./Results

# Learning Low-Dimensional Quadratic-Embeddings of High-Fidelity Nonlinear
Dynamics using Deep Learning

Pawan Goyal<EMAIL_ADDRESS>Peter Benner  benner@mpi-
magdeburg.mpg.de<EMAIL_ADDRESS>
###### Abstract

Learning dynamical models from data plays a vital role in engineering design,
optimization, and predictions. Building models describing dynamics of complex
processes (e.g., weather dynamics, or reactive flows) using empirical
knowledge or first principles are onerous or infeasible. Moreover, these
models are high-dimensional but spatially correlated. It is, however, observed
that the dynamics of high-fidelity models often evolve in low-dimensional
manifolds. Furthermore, it is also known that for sufficiently smooth vector
fields defining the nonlinear dynamics, a quadratic model can describe it
accurately in an appropriate coordinate system, conferring to the McCormick
relaxation idea in nonconvex optimization. Here, we aim at finding a low-
dimensional embedding of high-fidelity dynamical data, ensuring a simple
quadratic model to explain its dynamics. To that aim, this work leverages deep
learning to identify low-dimensional quadratic embeddings for high-fidelity
dynamical systems. Precisely, we identify the embedding of data using an
autoencoder to have the desired property of the embedding. We also embed a
Runge-Kutta method to avoid the time-derivative computations, which is often a
challenge. We illustrate the ability of the approach by a couple of examples,
arising in describing flow dynamics and the oscillatory tubular reactor model.

###### keywords:

Machine learning, deep learning, autoencoders, high-fidelity dynamical
systems, low-dimensional embedding, quadratic models.

  1. 1.

Learning low-dimensional quadratic-embeddings for high-dimensional dynamical
data using deep learning.

  2. 2.

Employing the power of neural networks, precisely, autoencoders, to identify
low-dimensional embeddings such that a simple quadratic model can explain the
dynamics of the embedding.

  3. 3.

Blending the fourth-order Runge-Kutta scheme to avoid time-derivative
computations.

  4. 4.

By virtue of nonlinear projection, which is an intrinsic interpretation of
autoencoders, low-dimensional representation can be found for data with a
slowly decaying _Kolmogorov $n$-width_.

  5. 5.

Showcasing the performance of the approach over finding the best possible
quadratic models for low-dimensional embeddings using linear projection (via
POD).

  6. 6.

Simple low-dimensional quadratic models should facilitate control prediction
and optimization of high-fidelity nonlinear dynamical processes.

##  1 Introduction

Inferring mathematical models describing the underlying dynamical behavior is
essential in building infrastructure to take technology forward. These models
allow us to understand the underlying dynamics and to perform engineering
studies, control, and predictions. Traditionally, such modeling is done based
on the first principles (e.g., conservation laws, gravitational laws) and
empirical knowledge by experts. However, for complex phenomenons and modern
engineering tools (e.g., advanced robotic dynamics, climate dynamics), the
first principle and empirical knowledge are not fully available to obtain
dynamics models accurately. However, with advancements in measurement
technology, data related to physical processes can be obtained, which can
support in uncovering the underlying dynamics. The paradigm of modeling using
solely data has got a lot of attention in last couple of decades [1, 2, 3, 4,
5, 6, 7, 8, 9, 10, 11, 12, 13]. A key towards learning dynamics models is to
obtain a parsimonious model which gives a good balanced between data fitting
and model complexity. Though dictionary-based sparsity methods are prone to
parsimonious models by enforcing sparsity, in our work, we impose parsimony
intending to determine obtaining low-dimensional models with simple non-
linearity, namely, quadratic-type. The motivation behind this is that high-
dimensional dynamical systems involve in low-dimensional manifolds [14, 15,
16, 17]; and, a nonlinear system with sufficient continuity can be written as
a quadratic nonlinear system [18, 19]. It is related to McCormick relaxation
[20].

Neural networks based learning of dynamical systems from time-series data has
surprisingly rich history [21, 22, 23, 24, 25, 26, 27, 28, 29]. Despite their
success, interpretability and generalizability remains primary concerns.
Parsimonious models, having the simplest model describing dynamics goes in the
opposite direction of neural networks as neural networks are typically high-
dimensional or highly parameterized. Towards identifying interpretable and
generalizable models, symbolic regression-based approaches have shown their
potential [1, 2, 3]. A seminal approach, the dictionary-based sparse
regression also emerged as a powerful method [30, 8]. It relies on
constructing a large feature dictionary followed by identifying the correct
few features from the dictionary that describe the system dynamics. The
approach is computationally efficient, but the success of the method highly
relies on the quality of the dictionary. The work [28] takes it one step
further and aims to identify the transformation of coordinate such that the
dynamics can be given sparsely in the constructed dictionary.

Furthermore, several complex dynamical phenomena are high-dimensional and
highly nonlinear. It makes analysis and control of those models _almost_
infeasible. However, a redeeming of these high-dimensional systems is that
their dynamics lie in a low-dimensional manifold. Principle component analysis
(PCA) is a widely used tool to obtain low-dimensional representation using
linear projection. Dynamic mode decomposition [5, 10, 31] aims at identifying
a linear dynamical models in that low-dimensional, but often it is not
sufficient to capture complex dynamics completely. Another popular method to
obtain nonlinear dynamical models in the low-dimensional data is operator
inference (OpInf) [6, 13]. The OpInf is also combined with hand-designed
features such that the dynamics in the low-dimensional dynamics can be given
by a quadratic model [19]. The OpInf is not fully data-driven as they require
the form of governing equations that may not be available. In any case, these
methods build on the linear projection of the high-dimensional data. However,
often, a linear projection does not give a good low-dimensional representation
(e.g., advection-dominant problems) due to slow decay of Kolmogorov $n$-width.
A nonlinear extension of PCA, the so-called autoencoder, is powered by neural
networks. The autoencoder has been widely used to obtain low-dimensional
representation, see, e.g., [32, 33], but they are often decoupled with the
course of learning dynamical models in the low-dimensional. We firmly
emphasize that there is not enough evidence that a simple model can explain
the dynamics of the lowest possible dimensional embedding of data. Towards
incorporating dynamics with autoencoder, the work [23] utilized the Koopman
theory [34] to identify the low-dimensional representation such that the
dynamics are almost linear in the representation. Although analysis and
engineering studies have become easier, linear models are not expressive
enough to capture complex dynamics completely. Furthermore, autoencoders are
combined with sparse regression in [28], where the idea is to find the
projection of the data such that dynamics of the projected data can be given
by selecting few features from the dictionary. Despite being a promising
approach, the dictionary-based approach quickly becomes intractable when the
number of variables becomes even slightly larger (e.g., for $10$ variables and
degree $3$ polynomial features, total features would be $286$). Additionally,
the approach [28] requires the derivative information, which is often a
challenging task.

In this work, we present a framework for learning a low-dimensional embedding
of data such that the dynamics in the low-dimensional embedding can be given a
simple quadratic model. It combines two main philosophies: (a) dynamics often
lies in a low-dimensional manifold [14, 16, 17], and (b) continuous-enough
dynamical systems can be written a quadratic model [20, 18]. We leverage the
autoencoder neural network framework to simultaneously identify the desired
low-dimensional embedding and the underlying quadratic model. It advocates a
good balance between the low-dimensional embedding of data and the complexity
of models, describing the dynamics of the embedding. Furthermore, to avoid
time-derivative computations, we fuse a numerical integration method, the so-
called _Runge-Kutta scheme_ to learn continuous quadratic models. We showcase
the approach with two examples: Burgers’ equations describing flow dynamics
and the tubular reactor model. These results expose the ability of our
approach to obtain low-dimensional parsimonious models that contain only
quadratic non-linearity.

The remainder of the paper is organized as follows. In Section 2, we explain
the writing of nonlinear dynamical systems as equivalent quadratic systems by
an appropriate transformation of coordinates–also known as lifting. Section 3
contains our core contributions. There, we present autoencoder network
architecture to identity low-dimensional embedding such that a simple
quadratic model can describe its dynamics. We also discuss how a Runge-Kutta
scheme can be fused in the course of learning encoding and decoding to compute
continuous quadratic models to describe the embedding dynamics. Thus, we can
avoid any computation related to time-derivatives. We demonstrate the approach
by means of two examples, namely, reactor tabular model and 2D Burgers’
equation in Section 4. We summarize our work and provide discussions with
future directions in Section 5.

