$

In this section we estimate $R(V,u)=(V-\widehat{V}(0))u$ in the
$X^{s,b}$-space.

######  Proposition 3.4.

Let $r\geq 0$ and $0<s<1+r$, and suppose that

$a\leq r,\quad a<1+r-s\quad\text{and}\quad a<1/2.$

Then, for every interval $I$,

$\|R(V,u)\|_{X_{I}^{s+a,b^{\prime}-1}}\lesssim\|V\|_{H^{r}}\|u\|_{X_{I}^{s,b}}$
(3.1)

provided that $b,b^{\prime}\in(\frac{1}{2},\frac{1}{2}+\epsilon)$ for an
$\epsilon>0$ small enough.

We first recall from [9, Lemma 3.3] the following simple lemma which is used
several times in proving the proposition.

######  Lemma 3.5.

Let us define, for $\beta\geq 0$,

$\phi_{\beta}(k):=\sum_{|n|\leq|k|}\frac{1}{\langle
n\rangle^{\beta}}\approx\begin{cases}1,&\beta>1,\\\ \log(1+\langle
k\rangle),&\beta=1,\\\ \langle k\rangle^{1-\beta},&\beta<1.\end{cases}$ (3.2)

If $\beta\geq\gamma\geq 0$ and $\beta+\gamma>1$, then

$\displaystyle\sum_{n}\frac{1}{\langle n-k_{1}\rangle^{\beta}\langle
n-k_{2}\rangle^{\gamma}}\approx\int_{\mathbb{R}}\frac{1}{\langle\tau-
k_{1}\rangle^{\beta}\langle\tau-
k_{2}\rangle^{\gamma}}d\tau\lesssim\frac{\phi_{\beta}(k_{1}-k_{2})}{\langle
k_{1}-k_{2}\rangle^{\gamma}}.$

###### Proof of Proposition 3.4.

First, let us prove (3.1) by replacing the restricted spaces
$X_{I}^{s+a,b^{\prime}-1}$ and $X_{I}^{s,b}$ with $X^{s+a,b^{\prime}-1}$ and
$X^{s,b}$, respectively. We write

$\displaystyle\|R(V,u)\|_{X^{s+a,b^{\prime}-1}}^{2}$ (3.3)
$\displaystyle=\int_{\mathbb{R}}\sum_{k\in\mathbb{Z}}\bigg{|}\sum_{l\in\mathbb{Z}\setminus\\{k\\}}\langle
k\rangle^{s+a}\langle\tau+k^{2}\rangle^{b^{\prime}-1}\widehat{V}(k-l)\widetilde{u}(\tau,l)\bigg{|}^{2}d\tau$
$\displaystyle=\int_{\mathbb{R}}\sum_{k\in\mathbb{Z}}\bigg{|}\sum_{l\in\mathbb{Z}\setminus\\{k\\}}M(k,l,\tau)\langle
k-l\rangle^{r}\widehat{V}(k-l)\langle
l\rangle^{s}\langle\tau+l^{2}\rangle^{b}\,\widetilde{u}(\tau,l)\bigg{|}^{2}d\tau,$

where

$M(k,l,\tau):=\langle
k\rangle^{s+a}\langle\tau+k^{2}\rangle^{b^{\prime}-1}\langle
l\rangle^{-s}\langle k-l\rangle^{-r}\langle\tau+l^{2}\rangle^{-b}.$

By the Cauchy–Schwarz inequality and Young’s inequality for the convolution,
(3.3) is bounded by

$\displaystyle\sup_{\tau,k}\sum_{l\in\mathbb{Z}\setminus\\{k\\}}M(k,l,\tau)^{2}\int\sum_{k\in\mathbb{Z}}\sum_{m\in\mathbb{Z}}\langle
k-m\rangle^{2r}|\widehat{V}(k-m)|^{2}\langle
m\rangle^{2s}\langle\tau+m^{2}\rangle^{2b}|\widetilde{u}(\tau,m)|^{2}d\tau$
$\displaystyle\leq\sup_{\tau,k}\sum_{l\in\mathbb{Z}\setminus\\{k\\}}M(k,l,\tau)^{2}\|V\|_{H^{r}}^{2}\|u\|_{X^{s,b}}^{2}.$

Hence it remains to prove that

$\sup_{\tau,k}\sum_{l\in\mathbb{Z}\setminus\\{k\\}}M(k,l,\tau)^{2}<\infty.$

From the assumption, it is clear that $2b\geq 2-2b^{\prime}$. Hence, by the
easy inequality
$\langle\tau+m\rangle\langle\tau+n\rangle\geq\frac{1}{2}\langle n-m\rangle$,
we have

$\displaystyle\sum_{l\in\mathbb{Z}\setminus\\{k\\}}M(k,l,\tau)^{2}$
$\displaystyle\leq\sum_{l\in\mathbb{Z}\setminus\\{k\\}}\frac{\langle
k\rangle^{2s+2a}\langle l\rangle^{-2s}\langle
k-l\rangle^{-2r}}{\langle\tau+k^{2}\rangle^{2-2b^{\prime}}\langle\tau+l^{2}\rangle^{2-2b^{\prime}}}$
$\displaystyle\lesssim\sum_{l\in\mathbb{Z}\setminus\\{k\\}}\frac{\langle
k\rangle^{2s+2a}\langle l\rangle^{-2s}\langle k-l\rangle^{-2r}}{\langle
l^{2}-k^{2}\rangle^{2-2b^{\prime}}}$
$\displaystyle\lesssim\sum_{l\in\mathbb{Z}\setminus\\{\pm k\\}}\frac{\langle
k\rangle^{2s+2a}\langle l\rangle^{-2s}\langle k-l\rangle^{-2r}}{\langle
l+k\rangle^{2-2b^{\prime}}\langle l-k\rangle^{2-2b^{\prime}}}+\langle
k\rangle^{2a-2r}.$ (3.4)

Since $a\leq r$ it is obvious that $\langle k\rangle^{2a-2r}\leq 1$, so we
need only to show the summation in (3.4) is bounded uniformly in
$k\in\mathbb{Z}$. For the purpose, we break the set of summation indices
$\mathbb{Z}\setminus\\{\pm k\\}$ and separately consider the contribution of

$A_{k}:=\\{l\in\mathbb{Z}\colon\,l\neq\pm k,\,|k-l|\geq|k|/2\\}\ \ \text{and}\
\ B_{k}:=\\{l\in\mathbb{Z}\colon\,l\neq\pm k,\,|l|>|k|/2\\}.$

It is obvious that $A_{k}\cup B_{k}=\mathbb{Z}\setminus\\{\pm k\\}$.

Let us first prove the boundedness of the summation in (3.4) over the set
$A_{k}$. If $l\in A_{k}$ then $\langle k-l\rangle\gtrsim\langle k\rangle$. It
is also clear from the assumptions on $s$ and $b^{\prime}$ that
$2-2b^{\prime}+2s>1$. Hence application of Lemma 3.5 yields

$\displaystyle\sum_{l\in A_{k}}\frac{\langle k\rangle^{2s+2a}\langle
l\rangle^{-2s}\langle k-l\rangle^{-2r}}{\langle
l+k\rangle^{2-2b^{\prime}}\langle l-k\rangle^{2-2b^{\prime}}}$
$\displaystyle\lesssim\langle k\rangle^{2s+2a-2r+2b^{\prime}-2}\sum_{l\in
A_{k}}\frac{1}{\langle l+k\rangle^{2-2b^{\prime}}\langle l\rangle^{2s}}$
$\displaystyle\lesssim\langle
k\rangle^{2s+2a-2r+2b^{\prime}-2}\frac{\phi_{\max\\{2s,2-2b^{\prime}\\}}(k)}{\langle
k\rangle^{\min\\{2s,2-2b^{\prime}\\}}}.$ (3.5)

In the case $s\geq\frac{1}{2}$, $\max\\{2s,2-2b^{\prime}\\}=2s$ since
$b^{\prime}>\frac{1}{2}$. It follows from (3.2) that

$\eqref{e:phk}=\langle
k\rangle^{2s+2a-2r+4b^{\prime}-4}\phi_{2s}(k)\lesssim\langle
k\rangle^{2s+2a-2r+4b^{\prime}-4}\log(1+\langle k\rangle).$

Since $b^{\prime}<\frac{1}{2}+\epsilon$, the last quantity is again bounded by

$\langle k\rangle^{2s+2a-2r-2+4\epsilon}\log(1+\langle k\rangle),$

which is bounded uniformly in $k$ since $a<1+r-s$ and $\epsilon$ is small
enough.

In the other case $0<s<\frac{1}{2}$, we have
$\max\\{2s,2-2b^{\prime}\\}=2-2b^{\prime}$ when $\epsilon$ is sufficiently
small. Thus, we get from (3.2) that

$\eqref{e:phk}=\langle
k\rangle^{2a-2r+2b^{\prime}-2}\phi_{2-2b^{\prime}}(k)\approx\langle
k\rangle^{2a-2r+4b^{\prime}-3}\leq\langle k\rangle^{2a-2r-1+4\epsilon},$

which is uniformly bounded since $a<\frac{1}{2}$ and $\epsilon$ is small
enough.

We now show the boundedness of the summation in (3.4) over the set $B_{k}$. We
note that $|k|\gtrsim|l|$ on $B_{k}$. Since $r\geq 0$ and
$b^{\prime}-\frac{1}{2}<\epsilon\ll 1$ we see that $2(2-2b^{\prime})+2r>1$.
Hence, application of Lemma 3.5 gives

$\displaystyle\sum_{l\in B_{k}}\frac{\langle k\rangle^{2s+2a}\langle
l\rangle^{-2s}\langle k-l\rangle^{-2r}}{\langle
l+k\rangle^{2-2b^{\prime}}\langle l-k\rangle^{2-2b^{\prime}}}$
$\displaystyle\lesssim\langle k\rangle^{2a}\sum_{l\in B_{k}}\frac{1}{\langle
l+k\rangle^{2-2b^{\prime}}\langle l-k\rangle^{2-2b^{\prime}+2r}}$
$\displaystyle\lesssim\langle
k\rangle^{2a+2b^{\prime}-2}\phi_{2-2b^{\prime}+2r}(k).$ (3.6)

By the estimate (3.2) we have

$\eqref{e:phk1}\lesssim\begin{cases}\langle
k\rangle^{2a+4b^{\prime}-3}&\text{if}\quad r=0,\\\ \langle
k\rangle^{2a+2b^{\prime}-2}&\text{if}\quad r>0,\end{cases}$

both of which are uniformly bounded since $a<\frac{1}{2}$ and $\epsilon$ is
sufficiently small.