##  2 Quadratic Modeling of Nonlinear Systems

We here briefly overview quadratic modeling of nonlinear systems and
demonstrate that the sufficiently continuous nonlinear systems can be
rewritten as quadratic systems. Such an approach is often employ to simplify
nonlinear optimization problems [20] or model reduction for nonlinear systems
[18, 35]. Let us consider a nonlinear system as follows:

$\dot{}\mathbf{x}(t)=\mathbf{f}(\mathbf{x}),$ (2.1)

where $\mathbf{x}\in\mathbb{R}^{n}$ and the function
$\mathbf{f}(\cdot):\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ with
$\mathbf{f}(\cdot)$ being sufficiently continuous. Then, there exit a lifting
mapping $\mathcal{L}:\mathbb{R}^{n}\rightarrow\mathbb{R}^{m}$, and its inverse
mapping $\mathcal{L}^{\sharp}:\mathbb{R}^{m}\rightarrow\mathbb{R}^{n}$,
resulting into

$\dot{}\mathbf{z}(t)=\mathcal{A}(\mathbf{z}(t))+\mathcal{H}(\mathbf{z}(t)),$
(2.2)

where $\mathbf{z}(t)=\mathcal{L}(\mathbf{x})$, and
$\mathcal{L}^{\sharp}\left(\mathcal{L}(\mathbf{x})\right)=\mathbf{x}$.
Moreover, $\mathcal{A}(\cdot)$ and $\mathcal{H}(\cdot)$ are linear and
quadratic operators, i.e., $\mathcal{A}(\mathbf{z}(t))=\mathbf{A}\mathbf{z}$
and
$\mathcal{H}(\mathbf{z}(t))=\mathbf{H}\left(\mathbf{z}(t)\otimes\mathbf{z}(t)\right)$
with $`\otimes^{\prime}$ denoting the Kronecker product [36]. We illustrate
the whole philosophy with a simple example nonlinear system, describing
dynamics of a simple pendulum:

$\begin{bmatrix}\dot{x}_{1}\\\
\dot{x}_{2}\end{bmatrix}=\begin{bmatrix}-\sin(x_{2})\\\ x_{1}\end{bmatrix}.$
(2.3)

For the above system, we define transformed or lifted coordinates and inverse
transformation as follows:

$\mathcal{L}:\begin{bmatrix}x_{1}\\\
x_{2}\end{bmatrix}\mapsto\begin{bmatrix}x_{1}\\\ x_{2}\\\ \sin(x_{2})\\\
\cos(x_{2})\end{bmatrix}=:\begin{bmatrix}z_{1}\\\ z_{2}\\\ z_{3}\\\
z_{4}\end{bmatrix},\qquad\mathcal{L}^{\sharp}:\begin{bmatrix}z_{1}\\\ z_{2}\\\
z_{3}\\\ z_{4}\end{bmatrix}\mapsto\begin{bmatrix}z_{1}\\\
z_{2}\end{bmatrix}\equiv\begin{bmatrix}x_{1}\\\ x_{2}\end{bmatrix}.$ (2.4)

Consequently, we can write dynamics in the variables $z_{i}$’s as a quadratic
system:

$\dfrac{d}{dt}\begin{bmatrix}z_{1}\\\ z_{2}\\\ z_{3}\\\
z_{4}\end{bmatrix}=\begin{bmatrix}-z_{3}\\\ z_{1}\\\ z_{1}z_{4}\\\
-z_{1}z_{3}\end{bmatrix}.$ (2.5)

The motivation for having quadratic models is that they ease analysis and
engineering studies compared to complex nonlinear systems. For more details on
lifting and writing nonlinear systems as quadratic systems, we refer to [20,
18, 35, 19]. However, we like to highlight that the lifting transformation to
rewrite the dynamics as quadratic systems is not unique, and the inverse
transformation is not either. Moreover, we need a nonlinear analytical system
to write a lifted quadratic system, which is visibly not available where our
aim itself is to find a model from data.

##  3 Autoencoders and Quadratic Embeddings for Learning Nonlinear Dynamics

As noted, nonlinear systems can often be written as quadratic systems.
However, it requires the information of lifting transformation and its inverse
transformation. In other words, it demands a coordinate change such that a
simple quadratic model can describe dynamics in the transformed coordinate.
Limiting ourselves to such a model, we obtain a parsimonious model prone to
interpretability and generalizability. Also, we focus on exploiting the fact
that the dynamics of high-fidelity models often lie in a low-dimensional
manifold. Thus, our work leverages the impressive approximation capabilities
of deep neural networks to obtain a low-dimensional embedding. Many low-
dimensional embedding may exist, but it is not necessary that a quadratic
model can explain the dynamics in the obtained low-dimensional embedding.
Hence, our primary goal is to discover a particular low-dimensional embedding
so that a simple quadratic model can explain the dynamics of the embedding.

Consider a high-fidelity nonlinear dynamical system of form (2.1). We seek to
identify a low-dimensional embedding or coordinate
$\mathbf{z}(t)\in\mathbb{R}^{\hat{n}}$ such that
$\mathbf{x}(t)\approx\Phi(\mathbf{z}(t))$, where
$\Phi:\mathbb{R}^{\hat{n}}\rightarrow\mathbb{R}^{n}$ is a nonlinear function.
Furthermore, $\mathbf{z}(t)$ satisfies

$\dfrac{d}{dt}\mathbf{z}(t)=\mathbf{A}\mathbf{z}(t)+\mathbf{H}\left(\mathbf{z}(t)\otimes\mathbf{z}(t)\right)+\mathbf{b},$
(3.1)

where
$\mathbf{A}\in\mathbb{R}^{\hat{n}\times\hat{n}},\mathbf{H}\in\mathbb{R}^{\hat{n}\times\hat{n}^{2}}$,
and $\mathbf{b}\in\mathbb{R}^{\hat{n}}$. Additionally, we require an inverse
mapping, reconstructing $\mathbf{z}$ from $\mathbf{x}$, and we do this by
another function $\Psi:\mathbb{R}^{n}\rightarrow\mathbb{R}^{\hat{n}}$, i.e.,
$\mathbf{z}(t)\approx\Psi(\mathbf{x}(t))$. Consequently, we have an intrinsic
autoencoder structure with a constraint on the low-dimensional embedding. We
aim to learn both functions, namely $\Phi$ and $\Psi$, using neural networks.
Thus, it can be seen as a nonlinear projection of the high-dimensional data
which can interpreted a non-linear generalization of PCA [37]. As a result,
our aim at find projections or functions $\Phi$ and $\Psi$, as well as, the
matrices $\mathbf{A},\mathbf{H}$ and $\mathbf{b}$, describing the dynamics of
the low-dimensional embedding.

$\mathbf{x}$Encoder$\mathbf{z}$Decoder$\mathbf{{x}}$$\dot{\mathbf{z}}=\mathbf{A}\mathbf{z}+\mathbf{H}\left(\mathbf{z}\otimes\mathbf{z}\right)+\mathbf{b}$
Figure 3.1: The figure depicts a schematic of the approach. It involve an
encoder, mapping the high-dimensional state $\mathbf{x}$ to a dimensional
state $\mathbf{z}$, and a decoder, mapping back $\mathbf{z}$ to $\mathbf{x}$.
We enforce a constraint of the low-dimensional variable $\mathbf{z}$—that is,
a simple quadratic model can explain the dynamics of the variable
$\mathbf{z}$.

One of major challenges in the approach is to estimate the time-derivative
information of $\mathbf{z}(t)$. If we were to access to the time-derivative of
$\mathbf{x}(t)$, then we can estimate $\dot{\mathbf{z}}(t)$ as
$\tfrac{d\Psi(\mathbf{x})}{d\mathbf{x}}\dot{}\mathbf{x}(t)$. But the time-
derivative information using time-series data is a challenging task for
$\mathbf{x}$ as well, and even if we manage to have it, computing
$\tfrac{d\Psi(\mathbf{x})}{d\mathbf{x}}$ is computationally expensive. As a
remedy, we propose to blend a Runge-Kutta scheme [38]—widely used method to
integrate differential equations with a high accuracy—to avoid computations or
estimate of the derivative information. Yet we obtain a continuous model for
the reduced latent variable $\mathbf{z}(t)$. Here, we focus on the fourth-
order Runge-Kutta (RK4) scheme. We briefly recap the RK4 scheme. For a
quadratic-nonlinear differential equation (3.1), we can estimate the variable
$\mathbf{z}$ at time $t_{i+1}$ using $\mathbf{z}$ at time $t_{i}$ as follows:

$\displaystyle\tilde{\mathbf{z}}_{1}$
$\displaystyle=\mathbf{g}(\mathbf{z}(t_{i})),\quad\tilde{\mathbf{z}}_{2}=\mathbf{g}\left(\mathbf{z}(t_{i})+\dfrac{h}{2}\tilde{\mathbf{z}}_{1}\right),\quad\tilde{\mathbf{z}}_{3}=\mathbf{g}\left(\mathbf{z}(t_{i})+\dfrac{h}{2}\tilde{\mathbf{z}}_{3}\right),\quad\tilde{\mathbf{z}}_{4}=\mathbf{g}\left(\mathbf{z}(t_{i})+h\tilde{\mathbf{z}}_{3}\right),$
(3.2a) $\displaystyle\mathbf{z}(t_{i+1})$
$\displaystyle\approx\mathbf{z}(t_{i})+\dfrac{h}{6}\left(\tilde{\mathbf{z}}_{1}+2\tilde{\mathbf{z}}_{2}+2\tilde{\mathbf{z}}_{3}+\tilde{\mathbf{z}}_{4}\right)=:\Pi_{\text{{{RK4}}}}(\mathbf{z}(t_{i})),$
(3.2b)

where $h=t_{i+1}-t_{i}$ and
$\mathbf{g}(\mathbf{z})=\mathbf{A}\mathbf{z}+\mathbf{H}\left(\mathbf{z}\otimes\mathbf{z}\right)+\mathbf{b}$.
The RK4 scheme is $\mathcal{O}(h^{4})$ accurate globally, thus for small $h$,
estimates are expected to be quite accurate. This allows us to write the
problem of identifying quadratic models without requiring derivative
information at any stage.

We depict our core network architecture in Figure 3.2. As indicated in the
figure, we aim at achieving the goals with our architecture, which are:
discovering a low-dimensional representation of $\mathbf{x}$, i.e.,
$\mathbf{z}=\Psi(\mathbf{x})$, together with dynamics of $\mathbf{z}$
satisfying
$\dot{}\mathbf{z}=\mathbf{A}\mathbf{z}+\mathbf{H}\left(\mathbf{z}\otimes\mathbf{z}\right)+\mathbf{b}$.
Also, we seek to identify a decoder, mapping $\mathbf{z}$ to $\mathbf{x}$,
i.e., $\mathbf{x}=\Phi(\mathbf{z})$. For this, we require two types of loss
functions to obtain encoder-decoder functions and a quadratic model describing
dynamics of the low-dimensional embedding:

  * •

Reconstruction loss: First, we seek to determine a low-dimensional embedding
$\mathbf{z}=\Psi(\mathbf{x})$, for which we construct a dynamical model, so
that the high-fidelity state $\mathbf{x}$ can again be reconstructed using
$\mathbf{z}$ using an inverse mapping, i.e., $\mathbf{x}=\Phi(\mathbf{z})$.
For this, we make use of an autoencoder neural network design as in Figure
3.2(a). Indeed, the dimension of $\mathbf{z}$ is a hyper-parameter but the
singular values of the data can provide a good indication of it. To training
networks or encoder/decoder, we penalize reconstruction inaccuracy by the
autoencoder through low-dimensional bottleneck as follows:

$\mathcal{L}_{\textsf{Rec}}:=\|\mathbf{x}-\Phi(\Psi(\mathbf{x}))\|,$ (3.3)

where $\|\cdot\|$ denotes the mean-squared error, averaging over all the
samples and dimensions.

  * •

Quadratic dynamics: Our second goal is to find the low-dimensional variable
$\mathbf{z}$ such that a quadratic model can describe its dynamics, i.e.,
$\dot{}\mathbf{z}(t)=\mathbf{A}\mathbf{z}(t)+\mathbf{H}\left(\mathbf{z}(t)\otimes\mathbf{z}(t)\right)+\mathbf{b}$.
To avoid time-derivative computational or estimates, we enforce the RK4 scheme
as illustrated in Figure 3.2(b). To learn the corresponding quadratic model,
we add the following loss function:

$\mathcal{L}_{\text{{{RK4}}}}:=\left\|\mathbf{z}(t_{i+1})-\Pi_{\text{{{RK4}}}}\left(\mathbf{z}(t_{i})\right)\right\|,$
(3.4)

where $\mathbf{z}(t_{i+1})$ and $\mathbf{z}(t_{i})$ are values at time
$t_{i+1}$ and $t_{i}$, respectively. Moreover, we can predict its past value
at time $t_{i-1}$ using present value of $\mathbf{z}(t_{i})$ by integrating
backward in time using the RK4 scheme, i.e.,

$\mathcal{L}^{\textsf{b}}_{\text{{{RK4}}}}:=\left\|\mathbf{z}(t_{i-1})-\Pi^{\textsf{back}}_{\text{{{RK4}}}}\left(\mathbf{z}(t_{i})\right)\right\|.$
(3.5)

One may add future predictions over $m$-steps using RK4 scheme, but it
severely affects training–potentially due to vanishing or exploding gradient
issues, and increases computational burden as well.

We combine these losses to train the autoencoder and a quadratic model
simultaneously, which is:

$\mathcal{L}_{\textsf{Total}}:=\mathcal{L}_{\textsf{Rec}}+\lambda\left(\mathcal{L}_{\text{{{RK4}}}}+\mathcal{L}^{\textsf{b}}_{\text{{{RK4}}}}\right),$
(3.6)

where $\lambda$ is a hyper-parameter.

Encoding: $\mathbf{z}_{k}=\Psi(\mathbf{x}_{k})$
---
Decoding: $\mathbf{x}_{k}=\Phi(\mathbf{z}_{k})$
Runge-Kutta Constraint:
$\mathbf{z}_{k+1}\approx\Pi_{\text{{{RK4}}}}(\mathbf{z}_{k})$
$\mathbf{g}(\mathbf{z})$:=
$\mathbf{A}\mathbf{z}+\mathbf{H}\left(\mathbf{z}\otimes\mathbf{z}\right)+\mathbf{b}$
ab Figure 3.2: The figure depicts the core principle of the approach to
identify low-dimensional quadratic-embeddings of high-fidelity dynamical
systems. Our architecture is inspired by autoencoder neural network and a
Runge-Kutta scheme. The (a) identifies low-dimensional coordinates
$z=\Psi(\mathbf{x})$ using encoder and recovers $\mathbf{x}=\Phi(\mathbf{z})$
by decoding it. The (b) shows the fourth-order Runge-Kutta scheme to predict
$z_{k+1}$ (the variable $\mathbf{z}$ at time $t_{k+1}$) from $z_{k}$ (the
variable $\mathbf{z}$ at time $t_{k}$) so that estimating derivatives can be
avoided.

######  Remark 1.

Here, we have focused on finding low-dimensional coordinate systems for high-
fidelity models that can describe the dynamics. Using the proposed
methodology, one can also discover an approximate coordinate for low-
dimensional nonlinear dynamical systems. The dynamics in the discovered
coordinate system can be described as a quadratic model.

##  4 Demonstration of the Approach

We demonstrate the realization of the proposed approach using two examples:
nonlinear tabular reactor models and 2D Burgers’ equations with a moving
shock. We have fixed the parameter $\lambda$ in (3.6) as
$\tfrac{1}{\text{{dt}}}$, where dt is the time interval between data. In our
experiments, we have taken the measurements at a regular interval, though the
approach is applicable when measurements are collected at an irregular
interval. All experiments were run on A$100$ Nvidia® GPU and have used Pytorch
[39] to train networks.

###  4.1 Tabular reactor model:

We first consider a 1D tubular reactor model that explains evolution of
concentration $\psi(x,t)$ and temperature $\theta(x,t)$. The governing
equations are given by partial differential equations [40]:

$\displaystyle\dfrac{\partial\psi}{\partial t}$
$\displaystyle=\dfrac{1}{\textsf{Pe}}\dfrac{\partial^{2}\psi}{\partial
x^{2}}-\dfrac{\partial\psi}{\partial
x}-\mathcal{D}\mathcal{F}(\psi,\theta;\gamma),$ (4.1)
$\displaystyle\dfrac{\partial\theta}{\partial t}$
$\displaystyle=\dfrac{1}{\textsf{Pe}}\dfrac{\partial^{2}\theta}{\partial
x^{2}}-\dfrac{\partial\theta}{\partial
x}-2.5\left(\theta-1\right)+0.5\cdot\mathcal{D}\mathcal{F}(\psi,\theta;\gamma),$

where $x$ being the spatial variable $x\in(0,1)$, time $t>0$, and Arrhenius
reaction term

$\mathcal{F}(\psi,\theta;\gamma)=\phi\exp\left(\gamma-\dfrac{\gamma}{\theta}\right).$
(4.2)

Moreover, $\mathcal{D},\textsf{Pe}$ and $\gamma$ denote Damköhler number,
Péclet number and the reactor rate, respectively, and we set
$D=0.167,\textsf{Pe}=5$, and $\gamma=25$ as discussed e.g., in [41, 13]. We
omit writing the boundary and initial conditions and refer to [41, 13] for
them. The model explains an oscillatory dynamics of the reactor. To collect
data, the partial differential equations are discretized using a finite-
element scheme by taking $99$ spatial degree of freedom. We gather data at
these points for the concentration and temperature in the time range
$t\in[0,60]$ at a regular time interval $\text{{dt}}=0.05$. A construction of
low-dimensional models of the rector has been considered, e.g., in [41, 42]
but they require knowledge of either equations or model.

(a) Encoder-decoder design for tabular example.

(b) Encoder-decoder design for Burger’s example.

Figure 4.1: The encoder and decoder architectures are shown in the figure.
_conv1D , $k$_ (_conv2D , $k$_) indicates a 1D(2D) convolution layer with $k$
kernels of size $5$, and likewise _convT1D , $k$_ (_convT2D , $k$_) is a
1D(2D) transpose convolution layer with $k$ transponse kernels of size $5$. We
have used _stride_ of $2$ to down-sample and up-sample; bn denotes batch-
normalization. We have use bn after each convolutional layer and final linear
layer in the encoder. We highlight the usage of bn after the last linear layer
in the encoder which makes better distribution of the low-dimensional
variable, thus improving training. As an activation function, we employed
exponential linear unit [43]. Below each block, we denote the size of output
of the block.

We construct a low-dimensional model for the reactor using only data. We
normalize the data between $0$ and $1$ before training and identifying a low-
dimensional quadratic embedding. We aim at identifying a low-dimensional model
with intrinsic dimension to $\hat{n}=2$ whose dynamics can be described by a
quadratic model. For this, we have used an encoder and decoder architecture
design, shown in Figure 4.1(a). We have employed using Adam optimizer [44] for
training and have trained for $15\leavevmode\nobreak\ 000$ epochs with an
initial learning rate $1\cdot 10^{-3}$ which is reduced by one-fifth after
every $2\leavevmode\nobreak\ 000$ epoch. Once trained encoder, decoder, and
the corresponding 2-dimensional quadratic model, we integrate the quadratic
model and use the decoder to reconstruct concentration and temperature on the
full grid. For comparison, we identify a 2-dimensional coordinate by
projecting the full-dimensional solution using the most dominant POD basis and
learn the best quadratic model, describing dynamics [6]. We plot the
reconstruction in Figure 4.2, where we observe that the proposed methodology
can discover a simple two-dimensional quadratic dynamics model which can
describe the dynamics of the high-fidelity system.

(a) Concentration over time in the domain.

(b) Temperature over time in the domain.

Figure 4.2:  The figure shows a comparison of the accuracy of the encoder-
decoder and 2-dimensional quadratic model, which are obtained using the
proposed methodology (in the middle) and operator inference in [6] (in the
rightmost). We plot the output of the decoders, which is nothing but the high-
fidelity solution, and the input to the decoders is the corresponding
2-dimensional evolution of the embeddings.