We now prove (3.1). Let $\chi_{I}\in C_{0}^{\infty}(\mathbb{R})$ be such that
$\chi_{I}=1$ on $I$. By the definition of the restricted Bourgain space and
the estimate what we have obtained, we see that

$\|R(V,u)\|_{X_{I}^{s+a,b^{\prime}-1}}\leq\|\chi_{I}R(V,u)\|_{X^{s+a,b^{\prime}-1}}\lesssim\|V\|_{H^{r}}\|\chi_{I}u\|_{X^{s,b}}.$

Taking infimum over all functions $w$ such that $w|_{I\times\mathbb{R}}=u$ on
the right side, we get the estimate (3.1). ∎

##  4\. Local well-posedness and smoothing estimate

In this section, we make use of Proposition 3.4 and employ the contraction
mapping principle to establish local well-posedness in the Sobolev spaces for
the initial value problem (1.2). We then prove Proposition 2.1.

Let $u$ be the solution to the equation (1.2) and set
$v(t,x):=e^{it\widehat{V}(0)}u(t,x)$. Since

$\displaystyle iv_{t}+v_{xx}$
$\displaystyle=e^{it\widehat{V}(0)}\big{(}iu_{t}-\widehat{V}(0)u+u_{xx}\big{)}$
$\displaystyle=e^{it\widehat{V}(0)}\big{(}V-\widehat{V}(0)\big{)}u=\big{(}V-\widehat{V}(0)\big{)}v=R(V,v),$

the initial value problem (1.2) is equivalent to the following:

$\begin{cases}iv_{t}+v_{xx}=R(V,v),\\\ v(0,x)=f(x).\end{cases}$ (4.1)

Let us recall the definition of $\mathcal{P}(t,x)$ from (2.1) and note that

$v(t,x)-e^{it\partial_{xx}}f=e^{it\widehat{V}(0)}\big{(}u(t,x)-e^{it(\partial_{xx}-\widehat{V}(0))}f\big{)}=e^{it\widehat{V}(0)}\mathcal{P}(t,x).$
(4.2)

Since
$\|\mathcal{P}(t,x)\|_{H^{s}}=\|e^{it\widehat{V}(0)}\mathcal{P}(t,x)\|_{H^{s}}$,
in order to prove Proposition 2.1 it is enough to prove the smoothing estimate
for $e^{it\widehat{V}(0)}\mathcal{P}(t,x)$ instead of $\mathcal{P}(t,x)$.
Therefore, in this section, we may and shall consider the equation (4.1)
instead of (1.2).

###  4.1. Local well-posedness

We now prove the initial value problem (4.1) is locally well-posed in $H^{s}$.
We use the notation $X^{s,b}_{\delta}:=X^{s,b}_{[0,\delta]}$.

######  Theorem 4.1.

Let $V\in H^{r}$ for $r\geq 0$ and suppose $0<s<1+r$ and
$\frac{1}{2}<b<\frac{1}{2}+\epsilon$ for some $\epsilon>0$ small enough. For
every $g\in H^{s}$, there exists a time $\delta>0$ and a unique solution
$v_{g}\in X_{\delta}^{s,b}$ for the integral equation

$v_{g}(t,x)=e^{it\partial_{xx}}g-i\int_{0}^{t}e^{i(t-t^{\prime})\partial_{xx}}R(V,v_{g})(t^{\prime})dt^{\prime}.$
(4.3)

Furthermore, the mapping $g\mapsto v_{g}$ is Lipschitz continuous with the
estimate

$\|v_{g}\|_{X_{\delta}^{s,b}}\lesssim\|g\|_{H^{s}}.$ (4.4)

######  Remark 4.2.

By Lemma 3.1 we have $X_{\delta}^{s,b}\hookrightarrow C([0,\delta],H^{s})$.
Thus, the theorem establishes the local well-posedness of (4.1) (hence (1.1))
in $H^{s}$, in the sense of [21, Definition 3.4].

######  Remark 4.3.

In fact, as can be seen in the proof (see (4.6) below), the small time
$\delta$ is independent of the initial data $f\in H^{s}$. Since the initial
value problem (4.1) is invariant under time translations, we can patch the
solution (4.3) along $t\in\mathbb{R}$ by repeatedly applying Theorem 4.1 to
obtain the global solution $v\in C(\mathbb{R},H^{s})$ to (4.1). Also, the
estimate (4.4) combined with Lemma 3.1 implies the following global bound

$\|v(t)\|_{H^{s}}\lesssim e^{|t|}\|f\|_{H^{s}},\quad\forall t\in\mathbb{R}.$
(4.5)

###### Proof of Theorem 4.1.

For each $g\in H^{s}$ we define

$\Gamma_{g}(v)(t,x):=e^{it\partial_{xx}}g-i\int_{0}^{t}e^{i(t-t^{\prime})\partial_{xx}}R(V,v)(t^{\prime})dt^{\prime}.$

We aim to show that, for some small time $0<\delta<1$ and $K>0$ to be chosen
later, the mapping $\Gamma_{g}$ is a contraction on the set

$X:=\big{\\{}w\in X_{\delta}^{s,b}\colon\|w\|_{X_{\delta}^{s,b}}\leq
K\|g\|_{H^{s}}\big{\\}}.$

Let us first show that the mapping $\Gamma_{g}\colon X\to X$ is well-defined.
Lemmas 3.2 and 3.3 imply that if $w\in X$, then

$\displaystyle\|\Gamma_{g}(w)\|_{X_{\delta}^{s,b}}$
$\displaystyle\leq\|e^{it\partial_{xx}}g\|_{X_{\delta}^{s,b}}+\Big{\|}\int_{0}^{t}e^{i(t-t^{\prime})\partial_{xx}}R(V,w)(t^{\prime})dt^{\prime}\Big{\|}_{X_{\delta}^{s,b}}$
$\displaystyle\leq C\|g\|_{H^{s}}+C\|R(V,w)\|_{X_{\delta}^{s,b-1}}$
$\displaystyle\leq
C\|g\|_{H^{s}}+C\delta^{b^{\prime}-b}\|R(V,w)\|_{X_{\delta}^{s,b^{\prime}-1}}$

whenever $0<\delta<1$ and $\frac{1}{2}<b<b^{\prime}<1$. We then apply
Proposition 3.4 (with $a=0$) to see that

$\|\Gamma_{g}(w)\|_{X_{\delta}^{s,b}}\leq
C\|g\|_{H^{s}}+C\delta^{b^{\prime}-b}\|V\|_{H^{r}}\|w\|_{X_{\delta}^{s,b}},$

provided that $\frac{1}{2}<b<b^{\prime}<\frac{1}{2}+\epsilon$ for $\epsilon>0$
small enough. Since $w\in X$ we get

$\|\Gamma_{g}(w)\|_{X_{\delta}^{s,b}}\leq
C_{0}\big{(}1+K\delta^{b^{\prime}-b}\|V\|_{H^{r}}\big{)}\|g\|_{H^{s}}.$

If we set $K\geq 2C_{0}$ and take $0<\delta<1$ so small that

$\delta^{b^{\prime}-b}<1/(1+K\|V\|_{H^{r}}),$ (4.6)

then it follows that

$\|\Gamma_{g}(w)\|_{X_{\delta}^{s,b}}\leq K\|g\|_{H^{s}}.$

Therefore, $\Gamma_{g}(X)\subset X$ for $g\in H^{s}$.

Secondly, let us prove that the map $\Gamma_{g}\colon X\to X$ is a
contraction. In a similar manner, by Lemmas 3.2 and 3.3 and Proposition 3.4 we
have

$\displaystyle\|\Gamma_{g}(w_{1})-\Gamma_{g}(w_{2})\|_{X_{\delta}^{s,b}}$
$\displaystyle\leq
C\delta^{b^{\prime}-b}\|R(V,w_{1}-w_{2})\|_{X_{\delta}^{s,b^{\prime}-1}}$
$\displaystyle\leq
C\delta^{b^{\prime}-b}\|V\|_{H^{r}}\|w_{1}-w_{2}\|_{X_{\delta}^{s,b}},$

which implies $\Gamma_{g}$ is a contraction on $X$ for $\delta$ satisfies
(4.6), if we set $K\geq 2C$ here. By applying the contraction mapping
principle, it follows that there exists a unique $v_{g}\in X$ solving the
equation $\Gamma_{g}(v_{g})=v_{g}$.

The continuity of the map $g\mapsto v_{g}$ also follows from utilizing Lemmas
3.2 and 3.3 and Proposition 3.4 as in the above. Indeed, by (4.3) we see that
for $g,h\in H^{s}$,

$\displaystyle\|v_{g}-v_{h}\|_{X_{\delta}^{s,b}}$
$\displaystyle\leq\|e^{it\partial_{xx}}(g-h)\|_{X_{\delta}^{s,b}}+\Big{\|}\int_{0}^{t}e^{i(t-t^{\prime})\partial_{xx}}\big{(}R(V,v_{g})(t^{\prime})-R(V,v_{h})(t^{\prime})\big{)}dt^{\prime}\Big{\|}_{X_{\delta}^{s,b}}$
$\displaystyle\leq
C\|g-h\|_{H^{s}}+C\delta^{b^{\prime}-b}\|V\|_{H^{r}}\|v_{g}-v_{h}\|_{X_{\delta}^{s,b}}$
$\displaystyle\leq
C\|g-h\|_{H^{s}}+\frac{1}{2}\|v_{g}-v_{h}\|_{X_{\delta}^{s,b}},$

from which it follows that
$\|v_{g}-v_{h}\|_{X_{\delta}^{s,b}}\lesssim\|g-h\|_{H^{s}}$. ∎

###  4.2. Smoothing estimate: Proof of Proposition 2.1

In this final section, we prove Proposition 2.1.

Let $f$ and $V$ be given as in the statement of Proposition 2.1. It is enough
to prove that for every $t_{\circ}\geq 0$ there exists a constant $C>0$ such
that

$\|\mathcal{P}(t_{\circ},x)\|_{H^{s+a}}\leq C\|V\|_{H^{r}}\|f\|_{H^{s}}.$
(4.7)

This gives regularity gain for $\mathcal{P}(t,\cdot)$ compared to the global
estimate (4.5) for the solution $v$. In order to prove (4.7), the bilinear
estimate (Proposition 3.4) as well as the local well-posedness (Theorem 4.1)
is crucial.

Let us invoke the $\delta>0$ in Theorem 4.1 and pick $m\in\mathbb{N}$ such
that $(m-1)\delta\leq t_{\circ}<m\delta$. We also set

$v_{j}(x)=v(j\delta,x)\quad\text{and}\quad I_{j}=[\delta j,\delta(j+1)],$
(4.8)

for $j\in\\{0,1,2,\cdots,m-1\\}$. Applying Theorem 4.1 with the initial data
$v_{j}$, it is possible to write the solution $v$ as

$v(t)=e^{i(t-\delta j)\partial_{xx}}v_{j}-i\int_{\delta
j}^{t}e^{i(t-t^{\prime})\partial_{xx}}R(V,v)(t^{\prime})dt^{\prime},\quad t\in
I_{j}.$

By Lemmas 3.1 and 3.3 we have, for $t\in I_{j}$,

$\displaystyle\|v(t)-e^{i(t-\delta j)\partial_{xx}}v_{j}\|_{H^{s+a}}$
$\displaystyle\lesssim\Big{\|}\int_{\delta
j}^{t}e^{i(t-t^{\prime})\partial_{xx}}R(V,v)(t^{\prime})dt^{\prime}\Big{\|}_{X_{I_{j}}^{s+a,b}}$
$\displaystyle\lesssim\|R(V,v)\|_{X_{I_{j}}^{s+a,b-1}}$

whenever $\frac{1}{2}<b\leq 1$. Let us choose $b$ sufficiently close to
$\frac{1}{2}$, and then apply Proposition 3.4 (with $b^{\prime}=b$) to see
that

$\|R(V,v)\|_{X_{I_{j}}^{s+a,b-1}}\lesssim\|V\|_{H^{r}}\|v\|_{X_{I_{j}}^{s,b}}.$

Hence, by the estimate (4.5), we conclude that

$\|v(t)-e^{i(t-\delta j)\partial_{xx}}v_{j}\|_{H^{s+a}}\leq
Ce^{t_{\circ}}\|V\|_{H^{r}}\|f\|_{H^{s}},\quad t\in I_{j},$ (4.9)

for $j\in\\{0,1,2,\cdots,m-1\\}$. We recall (4.2) and write

$\displaystyle
e^{it_{\circ}\widehat{V}(0)}\mathcal{P}(t_{\circ},x)=v(t_{\circ})-e^{it_{\circ}\partial_{xx}}f$
$\displaystyle=v(t_{\circ})-e^{i(t_{\circ}-\delta(m-1))\partial_{xx}}v_{m-1}+\sum_{j=1}^{m-1}e^{i(t_{\circ}-\delta
j)\partial_{xx}}\big{(}v_{j}-e^{i\delta\partial_{xx}}v_{j-1}\big{)}.$

Application of the estimate (4.9) gives

$\displaystyle\|\mathcal{P}(t_{\circ},\cdot)\|_{H^{s+a}}$
$\displaystyle\leq\|v(t_{\circ})-e^{i(t_{\circ}-\delta(m-1))\partial_{xx}}v_{m-1}\|_{H^{s+a}}+\sum_{j=1}^{m-1}\|v_{j}-e^{i\delta\partial_{xx}}v_{j-1}\|_{H^{s+a}}$
$\displaystyle\leq Cme^{t_{\circ}}\|V\|_{H^{r}}\|f\|_{H^{s}}.$

Therefore, we obtain the desired estimate (4.7).

## References

  * [1] J. Bergh and J. Löfström, Interpolation spaces. An introduction, Grundlehren der Mathematischen Wissenschaften, No. 223. Springer-Verlag, Berlin-New York, 1976.
  * [2] M. V. Berry, Quantum fractals in boxes, J. Phys. A: Math. Gen. 29 (1996), 6617–6629.
  * [3] M. V. Berry and S. Klein, Integer, fractional and fractal Talbot effects, J. Modern. Opt. 43 (1996), 2139–2164.
  * [4] M. V. Berry, I. Marzoli and W. Schleich, Quantum carpets, carpets of light, Phys. World 14 (6) (2001), 39–44.
  * [5] J. Bourgain, Global solutions of nonlinear Schrödinger equations, American Mathematical Society Colloquium Publications, 46. American Mathematical Society, Providence, RI, 1999.
  * [6] G. Chen and P. J. Olver, Numerical simulation of nonlinear dispersive quantization Discrete Contin. Dyn. Syst. 34 (2014), no. 3, 991–1008.
  * [7] V. Chousionis, M. B. Erdoğan and N. Tzirakis, Fractal solutions of linear and nonlinear dispersive partial differential equations, Proc. Lond. Math. Soc. (3) 110 (2015), no. 3, 543–564.
  * [8] A. Deliu and B. Jawerth, Geometrical dimension versus smoothness, Constr. Approx. 8 (1992), no. 2, 211–222.
  * [9] B. Erdoğan and N. Tzirakis, Smoothing and global attractors for the Zakharov system on the torus, Anal. PDE 6 (2013), no. 3, 723–750.
  * [10] M. B. Erdoğan and N. Tzirakis, _Dispersive partial differential equations_ , London Mathematical Society Student Texts, vol. 86, Cambridge University Press, Cambridge, 2016.
  * [11] K. Falconer, Fractal geometry, Mathematical foundations and applications. Third edition. John Wiley and Sons, Ltd., Chichester, 2014.
  * [12] U. S. Haslam-Jones, A Note on the Fourier Coefficients of Unbounded Functions, J. London Math. Soc. 2 (1927), no. 3, 151–154.
  * [13] L. Kapitanski and I. Rodnianski, Does a quantum particle know the time?, Emerging applications of number theory, IMA Volumes in Mathematics and its Applications 109 (eds D. Hejhal, J. Friedman, M. C. Gutzwiller and A. M. Odlyzko; Springer, New york, 1999), 355–371.
  * [14] C. Muscalu and W. Schlag, _Classical and multilinear harmonic analysis. Vol. I_ , Cambridge Studies in Advanced Mathematics, vol. 137, Cambridge University Press, Cambridge, 2013.
  * [15] P. J. Olver, Dispersive quantization, Amer. Math. Monthly 117 (2010), no. 7, 599–610.
  * [16] K. I. Oskolkov, A class of I. M. Vinogradov’s series and its applications in harmonic analysis, Progress in approximation theory (Tampa, FL, 1990), 353–402, Springer Ser. Comput. Math., 19, Springer, New York, 1992.
  * [17] L. Rayleigh, On copying diffraction-gratings, and on some phenomena connected therewith, Philos. Mag. 11 (1881), 196–205.
  * [18] I. Rodnianski, Fractal solutions of the Schrödinger equation, Contemp. Math. 255 (2000), 181–187.
  * [19] H.-J. Schmeisser and H. Triebel, _Topics in Fourier analysis and function spaces_ , A Wiley-Interscience Publication, John Wiley & Sons, Ltd., Chichester, 1987.
  * [20] H. F. Talbot, Facts relating to optical science. No. IV, Philos. Mag. 9 (1836), 401–407.
  * [21] T. Tao, Nonlinear dispersive equations; Local and global analysis, CBMS Regional Conference Series in Mathematics, 106. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2006.
  * [22] M. Taylor, The Schrödinger equation on spheres, Pacific J. Math. 209 (2003), 145–155.


#  QASE Enhanced PLMs: Improved Control in Text Generation for MRC

Lin Ai Zheng Hui Zizhou Liu Julia Hirschberg
Columbia University, New York, NY
{lin.ai<EMAIL_ADDRESS>
{zh2483, zl2889<EMAIL_ADDRESS>
###### Abstract

To address the challenges of out-of-control generation in generative models
for machine reading comprehension (MRC), we introduce the Question-Attended
Span Extraction (QASE) module. Integrated during the fine-tuning of pre-
trained generative language models (PLMs), QASE enables these PLMs to match
SOTA extractive methods and outperform leading LLMs like GPT-4 in MRC tasks,
without significant increases in computational costs. 111Our code is available
at this anonymous repo link.

QASE Enhanced PLMs: Improved Control in Text Generation for MRC

Lin Ai Zheng Hui Zizhou Liu Julia Hirschberg Columbia University, New York, NY
{lin.ai<EMAIL_ADDRESS>{zh2483, zl2889<EMAIL_ADDRESS>

##  1 Introduction

Machine Reading Comprehension (MRC) is a critical NLP challenge. Mainstream
approaches to MRC extract a relevant piece of text from the context in
response to a question Wang et al. (2018); Yan et al. (2019); Chen et al.
(2020), but in real-world application, the correct answers often span multiple
passages or are implicit Li et al. (2021). Exploring generative models, in
addition to extractive methods, is essential.

Generative models, however, underperform in MRC due to out-of-control
generation Li et al. (2021). This leads to two main challenges: (1) ill-formed
generated answers, containing incomplete or redundant phrases, and (2) factual
inconsistency in the generated answers deviating from the correct response. In
this paper, we address these by introducing a lightweight Question-Attended
Span Extraction (QASE) module. We fine-tune multiple open-source generative
pre-trained language models (PLMs) on various MRC datasets to assess the
module’s efficacy in guiding answer generation. Our contributions include: (1)
Developing QASE to improve fine-tuned generative PLMs’ quality and factual
consistency on MRC tasks, matching SOTA extractive methods and surpassing
GPT-4; (2) QASE boosts performance without significantly increasing
computational costs, benefiting researchers with limited resources.