###  4.2 2D Burgers’ equation

In our second example, we consider a 2D Burgers’ equation that explains
several flow-related dynamics. The dynamics is given by

$\dfrac{\partial u(x,t)}{\partial
t}+\left(\dfrac{1}{2},\dfrac{1}{2}\right)^{\top}\cdot\nabla
u(x,t)^{2}=0,\forall(x,t)\in\Omega\times[0,T].$ (4.3)

We consider a square block as an initial condition as in [45]—that is,

$u(x,t=0)=\begin{cases}1,\quad x\in[0,0.5]^{2}\\\
0,\quad\text{else}\end{cases}$ (4.4)

Like in [45], we also take $\Omega$ to be $(-0.1,1.5)^{2}$. Having discretized
the domain by taken $512$ points in $x$-direction as well as in $y$-direction;
thus, the system has $262\leavevmode\nobreak\ 144$ degrees of freedom. We take
$100$ data in the time interval $[0,1]$ at a regular time grid. We make the
encoder and decoder designs as shown in Figure 4.1(b) with bottleneck
dimension being only _one_. We have trained the networks using Adam for
$15\leavevmode\nobreak\ 000$ epochs. We set the initial learning rate $1\cdot
10^{-3}$, which is being reduced by one-fifth after each
$2\leavevmode\nobreak\ 000$ epochs. After trained, we identify a one-
dimensional quadratic model that encodes dynamics of complex Burger’s
equations. We obtain the evolution of the low-dimensional variable by
integrating with an appropriate initial condition and reconstruct the solution
of the full domain using the decoder. The results are shown in Figure 4.3. We
mention that we tried to obtain the best one-dimensional quadratic model using
linear projection, but the model could not capture any dynamics reasonably;
hence, we refrain from plotting them in the figure. It illustrates the
impressive capabilities of the neural networks to identify low-dimensional
coordinates such that a quadratic model can describe its dynamics. It allows
us to come up with low-dimensional using nonlinear projection (encoder-decoder
using neural networks) for data with a slow-decay of _Kolmogorov n-width_.
Hence, for complex high-fidelity nonlinear systems, we can identify a low-
dimensional quadratic model, encoding dynamics, thus easing engineering
studies, e.g., control, optimization.

Figure 4.3:  The transient behavior obtained from the learned models (encoder,
decoder, and quadratic model) with the ground truth is shown. The first row
shows the ground truth, and the second row is from the learned model,
indicating a good low-dimensional surrogate that faithfully captures the
original dynamics.

##  5 Discussion

In our work, we have utilized the power of deep learning to determine a low-
dimensional embedding of high-fidelity dynamical models so that the dynamics
of the low-dimensional embedding can be described by a simple quadratic model.
To that end, we have employed autoencoder architectures with a constraint of
the low-dimensional bottleneck embedding–that is, a quadratic model can
explain its dynamics. It is important to learn the low-dimensional
representation and the corresponding quadratic model to obtain interpretable
and parsimonious models. The proposed methodology addresses two main drawbacks
of earlier existing approaches, which are: (a) smooth-enough nonlinear systems
can be written as a quadratic model in the proper coordinates, but it is not
intuitive to find using only data, and (b) many applications such as
advection-dominant problems show a slow decay of the Kolmogorov $n$-width;
thus, a low-dimensional representation using a linear projection of data
displays a poor performance. In our approach, the usage of an autoencoder and
neural networks can be interpreted to find correct low-dimensional
representation using a nonlinear projection while ensuring that a quadratic
model can explain the dynamics of the representation. We have demonstrated our
approach using two examples to determine parsimonious models for complex high-
dimensional dynamical systems.

There are many open avenues for future research. An emerging field of research
in science and engineering is _scientific machine learning_ (see, e.g., [46]),
in which a primary goal is to infuse empirical knowledge and first-principles
rules of processes in the course of training neural networks. We hope that by
incorporating such information and physics while training, one learns models
using fewer data (otherwise, deep learning approaches are data-demanding).
Also, the autoencoder would be better interpretable and generalizable.
Furthermore, finding suitable designs of an encoder and decoder is a concern
though there are enough intuitions by deep learning experts to find reasonably
good designs. Moreover, we mention that despite our effort to avoid
computation of the time-derivatives—which is dubious for noisy measurements—to
obtain a continuous quadratic model by fusion with a Runge-Kutta method, our
approach still may show poor performance. However, one can execute a de-
noising step to remove noise from data using, e.g., the techniques in [47,
48]. Last but not least, finding the intrinsic dimension of the low-
dimensional embedding with the desired properties plays a crucial role in the
performance of our approach. So, to find a good estimate of it, thus leading
to the most parsimonious representation, would require further research.

## Acknowledgment

We would like to express our gratitude to Dr. Neeraj Sarna for providing data
for 2D Burgers’ example considered in Subsection 4.2.

## Funding Statement

Peter Benner was partially supported by the German Research Foundation (DFG)
Research Training Group 2297 “MathCoRe”, Magdeburg.