##  2 Related Work

Most current studies on MRC involve predicting the start and end positions of
the answer spans from a given context Ohsugi et al. (2019); Lan et al. (2019);
Bachina et al. (2021); Chen et al. (2022) using encoder-only PLM models such
as BERT and XLM-Roberta. To handle the multi-span setting, some studies frame
the problem as a sequence tagging task Segal et al. (2020), and others explore
ways to combine models with different tasks Hu et al. (2019); Lee et al.
(2023); Zhang et al. (2023). While these extractive-based methods mainly
utilize encoder-only models, there is also research focuses on using
generative language models Yang et al. (2020); Li et al. (2021); Su et al.
(2022).

Retrieval-augmented text generation (RAG) augments the input of PLMs with in-
domain Gu et al. (2018); Weston et al. (2018); Saha and Srihari (2023) or
external knowledge Su et al. (2021); Xiao et al. (2021) to control the quality
and factual consistency of generated content. It has become a new text
generation paradigm in many NLP tasks Li et al. (2022b), such as dialogue
response generation Wu et al. (2021); Liu et al. (2023b) and machine
translation He et al. (2021); Zhu et al. (2023). However, not much work
focuses on selective MRC. Our approach diverges from RAG as it directly fine-
tunes the weights of the PLMs rather than altering the input to the PLMs with
additional information.

##  3 Method

Question-Attended Span Extraction To guide text generation, we use QASE, a
question-attended span extraction module, during fine-tuning the generative
PLMs. QASE focuses model attention on potential answer spans within the
original context. We cast span extraction as a sequence tagging problem and
employ the Inside-Outside (IO) tagging schema, where each sequence token is
tagged as ‘inside’ (I) if part of a relevant span, or ‘outside’ (O) if not.
This schema works well for both single- and multi-span extraction settings,
achieving comparable or even better performance than the well-known BIO
tagging format Huang et al. (2015), as shown by Segal et al. (2020).

Figure 1: QASE-enhanced model architecture

The architecture of our model is shown in Figure 1. An input context and
question pair and an instruction are first tokenized and fed into the PLM. The
hidden states output from the PLM is then passed through projection layers to
produce embeddings $z_{i}=ReLU(W_{proj}v_{i}+b_{proj})$, where $v_{i}\in
R^{d}$ is the PLM output hidden state of the $i^{th}$ token.

To learn context tokens representations in relation to specific questions, we
employ a multi-head attention mechanism (MHA). Each head in MHA focuses to
different aspects of the context as it relates to the question, using question
embeddings as the query and context embeddings as key-value pairs. This
mechanism aligns the context token representations with the specifics of the
queried question. The projected embeddings $z_{i}$ are passed through MHA, and
subsequently channeled through a linear layer and a softmax layer to compute
$p_{i}=softmax(W_{lin}\cdot MHA(z_{i})+b_{lin})$, which denotes the
probability of the $i^{th}$ token being inside the answer spans. We then
compute the sequence tagging loss using the cross entropy loss
$L_{QASE}=-\frac{1}{N}\sum_{i=1}^{N}\sum_{j=0}^{1}y_{ij}log(p_{ij})$, where
$j\in{0,1}$ corresponds to class O and class I, and $y_{ij}$ is a binary value
indicating whether the $i^{th}$ token belongs to class $j$.

Fine-Tuning and Inference We fine-tune the PLMs using multi-task learning,
simultaneously optimizing both the language modeling loss and sequence tagging
loss: $L=L_{LML}+\beta L_{QASE}$, where $\beta$ is a hyper-parameter that
controls the weight of the span extraction task. This approach enhances the
PLMs’ ability to generate answers well-founded in the context and relevant
answer spans. During inference, only the generation component of the fine-
tuned model is employed.

##  4 Experiments

|  | Llama2 | Alpaca | Flan-T5-Small | Flan-T5-Base | Flan-T5-Large
---|---|---|---|---|---|---
SQuAD | no QASE | 36.68 | 47.06 | 27.88 | 43.95 | 77.33 | 85.51 | 82.09 | 89.56 | 83.16 | 90.71
(EM | F1) | QASE |  37.22 | 47.69 |  37.31 | 47.62 |  77.66 | 85.90 |  82.20 | 90.24 |  84.13 | 91.70
MultiSpanQA | no QASE | 50.93 | 68.14 |  52.73 | 69.10 |  59.13 | 76.49 | 64.66 | 81.41 |  67.41 | 83.09
(EM F1 | Overlap F1) | QASE |  51.75 | 70.39 | 52.20 | 70.01 | 59.08 | 77.10 |  64.87 | 81.50 | 66.92 | 84.22
Quoref | no QASE | 45.52 | 52.09 | - | 58.21 | 63.30 | 72.77 | 80.90 | 75.17 | 80.49
(EM | F1) | QASE |  54.28 | 60.44 | - |  60.70 | 66.88 |  75.17 | 81.18 |  76.19 | 82.13
Table 1: Performance of fine-tuned PLMs with or without QASE on each dataset.

Datasets and Metrics We utilize these 3 MRC datasets. (1) SQuAD Rajpurkar et
al. (2016): A benchmark dataset consisting of 100K+ questions with single-span
answers. We use SQuAD v1.1. Since the official evaluation on v1.1 has long
been ended, we report our results on the official v1.1 development set. (2)
MultiSpanQA Li et al. (2022a): This dataset consists of over 6.5k question-
answer pairs. Unlike most existing single-span answer MRC datasets,
MultiSpanQA focuses on multi-span answers. (3) Quoref Dasigi et al. (2019): A
benchmark dataset containing more than 24K questions, with most answers being
single-span and $\sim$10% being multi-span. Following the conventions of the
datasets’ official leaderboards (listed in A.1), we employ exact match (EM)
and partial match (Overlap) F1 scores as metrics on MultiSpanQA, and exact
match percentage and macro-averaged F1 score on SQuAD and Quoref.

Experimental Setup To evaluate the effectiveness of QASE independent of any
specific language model, we experiment with multiple open-source LLMs. These
include both decoder-only LLMs, such as Llama 2 Touvron et al. (2023) and
Alpaca Taori et al. (2023), and an encoder-decoder model, Flan-T5 Chung et al.
(2022). For Llama 2 and Alpaca, we fine-tune the pre-trained 7B version using
LoRA Hu et al. (2021) and instruction-tuning (see A.4 for instruction
templates). For Flan-T5 family models, we fine-tune the small, the base, and
the large versions. The trainable parameters for each model is provided in
Table 2.

| Trainable Parameters
---|---
| no QASE | QASE |  $\Delta$params
| Llama2/Alpaca
---
with LoRA
4.2M | 7.3M | 3.1M
Flan-T5-Small | 77.0M | 78.2M | 1.3M
Flan-T5-Base | 247.6M | 248.9M | 1.4M
Flan-T5-Large | 783.2M | 784.7M | 1.5M
Table 2: Trainable parameters of experimented models.

We set the hyper-parameters $\beta=1$ and the learning rate $lr=1e-4$. For
LoRA fine-tuning applied to Llama 2 and Alpaca models, we specify a rank
$r=8$, $\alpha=32$, and a dropout rate of $0.05$. The methodology for
selecting these hyper-parameters is detailed in A.2. We train all our models
on single GPUs, using a batch size of 2-4 depending on the VRAM of the
respective GPUs. We use four types of GPUs: A40, A10, A5500, and A100. Models
are trained for 3 epochs or until convergence.

Experiment Results To evaluate the efficacy of the QASE, we examine the
performance of various PLMs fine-tuned with and without QASE, as shown in
Table 1. Generally, models fine-tuned with QASE outperform those fine-tuned
without it. In particular, for SQuAD, QASE-enhanced model demonstrate an EM
percentage increase of up to 33.8% and an F1 score upsurge of up to 8.4% over
vanilla fine-tuned models. For MultiSpanQA, there is an improvement of up to
1.6% in the EM F1 and up to 3.3% in the overlap F1. Likewise, on Quoref, there
is an improvement of up to 19.2% in the EM percentage and up to 16.0% in the
F1 score. These results show that, by employing QASE, generative-based PLMs
can be fine-tuned to produce well-formed, context-grounded, and better-quality
answers in MRC tasks compared to the vanilla fine-tuning approach. For
reference, we also compare the fine-tuned PLMs to their corresponding PLMs in
zero-shot settings, as presented in Appendix A.3.

Computational Costs Table 2 shows that integrating QASE slightly raises the
number of trainable parameters in PLMs, with the increase dependent on the
models’ hidden sizes. Significantly, for the largest model, Flan-T5-Large,
QASE adds just 0.2% more parameters, indicating that QASE enhances the
capabilities of fine-tuned PLMs in MRC without major increase in computational
resources.

Model Comparisons Our top model, Flan-T5-LargeQASE, is further benchmarked
against leading models on each dataset’s official leaderboard, alongside zero-
shot GPT-3.5-Turbo and GPT-4. GPT-3.5-Turbo stands as one of OpenAI’s most
efficient models in terms of capability and cost, while GPT-4 shows superior
reasoning abilities Liu et al. (2023c). Studies indicate their superiority
over traditional fine-tuning methods in most logical reasoning benchmarks Liu
et al. (2023a). The prompts used to query the GPT variants are detailed in
Appendix A.4. On SQuAD, as showed in Table 3, Flan-T5-LargeQASE surpasses
human performance, equaling the NLNet model. Additionally, it surpasses GPT-4
by 113.8% on the exact match score and 32.6% on F1.

| EM |  F1 $\uparrow$
---|---|---
GPT-3.5-Turbo | 36.944 | 65.637
GPT-4 | 39.347 | 69.158
Human Performance | 82.304 | 91.221
BERT-Large Devlin et al. (2019) | 84.328 | 91.281
MSRA NLNet (ensemble) | 85.954 | 91.677
Flan-T5-LargeQASE | 84.125 | 91.701
Table 3: Flan-T5-LargeQASE and baselines on SQuAD.