## References

  * [1] J. Bongard and H. Lipson, “Automated reverse engineering of nonlinear dynamical systems,” Proc. Nat. Acad. Sci. U.S.A., vol. 104, no. 24, pp. 9943–9948, 2007.
  * [2] C. Yao and E. M. Bollt, “Modeling and nonlinear parameter estimation with Kronecker product representation for coupled oscillators and spatiotemporal systems,” Physica D: Nonlinear Phenomena, vol. 227, no. 1, pp. 78–99, 2007\.
  * [3] M. Schmidt and H. Lipson, “Distilling free-form natural laws from experimental data,” Science, vol. 324, no. 5923, pp. 81–85, 2009.
  * [4] C. W. Rowley, I. Mezić, S. Bagheri, P. Schlatter, and D. Henningson, “Spectral analysis of nonlinear flows,” J. Fluild Mech., vol. 641, no. 1, pp. 115–127, 2009.
  * [5] P. J. Schmid, “Dynamic mode decomposition of numerical and experimental data,” J. Fluild Mech., vol. 656, pp. 5–28, 2010.
  * [6] B. Peherstorfer and K. Willcox, “Data-driven operator inference for nonintrusive projection-based model reduction,” Comp. Meth. Appl. Mech. Eng., vol. 306, pp. 196–215, 2016.
  * [7] P. Benner, S. Gugercin, and K. Willcox, “A survey of projection-based model reduction methods for parametric dynamical systems,” SIAM Rev., vol. 57, no. 4, pp. 483–531, 2015.
  * [8] S. L. Brunton, J. L. Proctor, and J. N. Kutz, “Discovering governing equations from data by sparse identification of nonlinear dynamical systems,” Proc. Nat. Acad. Sci. U.S.A., vol. 113, no. 15, pp. 3932–3937, 2016.
  * [9] M. O. Williams, I. G. Kevrekidis, and C. W. Rowley, “A data–driven approximation of the Koopman operator: Extending dynamic mode decomposition,” J. Nonlinear Sci., vol. 25, no. 6, pp. 1307–1346, 2015\.
  * [10] J. N. Kutz, S. L. Brunton, B. W. Brunton, and J. L. Proctor, Dynamic Mode Decomposition: Data-Driven Modeling of Complex Systems.  Philadelphia, USA: Society of Industrial and Applied Mathematics, 2016\.
  * [11] S. H. Rudy, S. L. Brunton, J. L. Proctor, and J. N. Kutz, “Data-driven discovery of partial differential equations,” Sci. Adv., vol. 3, no. 4, p. e1602614, 2017.
  * [12] J. Pathak, B. Hunt, M. Girvan, Z. Lu, and E. Ott, “Model-free prediction of large spatiotemporally chaotic systems from data: A reservoir computing approach,” Physical Rev. Letters, vol. 120, no. 2, p. 024102, 2018.
  * [13] P. Benner, P. Goyal, B. Kramer, B. Peherstorfer, and K. Willcox, “Operator inference for non-intrusive model reduction of systems with non-polynomial nonlinear terms,” Comp. Meth. Appl. Mech. Eng., vol. 372, p. 113433, 2020\.
  * [14] P. Benner, V. Mehrmann, and D. C. Sorensen, Dimension Reduction of Large-Scale Systems, vol. 45 of Lect. Notes Comput. Sci. Eng. Berlin/Heidelberg, Germany: Springer-Verlag, 2005.
  * [15] A. T. Patera and G. Rozza, Reduced Basis Approximation and a Posteriori Error Estimation for Parametrized Partial Differential Equations.  MIT, 2007.  To appear in (tentative rubric) MIT Pappalardo Graduate Monographs in Mechanical Engineering.
  * [16] W. H. A. Schilders, H. A. van der Vorst, and J. Rommes, Model Order Reduction: Theory, Research Aspects and Applications.  Berlin, Heidelberg: Springer-Verlag, 2008.
  * [17] A. Quarteroni, A. Manzoni, and F. Negri, Reduced Basis Methods for Partial Differential Equations, vol. 92 of La Matematica per il 3+2.  Springer International Publishing, 2016.  ISBN: 978-3-319-15430-5.
  * [18] C. Gu, “QLMOR: A projection-based nonlinear model order reduction approach using quadratic-linear representation of nonlinear systems,” IEEE Trans. Comput. Aided Des. Integr. Circuits. Syst., vol. 30, no. 9, pp. 1307–1320, 2011.
  * [19] E. Qian, B. Kramer, B. Peherstorfer, and K. Willcox, “Lift & learn: Physics-informed machine learning for large-scale nonlinear dynamical systems,” Physica D: Nonlinear Phenomena, vol. 406, p. 132401, 2020.
  * [20] G. P. McCormick, “Computability of global solutions to factorable nonconvex programs: Part I—Convex underestimating problems,” Mathematical Programming, vol. 10, no. 1, pp. 147–175, 1976.
  * [21] J. A. Suykens, J. P. Vandewalle, and B. L. de Moor, Artificial Neural Networks for Modelling and Control of Non-Linear Systems.  Springer, 1996.
  * [22] S. N. Kumpati and P. Kannan, “Identification and control of dynamical systems using neural networks,” IEEE Trans. Neural Networks, vol. 1, no. 1, pp. 4–27, 1990.
  * [23] B. Lusch, J. N. Kutz, and S. L. Brunton, “Deep learning for universal linear embeddings of nonlinear dynamics,” Nature Commun., vol. 9, no. 1, pp. 1–10, 2018.
  * [24] A. Mardt, L. Pasquali, H. Wu, and F. Noé, “Vampnets for deep learning of molecular kinetics,” Nature Commun., vol. 9, no. 1, pp. 1–11, 2018.
  * [25] P. R. Vlachas, W. Byeon, Z. Y. Wan, T. P. Sapsis, and P. Koumoutsakos, “Data-driven forecasting of high-dimensional chaotic systems with long short-term memory networks,” Proc. Royal Society A: Mathematical, Physical and Engineering Sciences, vol. 474, no. 2213, p. 20170844, 2018.
  * [26] E. Yeung, S. Kundu, and N. Hodas, “Learning deep neural network representations for Koopman operators of nonlinear dynamical systems,” in American Control Conference, pp. 4832–4839, IEEE, 2019.
  * [27] N. Takeishi, Y. Kawahara, and T. Yairi, “Learning Koopman invariant subspaces for dynamic mode decomposition,” Adv. Neural Inform. Processing Systems, vol. 30, 2017.
  * [28] K. Champion, B. Lusch, J. N. Kutz, and S. L. Brunton, “Data-driven discovery of coordinates and governing equations,” Proc. Nat. Acad. Sci. U.S.A., vol. 116, no. 45, pp. 22445–22451, 2019.
  * [29] P. Goyal and P. Benner, “LQResNet: A deep neural network architecture for learning dynamic processes,” tech. rep., 2021.
  * [30] H. Schaeffer, R. Caflisch, C. D. Hauck, and S. Osher, “Sparse dynamics for partial differential equations,” Proc. Nat. Acad. Sci. U.S.A., vol. 110, no. 17, pp. 6634–6639, 2013.
  * [31] P. Benner, C. Himpe, and T. Mitchell, “On reduced input-output dynamic mode decomposition,” Adv. Comput. Math., vol. 44, no. 6, pp. 1821–1844, 2018\.
  * [32] K. Lee and K. T. Carlberg, “Model reduction of dynamical systems on nonlinear manifolds using deep convolutional autoencoders,” J. Comput. Phys., vol. 404, p. 108973, 2020.
  * [33] F. J. Gonzalez and M. Balajewicz, “Deep convolutional recurrent autoencoders for learning low-dimensional feature dynamics of fluid systems,” arXiv preprint arXiv:1808.01346, 2018.
  * [34] B. O. Koopman, “Hamiltonian systems and transformation in Hilbert space,” Proc. Nat. Acad. Sci. U.S.A., vol. 17, no. 5, p. 315, 1931.
  * [35] P. Benner and T. Breiten, “Two-sided projection methods for nonlinear model order reduction,” SIAM J. Sci. Comput., vol. 37, no. 2, pp. B239–B260, 2015.
  * [36] T. G. Kolda and B. W. Bader, “Tensor decompositions and applications,” SIAM Rev., vol. 51, no. 3, pp. 455–500, 2009.
  * [37] P. Baldi and K. Hornik, “Neural networks and principal component analysis: Learning from examples without local minima,” Neural networks, vol. 2, no. 1, pp. 53–58, 1989.
  * [38] A. Iserles, A First Course in the Numerical Analysis of Differential Equations.  No. 44, Cambridge University Press, 2009.
  * [39] A. Paszke, S. Gross, F. Massa, A. Lerer, J. Bradbury, G. Chanan, T. Killeen, Z. Lin, N. Gimelshein, L. Antiga, et al., “Pytorch: An imperative style, high-performance deep learning library,” Advances in Neural Information Processing Systems, vol. 32, pp. 8026–8037, 2019.
  * [40] R. F. Heinemann and A. B. Poore, “Multiplicity, stability, and oscillatory dynamics of the tubular reactor,” Chemical Engineering Science, vol. 36, no. 8, pp. 1411–1419, 1981.
  * [41] Y. B. Zhou, Model reduction for nonlinear dynamical systems with parametric uncertainties.  PhD thesis, Massachusetts Institute of Technology, 2012.
  * [42] P. Benner, P. Goyal, J. Heiland, and I. Pontes Duff, “Operator inference and physics-informed learning of low-dimensional models for incompressible flows,” Electron. Trans. Numer. Anal., 2021.  To appear.
  * [43] D.-A. Clevert, T. Unterthiner, and S. Hochreiter, “Fast and accurate deep network learning by exponential linear units (ELUs),” arXiv preprint arXiv:1511.07289, 2015.
  * [44] D. P. Kingma and J. Ba, “Adam: A method for stochastic optimization,” arXiv preprint arXiv:1412.6980, 2014.
  * [45] N. Sarna and P. Benner, “Data-driven model order reduction for problems with parameter-dependent jump-discontinuities,” arXiv preprint arXiv:2105.00547, 2021.
  * [46] K. E. Willcox, O. Ghattas, and P. Heimbach, “The imperative of physics-based modeling and inverse theory in computational science,” Nature Comput. Sci., vol. 1, no. 3, pp. 166–168, 2021.
  * [47] S. H. Rudy, J. N. Kutz, and S. L. Brunton, “Deep learning of dynamics and signal-noise decomposition with time-stepping constraints,” J. Comput. Phys., vol. 396, pp. 483–506, 2019.
  * [48] P. Goyal and P. Benner, “Learning dynamics from noisy measurements using deep learning with a Runge-Kutta constraint,” in Proc. The Symbiosis of Deep Learning and Differential Equations - NeurIPS, 2021.


# Game, Set, and Conflict: Evaluating Conflict and Game Frames in Indian
Election News Coverage

Tejasvi Chebrolu1, Rohan Chowdary1, N Harsha Vardhan1, Ponnurangam
Kumaraguru1, Ashwin Rajadesingan2

###### Abstract

News frames refer to how journalists organize and present information to
convey a particular message or perspective to their readers. When covering
elections, these frames shape how the public perceives electoral issues and
events. This study examines how news frames, especially conflict and game
frames, were employed by news organizations in India to cover the 2014 and
2019 general elections. We analyzed how the frames varied temporally, by
region, and by the party being featured in the articles. Key findings include
(i) conflict and games frames are employed more often in highly electorally
consequential states (higher legislative seats) than in other states (ii)
articles featuring challenger parties are more likely to have conflict and
game frame articles than those featuring incumbent parties (iii) the national
parties (BJP, Bharatiya Janata Party) and (INC, Indian National Congress)
disproportionately feature in articles having conflict frames. Overall, our
analysis highlights the evolving nature of election campaigns and how conflict
and game frames play a major part in them.

## Introduction

In democracies, citizens are expected to carefully scrutinize, evaluate, and
make informed decisions about which candidates to support and vote for.
Typically, news media coverage of the election campaigns in the form of
critical analyses and commentary, ground reporting, investigative reporting,
editorials, and opinion pieces help voters make sense of their political
choices. Given the vast amount of election campaign material available, news
organizations are tasked with deciding which aspects or frames to report on
for their readers. News framing is a deliberate and strategic communication
technique that involves carefully selecting specific aspects of an issue,
amplifying their significance to effectively convey a message (Entman 2006).
Frames affect how people perceive information. For example, episodic frames,
which focus on specific events, elicit emotional responses and simplify
complex issues by focusing on individual stories, whereas thematic frames,
which emphasize larger trends, provide a broader context and encourage a more
systemic understanding of the issues at hand (Iyengar 1994).

In this work, we primarily analyze conflict and game frames in Indian election
news reporting. News articles with a conflict frame emphasize a political
entity (political party or leader) initiating or responding to criticisms from
another political entity. For example, the headline “Congress mocking the
poor, can’t accept a tea-seller’s challenge: Narendra Modi” establishes a
conflict frame where Narendra Modi, the current Prime Minister and BJP leader,
attacks the Congress party. Similarly, news articles with a game frame
emphasize the ‘game’ aspects of the elections - winning and losing.
Additionally, these frames often employ references from war and sports
literature to reinforce this idea. For example, the headline “Why DMK may have
self-goaled with Alagiri suspension” uses the soccer terminology self-goaled,
which describes when a player unintentionally scores a goal for the opposing
team. Here, the term self-goaled implies that DMK (a political party in Tamil
Nadu) may have unintentionally harmed themselves by suspending Alagiri (a
prominent party member) using sports metaphors.

We choose to analyze these two frames as they are known to have negative
consequences for democracy. Conflict frames increase the salience of partisan
identities of the readers which in turn results in polarization (Kim and Zhou
2020). On the other hand, game frames, with their emphasis on opinion polls
and winning and losing, reduce the elections to mere contests to be won
(Aalberg, Strömbäck, and de Vreese 2012). Both frames distract citizens from
obtaining substantive policy or issue information, depressing political
knowledge, and increase cynicism (Cappella and Jamieson 1997). Additionally,
individuals often form their opinions based on readily available information
without deliberate contemplation (Higgins 1996). Therefore, exposure to
conflict and game frames in the news may encourage readers to stick to the
already present opinions and perceive politics as a competition or
confrontation rather than a means to effect policy changes and address
critical issues.

Figure 1: Here we display the methodology we followed, beginning with the
collection of news articles and culminating in their final classification into
frames in the form of a flowchart. A) We initiated the process by gathering
headlines from three news sources: NDTV, The Hindu, and The Times of India. In
total, we collected 371,885 article headlines from these three news sources.
These are the articles from January to May of 2014 and 2019. These headlines
were subsequently filtered based on the relevant election dates that were
under discussion. We employed regex matching with specific election-related
keywords to extract headlines related to elections, resulting in a total of
37,740 articles after this filtering process. These extracted articles
constitute our dataset. B) We then proceeded to assess whether these articles
could be categorized under the “conflict” frame. This assessment involved
examining whether the headline was a quote and whether it contained mentions
of multiple political parties. Additionally, we manually annotated 1,000
headlines according to their respective frames. C) For articles initially not
classified as “conflict,” we developed a classifier using manually annotated
data to categorize them into frames. Within our dataset, we also conducted an
analysis of sports and war terminology. These terms were sourced from websites
focused on sports and war-related topics.

We use a combination of content analysis and large-scale computational
analysis focused primarily on news headlines from three prominent Indian news
sources (N=37,740 articles) spanning the 2014 and 2019 election cycles. An
overview of our analysis can be seen in Figure 1. We focus primarily on
headlines in this study as most readers do not read articles beyond the
headlines (Gabielkov et al. 2016). Additionally, the prevalence of paywalls,
restricting access to complete articles, further elevates the significance of
headlines as readily accessible sources of information. This approach is also
similar to other pre-existing literature that effectively employed headlines
as a primary unit for frame analysis (Card et al. 2015; Akyürek et al. 2020).

To obtain a broad perspective of the kinds of frames employed by the news
media in covering the Indian elections, we performed a quantitative content
analysis of 1,000 headlines across the two election cycles. We found eight
frames (the definitions for which can be found in the Appendix section). The
most prominent frame was the election conduct frame, which occurs when a news
headline focuses on the logistics of conducting the elections or any general
non-partisan election news. The following two prominent frames were the
conflict and game frames, with about 30.3% of the election-related articles in
2014 and 34.4% of the articles in 2019 belonged to either the conflict or game
frames.

To perform a more in-depth analysis of conflict and game frames, we developed
a classifier to detect the two frames. Using the classifier, we make the
following observations on the regional, temporal and inter-party dynamics of
the two frames.

First, we analyzed the distribution of the frames state-wise and found that
for almost all articles mentioning Indian states, the percentages of conflict
frame articles are considerably more than the percentage of game frame
articles. Second, continuing our regional analysis, we found that news on
articles mentioning states with a larger number of constituencies, that is,
the most electorally consequential states, have a higher mean percentage of
conflict and game frame articles than articles mentioning states with fewer
constituencies. The articles mentioning states with a larger number of
constituencies have a mean percentage of 21.6% for conflict frames and 13.2%
for game frames. The articles mentioning states with a smaller number of
constituencies have percentages of 10.8% and 6.4% for conflict and game frames
respectively.

Third, analyzing variations in the frames featuring different parties, we
found that the two major national parties - the Bharatiya Janata Party (BJP)
and the Indian National Congress (INC) are mentioned disproportionately in
conflict frame articles. While 70.4% of the total articles in our dataset
which mention at least one party, mention either BJP or INC, 80.2% of the
conflict frame articles which mention at least one party, mention either of
these two parties. We analyzed the incumbent parties and the challenger
parties (parties that are currently not in power during the elections) and
found that challenger parties are present more often in conflict and game
frame articles when compared to incumbent parties, indicating their use of a
communication strategy involving the use of conflict frames. Roughly 10.54% of
the conflict frame articles, which have at most one state mentioned in the
article, mention the incumbent party, while about 15.55% mention the
challenger parties. Similarly, for the game frame articles, about 2.97%
mention the incumbent party, while 4.01% mention the challenger party.

We also performed a temporal analysis of frames to understand how these frames
were employed across two election cycles. We find that while the frequency of
game frame articles remain relatively constant, conflict frame articles peak
at the beginning and at the end of the campaign. Specifically analyzing the
use of sports and war terminology, we find that the frequency of such
references increase about a month before polling before peaking during the
first polling dates.

## Indian Electoral Context

In the Indian parliamentary system, the bicameral structure comprises the
Rajya Sabha and the Lok Sabha, each serving distinct yet interconnected roles
in the democratic governance of the nation. The Lok Sabha, with a maximum
limit of 545 seats, is directly connected to the populace as it represents the
people’s voice. The General elections, which are held once every five years,
are the mechanism through which individuals are elected to the Lok Sabha. The
Lok Sabha plays a pivotal role in forming the executive branch, as the
political party or coalition that secures the majority of seats in this house
is tasked with nominating the Prime Minister. In 2014, the incumbent party was
the United Progressive Alliance (a coalition led by INC). Since then, the
National Democratic Alliance (led by BJP) has been in power, winning both the
2014 and 2019 general elections. There are other regional parties of
importance such as the the Trinamool Congress (TMC), which have strongholds in
particular states.

In this paper, we analyzed the 2014 and 2019 general elections. In 2014, the
polling dates spanned a period from 7 April to 12 May, culminating in the
declaration of results on 16 May. This polling period was carried out in nine
phases. Similarly, in 2019, the polling dates extended from 11 April to 19
May, with the results officially announced on 23 May. There were seven phases
in this election. The data used in this work covers the campaigns across both
years, from January 1st of that year until the day before the results were
announced.

## Background

### News Framing

There have been various attempts at understanding the framing of news
pertaining to elections and politics in recent times. News frames in campaign
coverage have been found to affect individuals’ interpretation of campaigns
(Rhee 2006). Research suggests that media-framed election news on specific
public agendas determines the salience of these issues (Mccombs and Shaw
2017). The strategic game frame is used more frequently in the US coverage and
is correlated with the use of media analysts and campaign operatives
(Dimitrova and Strömbäck 2012). Additionally, different media sources in the
same country have different frames that are majorly present in the articles
which cover elections - For example, in the Canadian elections in 2000, most
National Post (a major media body in Canada) headlines adopted a game frame
whereas, in their headlines The Globe and Mail (other major media bodies in
Canada) adopted a classic strategic frame (Trimble and Sampert 2004).

However, there have been few attempts to understand election news framing in
India. An analysis of the content of seven mainstream TV channels in Hindi and
English in India found that development was a central theme of campaign
coverage (Mudgal 2015). However, the television channels prioritized star
campaigners and the distribution of political tickets over discussing
development policies. Furthermore, the pervasive impact of Twitter on the 2019
Indian General Elections is shown by studies like (Bedi et al. 2020), which
dissect the trends of Twitter usage across states and parties, providing
insights into the evolving communication strategies political entities adopt.
Our work aims to further this research in election news framing in India,
focusing on conflict and game frames.

Various methodologies have been developed to discern and analyze frames across
diverse linguistic contexts and media landscapes. Most of the computational
framing methods to analyse frames can be divided into - Topic modelling,
frequency modelling, neural network-based models, and cluster-based models
(Ali and Hassan 2022). Researchers have used topic modelling to classify
frames in large datasets (Nguyen 2015; Roberts et al. 2014). Similarly,
frequency modelling - where words, and phrases and their repetitions are used
to identify frames is also an existing method (Sanderink 2020). Lately, neural
models have also been used which use different architectures - LSTMs (Naderi
and Hirst 2017), and transformers-based language models (Mendelsohn, Budak,
and Jurgens 2021; Cabot et al. 2020; Khanehzar, Turpin, and Mikolajczak 2019)
to extract frames from a corpus. Finally, cluster-based models, where each
document is assigned to a particular cluster, have also been used to analyze
the frames present in a large corpus (Burscher, Vliegenthart, and Vreese
2016). Our work also uses neural-based models to perform frame analysis.

### Conflict and Game Framing

There has been significant work to understand the usage of conflict and game
frames in news articles. Previous research (Dunaway and Lawrence 2015)
emphasizes how news organizations choose to highlight game-frame election
stories based on contextual dependencies, elucidating the interplay between
news-making and political contexts. Additionally, contributions to this
discourse review research on the media’s framing of politics as a strategic
game, striving to enhance conceptual clarity and comparability in the field
(Aalberg, Strömbäck, and De Vreese 2012). Moreover, researchers have
identified influential factors, such as the competitive online environment and
content-related predictors, driving the use of the strategic game frame in
political news coverage during non-electoral periods (Schmuck et al. 2017).

Furthermore, investigations have produced notable findings regarding the
effects of game-framed news on political attitudes and behaviours. Studies
have uncovered the impact of game-framed news on political cynicism,
institutional trust, and media trust, underlining the influential role of news
framing in shaping citizens’ perceptions and trust in political institutions
and the media (Shehata 2014; Hopmann, Shehata, and Strömbäck 2015).
Additionally, researchers have examined the relationship between conflict news
framing, political participation, and the role of conflict avoidance (Bjarnøe,
de Vreese, and Albæk 2020). Finally, they have analyzed the interplay between
media routines, political sources, and journalistic perceptions of audience
preferences (Dekavalla 2018).

## Data

In this paper, we use news articles from three sources - The
Hindu,111https://www.thehindu.com/ NDTV,222https://www.ndtv.com/ and The Times
of India.333https://timesofindia.indiatimes.com/ These news sources are among
the most popular and influential English-language news media agencies in India
(Indian Readership Survey 2019). We collect the news articles from their
websites and focus on the articles related to the general elections of 2014
and 2019. Our analysis is limited to these two years as the online text
archives were not available for previous elections for NDTV which is one of
our three sources. We use keyword-based matching on article headlines to
identify election-related articles. We search for words such as election,
vote, polls, etc., and the names and acronyms of various political parties in
each article’s title and URL. We collect and analyze articles that match at
least one of these keywords. We limit our data to the articles that were
published from 1 January 2014 to 16 May 2014 and from 1 January 2019 to 23 May
2019. This covers the entire duration of the campaign for a particular
election. The final date for both years was the day the results of the
election were declared. No other headlines after these dates were analyzed
because the reporting of articles could change once the results were out which
could lead to inconsistent results because of the change in frames used. Our
analysis was only limited to the analyzing articles which were published in
the campaigning phase. We will be releasing our dataset, which follows
FAIR444Our dataset follows the FAIR data principles: Findable, Accessible,
Interoperableand Reusable, as follows: i) The entire dataset, including
article content and the frames we’ve predicted, will become accessible to the
public after the review process. This ensures that our data is findable and
accessible. ii) We store all the data in JSON format files because it
facilitates conversion to other formats like CSV (Comma Separated Values).
This makes our data Interoperable and Reusable. principles, after the peer
review.