On MultiSpanQA, Table 4 shows that Flan-T5-LargeQASE outperforms LIQUID Lee et
al. (2023), which currently ranks #1 on the leaderboard, with respect to the
overlap F1 score. Moreover, it surpasses GPT-4 by 4.5% on the exact match F1
and 1.5% on the overlap F1.

| EM F1 |  Overlap F1 $\uparrow$
---|---|---
GPT-3.5-Turbo | 59.766 | 81.866
GPT-4 | 64.027 | 82.731
LIQUID Lee et al. (2023) | 73.130 | 83.360
Flan-T5-LargeQASE | 66.918 | 84.221
Table 4: Performance of Flan-T5-LargeQASE and baselines on MultiSpanQA. | EM |  F1 $\uparrow$
---|---|---
GPT-3.5-Turbo | 50.22 | 59.51
GPT-4 | 68.07 | 78.34
CorefRoberta-Large Ye et al. (2020) | 75.80 | 82.81
Flan-T5-LargeQASE | 76.19 | 82.13
Table 5: Performance of Flan-T5-LargeQASE and baselines on Quoref.

On Quoref, Table 5 shows that Flan-T5-LargeQASE is comparable to CorefRoberta-
Large Ye et al. (2020), which ranks #9 on the leaderboard, with a 0.5% higher
exact match. Furthermore, it outperforms GPT-4 by 11.9% on the exact match and
4.8% on F1.

All top-performing models on these datasets’ leaderboards, equaling or
exceeding Flan-T5-LargeQASE, are encoder-only extractive models. Therefore,
these results demonstrate that QASE-enhanced generative PLMs can be fine-tuned
to match or exceed the capabilities of SOTA extractive models and outperform
leading LLMs in MRC.

Ablation Studies To demonstrate the superiority of the QASE architecture, we
compared Flan-T5-LargeQASE with vanilla fine-tuned Flan-T5-LargeFT and
Flan-T5-Largebaseline. The baseline span extraction module lacks the MHA
component, making it a standard architecture for fine-tuning pre-trained
encoders for downstream sequence tagging tasks. We also explored both
question-first (qf) and context-first prompting strategies, with further
details and analysis provided in Appendix A.5, where the model architecture is
also illustrated.

Table 6 shows that the baseline-embedded model performs better with a
question-first prompting strategy, as Flan-T5-Large${}_{baseline_{qf}}$
surpasses Flan-T5-Largebaseline and Flan-T5-Large${}_{FT_{qf}}$. Conversely,
the baseline span extraction module decreases performance in context-first
prompting, where Flan-T5-Largebaseline underperforms compared to
Flan-T5-LargeFT. This suggests that adding an auxiliary span extraction module
without careful design can negatively affect instruction fine-tuning.
Meanwhile, the QASE-enhanced model excels over both vanilla fine-tuned and
baseline-embedded models in both prompting scenarios, demonstrating its
architectural superiority. Specifically, in context-first setting,
Flan-T5-LargeQASE significantly outperforms Flan-T5-Largebaseline with a 4.3%
higher F1.

| EM |  F1 $\uparrow$
---|---|---
Flan-T5-Largebaseline | 79.877 | 87.918
Flan-T5-Large${}_{FT_{qf}}$ | 80.378 | 88.176
Flan-T5-Large${}_{baseline_{qf}}$ | 81.125 | 89.043
Flan-T5-Large${}_{QASE_{qf}}$ | 81.485 | 89.077
Flan-T5-LargeFT | 83.159 | 90.712
Flan-T5-LargeQASE | 84.125 | 91.701
Table 6: Performance of vanilla, baseline-, and QASE-enhanced fine-tuned
Flan-T5-Large on SQuAD.

Factual Consistency While token-based EM and F1 scores measure the structural
quality of generated text, they do not reflect factual accuracy relative to
the context. For this we used $Q^{2}$ Honovich et al. (2021), an automatic
metric for assessing factual consistency in generated text, which uses
question generation and answering methods over token-based matching. We
compared fine-tuned Flan-T5-Large with and without QASE in both single-span
(SQuAD) and multi-span (MultiSpanQA) answer settings. Table 7 shows that QASE-
enhanced models consistently outperform the vanilla fine-tuned model. On
SQuAD, $Q^{2}$ NLI score is improved by 1.0%, and on MultiSpanQA, it is
improved by 16.0%. Beyond the $Q^{2}$ statistical analysis, our detailed case
studies in Appendix A.6 highlight Flan-T5-LargeQASE’s improved performance.
These examples show the model’s better alignment with relevant context, its
enhanced understanding of complex sentences, its skill in synthesizing answers
from dispersed information, and its superior use of pre-existing real-world
knowledge in generating answers.

| Flan-T5-Large |  $\bm{Q^{2}}$ F1 |  $\bm{Q^{2}}$ NLI
---|---|---|---
SQuAD | no QASE | 42.927 | 44.983
QASE | 43.624 | 45.419
MultiSpanQA | no QASE | 32.889 | 31.433
QASE | 34.732 | 36.452
Table 7: $Q^{2}$ scores of fine-tuned Flan-T5-Large with or without $QASE$ on
each dataset.

##  5 Conclusion and Future Work

In this study, we address out-of-control text generation of generative PLMs in
MRC using QASE, a lightweight question-attended span extraction module, during
the fine-tuning of PLMs. Our experiments show that QASE-enhanced PLMs generate
better-quality responses with improved formality and factual consistency,
matching SOTA extractive models and outperforming GPT-4 by a significant
margin on all three MRC datasets. Importantly, QASE improves performance
without a significant increase in computational costs, benefiting researchers
with limited resources.

In the future, we plan to test our model on generative MRC datasets Nguyen et
al. (2016) to further assess its efficacy in more complex scenarios. Another
key focus will be evaluating the model’s general ability in answer generation,
particularly from the perspective of human perception. This will involve
incorporating human annotators in addition to automatic metrics. For a long-
term goal, we are looking to expand our work to explore solutions for
addressing input- and context-conflicting hallucinations in LLMs.

## Limitations

Due to our limited computational resources, we have been able to perform our
experiments on models no larger than Flan-T5-Large. This same constraint led
us to only fine-tuning of Llama 2 and Alpaca with LoRA. We note that models
based on Llama 2 and Alpaca generally underperform those based on Flan-T5.
Apart from the inherent distinctions between decoder-only and encoder-decoder
models, and their suitability for different tasks (as seen from the models’
zero-shot performance), a possible factor could be the number of trainable
parameters during fine-tuning. Specifically, fine-tuning Llama 2 and Alpaca
with LoRA results in only 4.2M trainable parameters, while even the smallest
Flan-T5 model provides 77.0M trainable parameters, as shown in Table 2. We
acknowledge that many researchers face similar computational resource
limitations. Therefore, our research should be very useful, proposing this
lightweight module capable of enhancing smaller PLMs to outperform leading
LLMs on MRC tasks like these, achieving a balance of effectiveness and
affordability.

One foreseeable limitation of our work is the dependency of the fine-tuning
process on answer span annotations, since QASE works as an auxiliary
supervised span extraction module. This reliance on annotated data could
potentially limit the model’s broader applicability. A prospective exciting
future direction to address this limitation is to develop a semi- or
unsupervised module that focuses on selecting relevant spans or rationales
within a given context. By integrating this module with our current model, we
could significantly improve its generalization capabilities, thereby making it
more adaptable and effective across a wider range of scenarios.

One popular method to enhance the formality of answers generated by LLMs is
through prompt engineering, paired with few-shot or in-context learning
techniques. While these strategies offer great advantages, our ultimate goal
is to create a system with broad domain generalization, one that minimizes the
need for extensive, calibrated prompt engineering and sample selections for
task adaptation. Although developing a robust prompt engineering framework or
paradigm is an appealing direction, our current focus diverges from this path.
As a long-term goal, we aim for a solution that handles diverse tasks with
minimal task-specific tuning.

## References

  * Bachina et al. (2021) Sony Bachina, Spandana Balumuri, and Sowmya Kamath S. 2021.  Ensemble ALBERT and RoBERTa for span prediction in question answering.  In _Proceedings of the 1st Workshop on Document-grounded Dialogue and Conversational Question Answering (DialDoc 2021)_ , pages 63–68, Online. Association for Computational Linguistics.
  * Chen et al. (2020) Kunlong Chen, Weidi Xu, Xingyi Cheng, Zou Xiaochuan, Yuyu Zhang, Le Song, Taifeng Wang, Yuan Qi, and Wei Chu. 2020.  Question directed graph attention network for numerical reasoning over text.  _arXiv preprint arXiv:2009.07448_.
  * Chen et al. (2022) Nuo Chen, Linjun Shou, Ming Gong, and Jian Pei. 2022.  From good to best: Two-stage training for cross-lingual machine reading comprehension.  In _Proceedings of the AAAI Conference on Artificial Intelligence_ , volume 36, pages 10501–10508.
  * Chung et al. (2022) Hyung Won Chung, Le Hou, Shayne Longpre, Barret Zoph, Yi Tay, William Fedus, Eric Li, Xuezhi Wang, Mostafa Dehghani, Siddhartha Brahma, et al. 2022.  Scaling instruction-finetuned language models.  _arXiv preprint arXiv:2210.11416_.
  * Dasigi et al. (2019) Pradeep Dasigi, Nelson F. Liu, Ana Marasović, Noah A. Smith, and Matt Gardner. 2019.  Quoref: A reading comprehension dataset with questions requiring coreferential reasoning.  In _Proceedings of the 2019 Conference on Empirical Methods in Natural Language Processing and the 9th International Joint Conference on Natural Language Processing (EMNLP-IJCNLP)_ , pages 5925–5932, Hong Kong, China. Association for Computational Linguistics.
  * Devlin et al. (2019) Jacob Devlin, Ming-Wei Chang, Kenton Lee, and Kristina Toutanova. 2019.  BERT: Pre-training of deep bidirectional transformers for language understanding.  In _Proceedings of the 2019 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, Volume 1 (Long and Short Papers)_ , pages 4171–4186, Minneapolis, Minnesota. Association for Computational Linguistics.
  * Gu et al. (2018) Jiatao Gu, Yong Wang, Kyunghyun Cho, and Victor OK Li. 2018.  Search engine guided neural machine translation.  In _Proceedings of the AAAI Conference on Artificial Intelligence_ , volume 32.
  * He et al. (2021) Qiuxiang He, Guoping Huang, Qu Cui, Li Li, and Lemao Liu. 2021.  Fast and accurate neural machine translation with translation memory.  In _Proceedings of the 59th Annual Meeting of the Association for Computational Linguistics and the 11th International Joint Conference on Natural Language Processing (Volume 1: Long Papers)_ , pages 3170–3180, Online. Association for Computational Linguistics.
  * Honovich et al. (2021) Or Honovich, Leshem Choshen, Roee Aharoni, Ella Neeman, Idan Szpektor, and Omri Abend. 2021.  ${Q^{2}}$: Evaluating factual consistency in knowledge-grounded dialogues via question generation and question answering.  _arXiv preprint arXiv:2104.08202_.
  * Hu et al. (2021) Edward J Hu, Yelong Shen, Phillip Wallis, Zeyuan Allen-Zhu, Yuanzhi Li, Shean Wang, Lu Wang, and Weizhu Chen. 2021.  Lora: Low-rank adaptation of large language models.  _arXiv preprint arXiv:2106.09685_.
  * Hu et al. (2019) Minghao Hu, Yuxing Peng, Zhen Huang, and Dongsheng Li. 2019.  A multi-type multi-span network for reading comprehension that requires discrete reasoning.  In _Proceedings of the 2019 Conference on Empirical Methods in Natural Language Processing and the 9th International Joint Conference on Natural Language Processing (EMNLP-IJCNLP)_ , pages 1596–1606, Hong Kong, China. Association for Computational Linguistics.
  * Huang et al. (2015) Zhiheng Huang, Wei Xu, and Kai Yu. 2015.  Bidirectional lstm-crf models for sequence tagging. arxiv 2015.  _arXiv preprint arXiv:1508.01991_.
  * Lan et al. (2019) Zhenzhong Lan, Mingda Chen, Sebastian Goodman, Kevin Gimpel, Piyush Sharma, and Radu Soricut. 2019.  Albert: A lite bert for self-supervised learning of language representations.  _arXiv preprint arXiv:1909.11942_.
  * Lee et al. (2023) Seongyun Lee, Hyunjae Kim, and Jaewoo Kang. 2023.  Liquid: A framework for list question answering dataset generation.  _arXiv preprint arXiv:2302.01691_.
  * Li et al. (2021) Chenliang Li, Bin Bi, Ming Yan, Wei Wang, and Songfang Huang. 2021.  Addressing semantic drift in generative question answering with auxiliary extraction.  In _Proceedings of the 59th Annual Meeting of the Association for Computational Linguistics and the 11th International Joint Conference on Natural Language Processing (Volume 2: Short Papers)_ , pages 942–947, Online. Association for Computational Linguistics.
  * Li et al. (2022a) Haonan Li, Martin Tomko, Maria Vasardani, and Timothy Baldwin. 2022a.  MultiSpanQA: A dataset for multi-span question answering.  In _Proceedings of the 2022 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies_ , pages 1250–1260, Seattle, United States. Association for Computational Linguistics.
  * Li et al. (2022b) Huayang Li, Yixuan Su, Deng Cai, Yan Wang, and Lemao Liu. 2022b.  A survey on retrieval-augmented text generation.  _arXiv preprint arXiv:2202.01110_.
  * Liu et al. (2023a) Hanmeng Liu, Ruoxi Ning, Zhiyang Teng, Jian Liu, Qiji Zhou, and Yue Zhang. 2023a.  Evaluating the logical reasoning ability of chatgpt and gpt-4.  _arXiv preprint arXiv:2304.03439_.
  * Liu et al. (2023b) Shuai Liu, Hyundong Cho, Marjorie Freedman, Xuezhe Ma, and Jonathan May. 2023b.  RECAP: Retrieval-enhanced context-aware prefix encoder for personalized dialogue response generation.  In _Proceedings of the 61st Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers)_ , pages 8404–8419, Toronto, Canada. Association for Computational Linguistics.
  * Liu et al. (2023c) Xiao Liu, Junfeng Yu, Yibo He, Lujun Zhang, Kaiyichen Wei, Hongbo Sun, and Gang Tu. 2023c.  System report for CCL23-eval task 9: HUST1037 explore proper prompt strategy for LLM in MRC task.  In _Proceedings of the 22nd Chinese National Conference on Computational Linguistics (Volume 3: Evaluations)_ , pages 310–319, Harbin, China. Chinese Information Processing Society of China.
  * Nguyen et al. (2016) Tri Nguyen, Mir Rosenberg, Xia Song, Jianfeng Gao, Saurabh Tiwary, Rangan Majumder, and Li Deng. 2016.  Ms marco: A human generated machine reading comprehension dataset.  _choice_ , 2640:660.
  * Ohsugi et al. (2019) Yasuhito Ohsugi, Itsumi Saito, Kyosuke Nishida, Hisako Asano, and Junji Tomita. 2019.  A simple but effective method to incorporate multi-turn context with BERT for conversational machine comprehension.  In _Proceedings of the First Workshop on NLP for Conversational AI_ , pages 11–17, Florence, Italy. Association for Computational Linguistics.
  * Rajpurkar et al. (2016) Pranav Rajpurkar, Jian Zhang, Konstantin Lopyrev, and Percy Liang. 2016.  SQuAD: 100,000+ questions for machine comprehension of text.  In _Proceedings of the 2016 Conference on Empirical Methods in Natural Language Processing_ , pages 2383–2392, Austin, Texas. Association for Computational Linguistics.
  * Saha and Srihari (2023) Sougata Saha and Rohini Srihari. 2023.  ArgU: A controllable factual argument generator.  In _Proceedings of the 61st Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers)_ , pages 8373–8388, Toronto, Canada. Association for Computational Linguistics.
  * Segal et al. (2020) Elad Segal, Avia Efrat, Mor Shoham, Amir Globerson, and Jonathan Berant. 2020.  A simple and effective model for answering multi-span questions.  In _Proceedings of the 2020 Conference on Empirical Methods in Natural Language Processing (EMNLP)_ , pages 3074–3080, Online. Association for Computational Linguistics.
  * Su et al. (2022) Dan Su, Xiaoguang Li, Jindi Zhang, Lifeng Shang, Xin Jiang, Qun Liu, and Pascale Fung. 2022.  Read before generate! faithful long form question answering with machine reading.  In _Findings of the Association for Computational Linguistics: ACL 2022_ , pages 744–756, Dublin, Ireland. Association for Computational Linguistics.
  * Su et al. (2021) Yixuan Su, Yan Wang, Deng Cai, Simon Baker, Anna Korhonen, and Nigel Collier. 2021.  Prototype-to-style: Dialogue generation with style-aware editing on retrieval memory.  _IEEE/ACM Transactions on Audio, Speech, and Language Processing_ , 29:2152–2161.
  * Taori et al. (2023) Rohan Taori, Ishaan Gulrajani, Tianyi Zhang, Yann Dubois, Xuechen Li, Carlos Guestrin, Percy Liang, and Tatsunori B. Hashimoto. 2023.  Stanford alpaca: An instruction-following llama model.  https://github.com/tatsu-lab/stanford_alpaca.
  * Touvron et al. (2023) Hugo Touvron, Louis Martin, Kevin Stone, Peter Albert, Amjad Almahairi, Yasmine Babaei, Nikolay Bashlykov, Soumya Batra, Prajjwal Bhargava, Shruti Bhosale, et al. 2023.  Llama 2: Open foundation and fine-tuned chat models.  _arXiv preprint arXiv:2307.09288_.
  * Wang et al. (2018) Wei Wang, Ming Yan, and Chen Wu. 2018.  Multi-granularity hierarchical attention fusion networks for reading comprehension and question answering.  In _Proceedings of the 56th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers)_ , pages 1705–1714, Melbourne, Australia. Association for Computational Linguistics.
  * Weston et al. (2018) Jason Weston, Emily Dinan, and Alexander Miller. 2018.  Retrieve and refine: Improved sequence generation models for dialogue.  In _Proceedings of the 2018 EMNLP Workshop SCAI: The 2nd International Workshop on Search-Oriented Conversational AI_ , pages 87–92, Brussels, Belgium. Association for Computational Linguistics.
  * Wu et al. (2021) Zeqiu Wu, Michel Galley, Chris Brockett, Yizhe Zhang, Xiang Gao, Chris Quirk, Rik Koncel-Kedziorski, Jianfeng Gao, Hannaneh Hajishirzi, Mari Ostendorf, et al. 2021.  A controllable model of grounded response generation.  In _Proceedings of the AAAI Conference on Artificial Intelligence_ , volume 35, pages 14085–14093.
  * Xiao et al. (2021) Fei Xiao, Liang Pang, Yanyan Lan, Yan Wang, Huawei Shen, and Xueqi Cheng. 2021.  Transductive learning for unsupervised text style transfer.  In _Proceedings of the 2021 Conference on Empirical Methods in Natural Language Processing_ , pages 2510–2521, Online and Punta Cana, Dominican Republic. Association for Computational Linguistics.
  * Yan et al. (2019) Ming Yan, Jiangnan Xia, Chen Wu, Bin Bi, Zhongzhou Zhao, Ji Zhang, Luo Si, Rui Wang, Wei Wang, and Haiqing Chen. 2019.  A deep cascade model for multi-document reading comprehension.  In _Proceedings of the AAAI conference on artificial intelligence_ , volume 33, pages 7354–7361.
  * Yang et al. (2020) Junjie Yang, Zhuosheng Zhang, and Hai Zhao. 2020.  Multi-span style extraction for generative reading comprehension.  _arXiv preprint arXiv:2009.07382_.
  * Ye et al. (2020) Deming Ye, Yankai Lin, Jiaju Du, Zhenghao Liu, Peng Li, Maosong Sun, and Zhiyuan Liu. 2020.  Coreferential Reasoning Learning for Language Representation.  In _Proceedings of the 2020 Conference on Empirical Methods in Natural Language Processing (EMNLP)_ , pages 7170–7186, Online. Association for Computational Linguistics.
  * Zhang et al. (2023) Chen Zhang, Jiuheng Lin, Xiao Liu, Yuxuan Lai, Yansong Feng, and Dongyan Zhao. 2023.  How many answers should i give? an empirical study of multi-answer reading comprehension.  _arXiv preprint arXiv:2306.00435_.
  * Zhu et al. (2023) Wenhao Zhu, Jingjing Xu, Shujian Huang, Lingpeng Kong, and Jiajun Chen. 2023.  INK: Injecting kNN knowledge in nearest neighbor machine translation.  In _Proceedings of the 61st Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers)_ , pages 15948–15959, Toronto, Canada. Association for Computational Linguistics.