## Conflict and Game Frame definitions

We define the conflict and game frame as follows:

  1. 1.

Conflict Frame: Any headline that sees one political entity either attacking
(directly or indirectly) or defending an attack from another political entity.
Example: “Arvind Kejriwal Warns People Against Voting Congress In Lok Sabha
Polls.”

  2. 2.

Game Frame: Any headline dealing with political entities concerning potential
election outcomes or using sports or war terminology.
Example: “Congress will be an underdog in the polls: Chidambaram.”

We also identified other frames through manual content analysis (as described
in the next section). While a substantial portion of these frames draws from
existing literature, we have identified certain India-specific frames such
coalition. These frames and definitions can be seen in the Appendix section.

## Methods

### Content Analysis

To identify the different frames used in election news coverage, we performed
a quantitative content analysis on a dataset comprising 1,000 randomly sampled
article headlines. A mixture of inductive labeling, wherein the headlines were
analyzed with a certain set of frames defined a priori (Gamson 1992; Neuman,
Just, and Crigler 1992) and deductive labeling, where the frames were deduced
after analyzing the headlines were used to select the frames for our analysis.
Initially, two annotators labeled the same 500 articles. The annotators
labeled these 500 articles in two sets. First, they annotated 200 articles,
and later, the remaining 300. Since headlines can have more than 1 frame, they
also annotated up to 2 frames for each article. After each set, annotators
discussed their labels and resolved disagreements. An inter-rater agreement
score of 0.67 (Cohen’s Kappa) was achieved on these 500 articles. The
annotators later independently labeled 250 articles each.

Eight frames were identified through this process. The distribution of the
frames can be seen in Table 1.555The values in the table do not add up to 100%
as headlines can have up to two frames. For example, “‘Controlled From
Nagpur’: HD Kumaraswamy’s Comeback On PM’s Jibe.” This headline is labeled as
having two frames: game and conflict. It fits the game frame because it uses
‘comeback’ which is a sports reference. It also fits the conflict frame
because Kumaraswamy, a leader of the Janata Dal party, is attacking the PM who
belongs to a different party. Notably, approximately 28% of the articles
centred around the Election Conduct frame, shedding light on the operational
facets and challenges of the electoral process. The next two most prominent
frames in the data were conflict and game frames.

Frame | Frequency
---|---
Election Conduct | 28.7%
Conflict | 18.1%
Game | 15.3%
Strategy | 14.4%
Miscellaneous | 14.0%
Issue | 12.4%
Intra Party Politics | 11.0%
Coalition | 8.0%
Table 1: Distribution of the manually annotated labels. The combined values of
conflict (18.1%) and game frame (15.3%) make up almost one-third of the
labeled data.

### Frame Classifier

We developed a classifier to identify which frame a particular headline
belongs to. Although sampling and hand coding gave us an important overview of
the distribution, it was not possible to repeat this process for the entire
dataset due to the huge amount of manual labor needed. Also, a classifier
would help us accurately answer more specific questions about when conflict
frames were used and which parties are being mentioned in what kinds of
frames. Furthermore, using a classifier could also help us analyze these
frames temporally across both election years and perform statistical tests. To
construct our classifier, we used a fine-tuned distilbert-base-uncased (Sanh
et al. 2019) classification model. This base model was originally fine-tuned
on the sst-2 English dataset. AdamW was used as the optimizer, with a learning
rate of 2e-5. Given our focus on game and conflict frames, the hand-labeled
dataset which consisted of 1,000 headlines was then re-labeled to change all
frames, excluding conflict and game to Others to be used to train a 3-class
classifier that labels a headline as conflict, game or other frame. We used a
train-test split of 80-20, by training each fold with 800 headlines and
testing with 200. The overall accuracy for the model was 84.5%, and the 5-fold
cross-validation accuracy for each class can be seen in Table 2. We have also
reported the error bars in the table.

We noticed a clear pattern in news headlines during the construction of our
classifier: headlines with quotes mentioning more than one political entity
often belonged to the conflict frame, even if they didn’t contain aggressive
language. To account for this phenomenon, we use a simple heuristic.

We first perform a quote classification - when given a headline we predict if
it is a quote or not; before classifying the headlines into frames using the
frame classifier. If a headline contains a quote, we check if it mentions
multiple political parties. If it does, we classify it as a conflict-frame
headline. If not, we proceed with the frame classification using our frame
classifier. The frame classifier will only be used to classify the headline if
the headline does not contain a quote mentioning multiple parties or party
leaders belonging to multiple parties.

We used the distilbert-base-uncased model (Sanh et al. 2019) to build the
quote classifier that distinguishes headlines with quotes from those without.
We generated training data by prompting a GPT-3.5 model (ChatGPT) (OpenAI
2023) 666The link
https://chat.openai.com/share/03aa9180-0bd5-4087-9559-a75879a149b9 contains
the chat with the prompts used to generate headlines for the quote
classifier., with examples of both sentence types from our dataset similar to
(Lee et al. 2021). The training data consisted of 700 sentences evenly
distributed between quotes and not-quotes.

Additionally, out of the many parties that contested, we selected a total of
27 parties for our analysis, considering their performance in the 2014 and
2019 elections. These parties were selected based on the number of seats they
won in previous elections and their popularity. This list includes party
names, party leader names, and their name variations. For example, INC is
referred to in articles with names such as ‘Congress’, ‘Cong’. Subsequently,
we collected information about the party leaders’ names and their variations
from various Internet sources. We used ChatGPT (OpenAI 2023) to enhance this
list, making it more comprehensive. We used this for our subsequent analyses
also, including the incumbency analysis discussed later in the paper.

We find that including the additional quote classifier, in addition to
checking for multiple party mentions before applying the frame classifier,
leads to an increase in the accuracy of predicting both game and conflict
frames, as seen in Table 2.

Class |  | CVA having
---
only frame classifier
| CVA after
---
adding quote classifier
Conflict | 0.7529 | 0.8588
Game | 0.6934 | 0.7299
Other | 0.8975 | 0.8644
Table 2: Cross-Validation accuracy (CVA) per class using only the frame
classifier compared to CVA per class after incorporating the quote classifier
and multiple party mentions checker before the frame classifier. A significant
increase in the accuracy of conflict frames can be seen due to adding this
additional classifier. After performing the 5-fold cross-validation over 20
iterations, we found the standard deviation for the accuracy to be 0.035 and
the 95% confidence interval to be 0.775 to 0.81 when we have only the frame
classifier and standard deviation to be 0.028 and 95% confidence interval to
be 0.831 to 0.859 after adding the quote classifier.

## Analysis and Discussion

### Regional variations

Here, we discuss the distribution of conflict and game frames identified using
the frame classifier in particular states. First, we look at the distribution
across all the states. Then, we analyze the occurrence of these frames in
articles mentioning large versus small states in the country.

#### State Analysis

We examine the distribution of the frames across the states in India (Figure
2). We utilized a list of 1,482 Indian cities along with their respective
states, which we obtained from public lists available on the
Internet.777https://github.com/thatisuday/indian-cities-
database/blob/master/cities.json is one such list we used This list was
subsequently updated to accommodate our specific requirements, such as the
inclusion of cities from Telangana (which was established in 2014 and was not
originally present in the list). Our approach involved conducting a regex
match on the main text of the articles to identify mentions of states.
Additionally, we employed Spacy’s NER model to ensure that only locations were
being identified. Our analysis of percentages of game and conflict frames in
articles mentioning various Indian states reveals that there are a lot more
conflict frame articles than game frame articles in all the headlines
mentioning states. 888with the exception of Sikkim, which has around 5%
conflict frame articles and 6% game frame articles, that is, 6% of all
articles that mentioned Sikkim were classified as a game frame. This trend
remains the same in both the election cycles.

Figure 2: Distribution of Frames in India for 2014 and 2019 with darker colors
indicating states with a higher percentage of articles belonging to that
particular frame. The map on the right side (in red) of the graph corresponds
to the percentage of conflict frame articles which mention the particular
state. The map on the left (in green) corresponds to the percentage of game
frame articles which mention the particular state.

#### Large States

We compare the frequency of conflict and game frames in articles mentioning
large states to those mentioning small states. A large state is a state with a
number of constituencies greater than or equal to 10. According to this
metric, some large states are Bihar, Andhra Pradesh, etc., while others
include Tripura, Goa, etc., where the number of constituencies present in the
state is less than 10. We perform a statistical analysis to compare the
frequency of the conflict and game frames of articles mentioning large states
to those articles mentioning other states. We found 18 large states and union
territories and 17 other states and union territories. A t-test was performed
after appending percentages of these states across both years. We found (with
statistical significance ($p<0.01$) for conflict and game frames respectively)
that headlines mentioning large states have a higher mean percentage of
conflict and game frame articles.

#### Discussion:

The observation that large states with a greater number of constituencies and
thus a more substantial role in determining the ruling party tend to have
titles mentioning these states with higher mean percentages of conflict and
game frame articles emphasizes the strategic use of these frames in pivotal
electoral battlegrounds. It potentially reflects that parties, especially
those vying for national power, recognize these states’ heightened
significance and adopt a more aggressive and confrontational approach in their
campaigning.

### Party variations

We examine how often election news headlines mentioning specific parties are
classified either as conflict or game frames. First, we explore the frequency
of these parties being mentioned in these frames. We then divide the parties
into incumbent and challenger parties and check for any potential statistical
trends.