##  Appendix A Detailed Experiment Setup and Results

###  A.1 Dataset Leaderboard

Below are the official leaderboards all the datasets we refer to:

SQuAD |  https://rajpurkar.github.io/SQuAD-explorer/
---|---
MultiSpanQA |  https://multi-span.github.io/
Quoref |  https://leaderboard.allenai.org/quoref/submissions/public
Table 8: Dataset official leaderboards.

###  A.2 Hyper-Parameter Selection

In this section, we outline the process for selecting the hyper-parameter
$\beta$ and detail our approach to LoRA fine-tuning.

For selecting $\beta$, we use a grid search method, exploring values from 0.5
to 2 in increments of 0.1, on 30% of the MultiSpanQA training dataset. This
process leads to the determination that $\beta=1$ empirically yield the best
performance, hence it is selected for use in our experiments.

To select the learning rate $lr$, we conduct a grid search, testing values
from $\\{1e-5,5e-5,1e-4,5e-4,1e-3\\}$ on 30% of the MultiSpanQA training
dataset. Empirically, the value $1e-4$ demonstrates the best performance and
is therefore chosen for our experiments. This selection is in agreement with
the default $lr$ value used in Meta’s official Llama 2 fine-tuning
recipe222Link to the fine-tuning configuration of Meta’s official Llama 2
recipe..

In the case of LoRA fine-tuning, we follow the established methodology as
outlined by Hu et al. (2021). This involves applying LoRA to Llama 2 and the
pre-trained Alpaca models by freezing their pre-trained weights and
integrating trainable rank decomposition matrices at every layer of their
Transformer structures, aimed at reducing the number of trainable parameters
to enhance computational efficiency. We implement this using the PEFT
package333Link to the Hugging Face PEFT implementation.. The fine-tuning
hyper-parameters for LoRA are set according to the default settings specified
in Meta’s official Llama 2 fine-tuning recipe444Link to the LoRA hyper-
parameter configuration of Meta’s official Llama 2 recipe., which include a
rank $r=8$, $\alpha=32$, and a dropout rate of $0.05$.

###  A.3 Full Experiment Results

In addition to the highlighted results presented in Section 4, we also compare
the fine-tuned PLMs to their corresponding base PLMs in zero-shot settings.
The results, presented in Table 9, show that fine-tuning with QASE improves
performance across all datasets. Specifically, on the SQuAD dataset, models
using QASE perform up to 5.6 times better in exact match and 3.0 times better
in F1 score compared to the original models. On the MultiSpanQA dataset, the
exact match improves by up to 124.4 times, and F1 score by up to 3.4 times.
Similarly, on the Quoref dataset, the exact match improves by up to 38.4
times, and F1 score by up to 11.2 times with QASE.

| MultiSpanQA | SQuAD | Quoref
---|---|---|---
| EM F1 | Overlap F1 | EM | F1 | EM | F1
Llama2 | 7.354 | 34.031 | 13.443 | 28.931 | 5.02 | 28.91
Llama2FT | 50.934 | 68.140 | 36.679 | 47.055 | 45.52 | 52.09
Llama2QASE | 51.748 | 70.389 | 37.219 | 47.686 | 54.28 | 60.44
Alpaca | 15.201 | 42.759 | 18.259 | 33.871 | - | -
AlpacaFT | 52.730 | 69.099 | 27.881 | 43.950 | - | -
AlpacaQASE | 52.196 | 70.008 | 37.313 | 47.622 | - | -
Flan-T5-Small | 0.475 | 22.539 | 13.878 | 28.710 | 1.58 | 5.96
Flan-T5-SmallFT | 59.128 | 76.494 | 77.332 | 85.513 | 58.21 | 63.30
Flan-T5-SmallQASE | 59.080 | 77.103 | 77.663 | 85.901 | 60.70 | 66.88
Flan-T5-Base | 4.113 | 37.694 | 37.596 | 51.747 | 27.08 | 34.38
Flan-T5-BaseFT | 64.659 | 81.408 | 82.090 | 89.558 | 72.77 | 80.90
Flan-T5-BaseQASE | 64.874 | 81.498 | 82.204 | 90.240 | 75.17 | 81.18
Flan-T5-Large | 13.907 | 51.501 | 16.149 | 37.691 | 15.96 | 24.10
Flan-T5-LargeFT | 67.408 | 83.094 | 83.159 | 90.712 | 75.17 | 80.49
Flan-T5-LargeQASE | 66.918 | 84.221 | 84.125 | 91.701 | 76.19 | 82.13
Table 9: Performance of zero-shot PLMs and fined-tuned PLMs with and without
QASE.

###  A.4 Instruction Templates and Model Prompts

Table 10 provides the instruction and prompt templates used for fine-tuning
the PLMs and for zero-shot querying of PLMs and GPT variants across both
single- and multi-span answer datasets.

Fine-tuning PLMs |  Instruction: Using the provided context, answer the question with exact phrases and avoid explanations.
\- - -
Context: {context}
\- - -
Question: {question}
\- - -
Answer:
---|---
Zero-shot prompting PLMs and GPT variants on single-span answer dataset, SQuAD |  Instruction: Using the provided context, answer the question with exact phrases and avoid explanations.
\- - -
Context: {context}
\- - -
Question: {question}
\- - -
Answer:
Zero-shot prompting PLMs and GPT variants on multi-span answer datasets, MultiSpanQA and Quoref |  Instruction: Using the provided context, answer the question with exact phrases and avoid explanations. Format the response as follows: ["answer1", "answer2", …].
\- - -
Context: {context}
\- - -
Question: {question}
\- - -
Answer:
Table 10: Templates for fine-tuning instructions and zero-shot query prompts

###  A.5 Ablation Studies Details

Figure 2 depicts the architecture of the model we use for the ablation
studies, with a baseline span extraction module. The baseline span extraction
module omits the MHA component, typifying a standard architecture for fine-
tuning pre-trained encoders for downstream sequence tagging tasks. The
baseline-embedded Flan-T5-Large models are fine-tuned with the same
configurations as Flan-T5-LargeQASE including learning rate, weight decay,
batch size, epoch number, and GPU type.

Figure 2: Baseline-embedded model architecture

We experiment with 2 prompting strategies for ablation studies:

  * •

Context-first prompting: The default prompting strategy we utilize for fine-
tuning PLMs, both with and without QASE. In this setting, the prompt is
ordered as "<instruction tokens> <context tokens> <question tokens>".

  * •

Question-first prompting (qf): Following BERT’s standard fine-tuning
procedures. In this setting, the prompt is ordered as "<instruction tokens>
<question tokens> <SEP> <context tokens>". <SEP> is a special separator token.

###  A.6 Factual Consistency Case Studies

In Section 4, we demonstrate that the Flan-T5-Large model, when fine-tuned
with QASE, produces answers with greater factual accuracy in relation to the
context compared to its counterpart fine-tuned without QASE. Specifically, we
observe a 1.0% improvement in the $Q^{2}$ score on the SQuAD dataset and a
significant 16.0% increase on MultiSpanQA. This section includes examples to
further illustrate QASE’s effectiveness.

Table 11 showcases that Flan-T5-LargeQASE more accurately identifies the key
focus of the question and locates the pertinent factual information within the
context, with the aid of the QASE module. For instance, in Sample 1,
Flan-T5-LargeQASE correctly interprets the question as seeking the age
difference between Newton and Manning, rather than the age of either
individual, and accordingly provides the accurate answer. In contrast,
Flan-T5-LargeFT mistakenly provides Newton’s age as the answer. Similarly, in
Sample 2, Flan-T5-LargeQASE accurately discerns that the question pertains to
Thoreau’s claim regarding the majority, generating in the correct answer,
whereas Flan-T5-LargeFT misguidedly responds with Thoreau’s political
philosophy.

Sample 1
---
Context: This was the first Super Bowl to feature a quarterback on both teams
who was the #1 pick in their draft classes. Manning was the #1 selection of
the 1998 NFL draft, while Newton was picked first in 2011. The matchup also
pits the top two picks of the 2011 draft against each other: Newton for
Carolina and Von Miller for Denver. Manning and Newton also set the record for
the largest age difference between opposing Super Bowl quarterbacks at 13
years and 48 days (Manning was 39, Newton was 26).
Question: What was the age difference between Newton and Manning in Super Bowl
50?
Gold Answer: 13 years and 48 days
| Flan-T5-LargeQASE
---
Generation
13 years and 48 days
| Flan-T5-LargeFT
---
Generation
26
Sample 2
Context: However, this definition is disputed by Thoreau’s political
philosophy, which contrasts the conscience with the collective. The individual
is the ultimate arbiter of right and wrong. Beyond this, since only
individuals act, only they can commit injustices. When the government knocks
on the door, it is an individual in the guise of a postman or tax collector
whose hand meets the wood. Before Thoreau’s imprisonment, when a perplexed tax
collector openly pondered how to deal with his refusal to pay, Thoreau had
advised, "Resign." If a man chose to be an agent of injustice, then Thoreau
insisted on confronting him with the reality that he was making a choice. But
if the government is "the voice of the people," as often claimed, shouldn’t
that voice be heeded? Thoreau acknowledges that the government may represent
the will of the majority but it might also merely reflect the desires of elite
politicians. Even a good government is "liable to be abused and perverted
before the people can act through it." Furthermore, even if a government did
express the voice of the people, this fact would not obligate the obedience of
individuals who dissent. The majority may be powerful but it is not
necessarily right. What, then, is the appropriate relationship between the
individual and the government?
Question: What did Thoreau claim about the majority?
Gold Answer: not necessarily right
| Flan-T5-LargeQASE
---
Generation
it is not necessarily right
| Flan-T5-LargeFT
---
Generation
conscience vs. the collective
Table 11: Comparisons of model attention alignment with question key aspects
and relevant factual context between Flan-T5-LargeQASE and Flan-T5-LargeFT.

Flan-T5-LargeQASE also shows a notable improvement in comprehending complex,
lengthy sentences and synthesizing answers from information that is sparsely
distributed across multiple spans requiring logical processing. This
capability is particularly valuable when the answer to a question does not
directly stem from a single phrase. Table 12 provides examples of such
instances. In Sample 3, the model needs to recognize that ESPN Deportes is the
exclusive broadcaster in Spanish and that CBS, although mentioned, does not
offer Spanish-language broadcasting. Combining these facts leads to the
correct answer, that ESPN Deportes is the network that broadcast the game in
Spanish. Flan-T5-LargeQASE accurately generates this answer, whereas
Flan-T5-LargeFT incorrectly answers with "CBS", likely due to confusion caused
by the complex sentence structures and dispersed information. Similarly, in
Sample 4, Flan-T5-LargeQASE correctly identifies the question as seeking the
name of the force related to a potential field between two locations. It
successfully locates the relevant long sentence, deconstructs, and comprehends
it to produce the correct answer, in contrast to Flan-T5-LargeFT, which
incorrectly selects the first phrase mentioning "force". In Sample 5, the
question asks for the class most commonly not ascribed to the graph
isomorphism problem. The model needs to deduce from the context that "it is
widely believed that the polynomial hierarchy does not collapse to any finite
level", implying "graph isomorphism is not NP-complete". Once again,
Flan-T5-LargeQASE arrives at the correct conclusion, while Flan-T5-LargeFT
does not.

Sample 3
---
Context: On December 28, 2015, ESPN Deportes announced that they had reached
an agreement with CBS and the NFL to be the exclusive Spanish-language
broadcaster of the game, marking the third dedicated Spanish-language
broadcast of the Super Bowl. Unlike NBC and Fox, CBS does not have a Spanish-
language outlet of its own that could broadcast the game (though per league
policy, a separate Spanish play-by-play call was carried on CBS’s second audio
program channel for over-the-air viewers). The game was called by ESPN
Deportes’ Monday Night Football commentary crew of Alvaro Martin and Raul
Allegre, and sideline reporter John Sutcliffe. ESPN Deportes broadcast pre-
game and post-game coverage, while Martin, Allegre, and Sutcliffe contributed
English-language reports for ESPN’s SportsCenter and Mike & Mike.
Question: Which network broadcast the game in Spanish?
Gold Answer: ESPN Deportes
| Flan-T5-LargeQASE
---
Generation
ESPN Deportes
| Flan-T5-LargeFT
---
Generation
CBS
Sample 4
Context: A conservative force that acts on a closed system has an associated
mechanical work that allows energy to convert only between kinetic or
potential forms. This means that for a closed system, the net mechanical
energy is conserved whenever a conservative force acts on the system. The
force, therefore, is related directly to the difference in potential energy
between two different locations in space, and can be considered to be an
artifact of the potential field in the same way that the direction and amount
of a flow of water can be considered to be an artifact of the contour map of
the elevation of an area.
Question: What is the force called regarding a potential field between two
locations?
Gold Answer: an artifact
| Flan-T5-LargeQASE
---
Generation
an artifact
| Flan-T5-LargeFT
---
Generation
conservative force
Sample 5
Context: The graph isomorphism problem is the computational problem of
determining whether two finite graphs are isomorphic. An important unsolved
problem in complexity theory is whether the graph isomorphism problem is in P,
NP-complete, or NP-intermediate. The answer is not known, but it is believed
that the problem is at least not NP-complete. If graph isomorphism is NP-
complete, the polynomial time hierarchy collapses to its second level. Since
it is widely believed that the polynomial hierarchy does not collapse to any
finite level, it is believed that graph isomorphism is not NP-complete. The
best algorithm for this problem, due to Laszlo Babai and Eugene Luks has run
time $2O(\sqrt{nlog(n)})$ for graphs with n vertices.
Question: What class is most commonly not ascribed to the graph isomorphism
problem in spite of definitive determination?
Gold Answer: NP-complete
| Flan-T5-LargeQASE
---
Generation
NP-complete
| Flan-T5-LargeFT
---
Generation
NP-intermediate
Table 12: Comparison of Flan-T5-LargeQASE and Flan-T5-LargeFT in understanding
complex sentence structures.

While our primary evaluation focuses on the model’s proficiency in deriving
answers from provided contexts, we also note that QASE enhances the model’s
capacity to leverage real-world knowledge acquired during its pre-training
phase. This improvement is attributed to QASE’s ability to better align the
model’s focus on parts of the context that are relevant to the questions
asked. Table 13 presents an example of this phenomenon. In Sample 6, when
asked about the California venue considered for the Super Bowl,
Flan-T5-LargeQASE correctly associates the San Francisco Bay Area with
California, thus producing the accurate answer. On the other hand,
Flan-T5-LargeFT erroneously identifies a stadium in Miami as the answer. This
example illustrates how QASE not only improves context-based answer generation
but also the model’s application of pre-existing real-world knowledge to the
questions posed.

Sample 6
---
Context: The league eventually narrowed the bids to three sites: New Orleans’
Mercedes-Benz Superdome, Miami’s Sun Life Stadium, and the San Francisco Bay
Area’s Levi’s Stadium.
Question: Which California venue was one of three considered for Super Bowl
50?
Gold Answer: San Francisco Bay Area’s Levi’s Stadium
| Flan-T5-LargeQASE
---
Generation
San Francisco Bay Area’s Levi’s Stadium
| Flan-T5-LargeFT
---
Generation
Sun Life Stadium
Table 13: Comparison of Flan-T5-LargeQASE and Flan-T5-LargeFT in utilizing
real-world knowledge.

##  Appendix B Extended Discussion on Model Performance

In this section, we engage in a detailed discussion on the performance of the
Flan-T5 family of models and Llama 2 in MRC tasks. Our aim is to gain insights
into the reasons behind the modest zero-shot performance of these large PLMs
on MRC tasks, despite their adeptness at handling other complex NLP tasks such
as dialogue generation and summarization. Although a comprehensive analysis
falls outside the scope of our current study, exploring these performance
nuances can provide valuable perspectives on how to potentially enhance the
effectiveness of these PLMs on similar tasks.

###  B.1 Discussion on Flan-T5 Zero-Shot Performance

We observe that the zero-shot performance of Flan-T5 models across all
datasets, including SQuAD, remains low as shown in Table 9, despite being
instruct-tuned on the SQuAD dataset during the pre-training phase. This
underperformance might stem from the fact that Flan-T5 models, although
trained on the <SQuAD, Extractive QA> task, are also trained on a broad
spectrum of 1,836 tasks, predominantly focusing on free-form generation, QA,
and reasoning tasks Chung et al. (2022). Consequently, these models are not
finely optimized for extractive QA tasks like MRC, especially under metrics
like exact match and F1, particularly for the smaller to larger variants under
study. The larger XL and XXL variants may exhibit better performance in these
tasks. Furthermore, as discussed in the previous sections, generative models,
including Llama 2, Alpaca, and GPT variants, generally show limited
effectiveness in MRC tasks in zero-shot settings, underscored by their poorer
performance despite having significantly larger model parameters compared to
the Flan-T5 variants we experiment with.

To ensure that our zero-shot experiment’s prompts do not adversely affect
Flan-T5’s performance, we compare our prompt template, detailed in Table 10,
with those Google released for Flan-T5’s instruct-tuning on the SQuAD v1
dataset555Link to Flan-T5 instruct-tuning prompt templates.. Our template,
similar to Google’s, differs mainly by including "with exact phrases and avoid
explanations." This difference could potentially affect performance, yet our
subsequent experiments demonstrate otherwise.

We conduct a series of experiments to assess the zero-shot performance of
Flan-T5-Large on SQuAD, using Google released templates for Flan-T5 instruct-
tuning. We select three templates of varying complexities, as listed in Table
14. Our results, detailed in Table 14, reveal that our template achieves the
highest F1 score. This indicates the lower performance of zero-shot Flan-T5 on
SQuAD and similar MRC datasets is expected, even with the original instruct-
tuning templates. It supports our hypothesis that, although Flan-T5 is
instruct-tuned on SQuAD, its primary strengths are in broader generative
question answering and reasoning, rather than specific extractive QA tasks
such as MRC, particularly when evaluated by exact match and F1 metrics.

| SQuAD Performance
---|---
Prompt Template | EM | F1
Article: {context} Question: {question} Answer: | 7.001 | 21.717
Answer a question about this article. Article: {context} Question: {question} Answer: | 15.875 | 33.375
Here is a question about this article: Article: {context} What is the answer to this question: Question: {question} Answer: | 16.764 | 35.304
Our Template See Table 10 | 16.149 | 37.691
Table 14: Flan-T5-Large zero-shot performance on SQuAD with different prompt
templates.

###  B.2 Discussion on Llama 2 Performance

We observe that models based on Llama 2 and Alpaca generally underperform
compared to those based on Flan-T5, in both zero-shot and fine-tuned
scenarios, with or without QASE. This section delves into a detailed
discussion of the potential reasons behind this trend.

Firstly, the discrepancy in performance may stem from the inherent structural
differences between decoder-only models (Llama 2 and Alpaca) and encoder-
decoder models (Flan-T5). Encoder-decoder models are better equipped for tasks
that require extensive input processing, such as MRC, making them more apt for
these tasks than decoder-only models, which are typically more suited to open-
ended QA scenarios. This fundamental distinction partially accounts for
Flan-T5’s superior performance in context-based question answering across both
zero-shot and fine-tuned settings.

Additionally, the difference in the number of trainable parameters during
fine-tuning might contribute to the observed performance gap. Table 2
indicates that fine-tuning Llama 2 and Alpaca with LoRA leads to a
significantly lower count of trainable parameters (4.2M) compared to even the
smallest Flan-T5 model (77.0M). This disparity in trainable parameters is a
crucial factor in explaining why fine-tuned Flan-T5 models, irrespective of
the use of QASE, outperform Llama 2 and Alpaca models.

While we address these factors, conducting a comprehensive comparison and
analysis of different generative model architectures in MRC tasks exceeds the
scope of our current study. Nonetheless, we acknowledge that additional
factors, such as the specific instruct-fine-tuning of Flan-T5 models on MRC
datasets like SQuAD, might also play a role in their enhanced performance over
Llama 2 and Alpaca.