#### Mentions

We investigate mentions of specific political parties in the dataset. We
employ a regex matching method to scan the headlines for party or politician
names. We then map each politician to their corresponding party using the
party list we generated for building the classifier. We also check how often
they were mentioned in the dataset for articles which mention at least one
party compared to their mentions in conflict frame articles. We find that INC
and BJP, the major national parties, are disproportionately mentioned in
conflict frame articles mentioning at least one party compared to their
overall mention in the dataset for articles which mention at least one party.
We see that the BJP is mentioned in 47.4% of the total articles, compared to
53.9% of the conflict frame articles. Similarly, INC is mentioned in 34.8% of
the total articles, compared to 39.2% of the conflict frame articles. This is
a trend that is present in both 2014 and 2019. The results can be seen in
Figure 3.999The percentage of mentions of parties in the conflict frame
articles adds up to more than 100% because multiple parties can be mentioned
in a single article. The parties that are visualized are parties belonging to
the states with the most number of legislative seats, along with Delhi where
New Delhi (capital of the country) is located.

Figure 3: Percentage mentioned in the entire dataset for articles mentioning
at least one party versus percentage mentioned in conflict frame articles
mentioning at least one party for specific political parties. The y-axis
represents the percentage mentioned in the dataset or the conflict frame
percentage. The two national parties the BJP and INC are disproportionately
mentioned in conflict frame articles when compared to the entire dataset.

#### Discussion:

The prevalence of conflict frame articles involving these parties emphasizes
the competitive nature of Indian politics. The rivalry between these two major
parties often intensifies during election periods as they vie for power and
control at various levels of government. As a result, their confrontations and
disagreements become newsworthy (Price and Zaller 1993), and the media is more
inclined to cover these disputes in conflict frame articles to capture the
drama and conflict, which tend to attract the audience’s attention. Further,
As the BJP and INC are the biggest parties in the nation, their
disproportionate presence in conflict frame articles likely contribute to
citizens misconceiving politics to be entirely conflict-ridden.

#### Incumbency Analysis

We now investigate whether there is a difference in how the media portrays the
incumbent parties versus the challenger parties. First, we compile a list of
all incumbent parties in each state101010barring a few states where either
there was no clear majority for a single party/alliance and Telangana which
didn’t exist in 2014. for the 2014 and 2019 elections, and the other parties
are considered as challengers. We also filter out articles with multiple
states mentioned and only consider those articles where either one state or
zero states (National). We then analyze the percentage of conflict and game-
framed articles that mention the incumbent and challenger. We find that the
average percentage of times the incumbent parties were mentioned in conflict
frame articles was 10.54%, whereas the challenger parties were mentioned
15.55% ($p<0.05$).

This trend also follows for game frame articles where the incumbent parties
are mentioned 2.97% and the challenger parties are mentioned 4.01% on average
($p<0.05$). As seen earlier, this trend remains the same for both campaign
years. This analysis was performed at a state level (articles for which the
state was known) and also at a national level (articles for which the state
was not clearly defined). In all of these analyses, we find that the
challenger parties are mentioned more often in conflict and game frame
articles when compared to the incumbent party.

#### Discussion:

Challenger parties are mentioned more frequently in conflict frame articles
suggests a dynamic where these parties actively seek to disrupt the status quo
and challenge the incumbents. In a democratic setting like India, where
multiparty competition is the norm (Yadav 1996), challenger parties could
often adopt more aggressive communication strategies to draw attention to
themselves and their policy positions. As they strive to break into the
political mainstream, they may find it advantageous to employ conflict frames
to differentiate themselves from incumbents (Hänggli and Kriesi 2010).
Furthermore, the analysis performed at both the state and national levels
highlights the universality of these trends. Whether at the state or national
level, the dynamics of challenger parties remain the same - actively seeking
recognition and engagement with the media remain consistent.

### Temporal Analysis

We analyze the conflict and game frames temporally. Additionally, we also
analyze the temporal distribution of sports and war terminologies, comparing
their frequencies with polling dates to investigate any potential trends.

#### Temporal Analysis for Frames

To examine the distribution of the occurrence of frames across time, we plot
Figure 4, where the y-axis represents the percentage occurrence of the
particular frame in that month. The lines representing the game frames across
the two years almost overlap, indicating similar trends across both years.
Similarly, the lines for the conflict frame have similar trends but are
laterally shifted up indicating the increase in conflict frame articles in
2019 compared to 2014. Furthermore, a clear dip can be seen for the conflict
frame articles in the month of March when compared to the beginning and end of
the campaign timeline.

Figure 4: Distribution of Frames Across Election Years with the y-axis
denoting the percentage of these frames across the entire dataset. The lines
look similar in both election years with there being a slight increase with
time for game frames and a dip around March for conflict frames.

#### Discussion:

First, the observation that the lines (in Figure 4) representing game frames
almost overlap across the two years suggests a consistent trend in the media
using game frames to talk about the elections. The lines (in Figure 4)
representing conflict frames displaying similar trends but with a notable
shift in 2019 highlight the constantly increasing nature of these frames. The
increase in conflict frame articles during the later phase of the election
campaign could signify a heightened competitive atmosphere and a more
aggressive tone in political messaging. This shift reflects parties’
adaptability to changing electoral dynamics, demonstrating that they
strategically intensify their use of conflict frames when it matters most,
just before the final voting as conflict frames have been shown to mobilise
voters (Schuck, Vliegenthart, and De Vreese 2016). Furthermore, the
discernible dip in conflict frame articles during the month of March (right
before polling starts), and parties may temporarily reduce conflict frames to
avoid alienating undecided voters or introduce a more positive narrative to
garner support as citizens typically react negatively to disagreements between
politicians (Hibbing and Theiss-Morse 2002).

#### War and Sports Terms

We analyze game framing using war and sports terms by initially curating a set
of keywords that are specifically related to sports/war but can also be used
in a general context. The keywords for sports were primarily sourced from
websites related to Basketball, Boxing, Cricket, Football, Hockey, and War.
Most of the keywords were directly sourced from online glossaries and filtered
to discard words that were not specific and are commonly used in everyday
language. Some examples of keywords used are

  * •

Sports: Fielding, knockout, bouncer, violation, foul, points, verdict, etc.

  * •

War: Alliance, fight, battle, attack, violence, clash, etc.

Figure 5: Sports and War terms Percentage along with Polling Dates. The x-axis
represents the dates, and the y-axis gives us the percentage of headlines with
such terms present in them. The polling dates for both 2014 and 2019 are also
included. The peaks of all the graphs are very close to the first day of
polling.

We found that 10.9% of the articles contained sports references, and 11.2% of
the articles contained war references. In the context of sports references, it
is noteworthy that Cricket, Football, and Boxing stand out as the largest
contributors, accounting for 4.58%, 3.10%, and 2.03%, respectively. This
observation aligns with the fact that Cricket, Football, and Boxing are the
most watched sports in India (Shetty 2023). The temporal analysis also reveals
that the mentions of such phrases increase rapidly about a month before the
first polling date and peak around the first polling date for both election
years.

#### Discussion:

One plausible explanation for this trend could be the timing of significant
sporting events, such as the 2014 and 2019 Cricket World Cup, which overlapped
with the election cycle. Major sports tournaments often captivate the nation’s
attention, drawing substantial media coverage and public interest (Aalberg, De
Vreese, and Strömbäck 2016). During such events, the lines between political
and sports discourse can blur as politicians and media outlets seize the
opportunity to draw parallels between political contests and sports
competitions (Schmuck et al. 2017). Sporting terms are frequently employed in
political communication to frame elections as a competitive “game,” with
candidates as “players” and voters as “spectators.” This framing engages the
public and helps create a sense of urgency and excitement around the election.

## Limitations and future work

There are several promising avenues for extending the findings of this study
and further advancing our understanding of news framing in the election
context. In an era where multimedia content plays a significant role in
political discourse, future research could explore how visual elements, such
as images and videos, are used to convey frames in political communication.
Furthermore, our analysis was limited to conflict and game frames. Future work
could analyze various other frames mentioned in the paper, like strategy,
issue, or coalition frames. Our analysis relied on a specific dataset of media
articles, which may not capture the entirety of political communication.
Additionally, the dataset’s coverage regarding regional and linguistic
diversity could be limited as we have only limited ourselves to articles
written in English. Furthermore, it is possible that a headline might have
utilised multiple frames, however, currently, our classifier can only predict
one frame. Also, all the analysis was made using the frames identified by the
classifier, so the results could be improved with a better classifier.

## Appendix

Definitions for additional frames used in the paper.

  1. 1.

Election Conduct: Any headline that discusses the logistics of the election or
any non-partisan election news.
Example: “LS polls: Turnout in 30% of seats was lower than in 2014.””

  2. 2.

Strategy: Any headline that discusses the strategies of a political party in
winning elections, debates, etc.
Example: “GSM seeks ’alms’ to fund its poll campaign”

  3. 3.

Issue: Any headline that discusses issues pertaining to the people in the
country or policies to be implemented.
Example: “Mamata for poll reforms, empowering States”

  4. 4.

Intra Party Politics: Any headline that discusses intra-party politics like
defection, ticket allocation, etc.
Example: “Now, AAP workers target their own senior leaders”

  5. 5.

Coalition: Any headline that discusses a coalition between multiple political
parties.
Example: “JDS, Congress Work On Unity As Template For Opposition”

  6. 6.

Miscellaneous: Any headline that could not be classified into the
aforementioned frames - these include headlines that talk about specific
politicians in the elections, or any other miscellaneous election news.
Example: “Modi factor or voter anger? What tilts the scales in U.P.?”

The prompts provided to ChatGPT for performing the following tasks:

  1. 1.

Generating quotes: “Based on the examples given, generate 250 examples of
Indian news headlines which are quotes. They should all be very different from
each other. Use as much variation as possible.”

  2. 2.

Generating variations in party, leader names: “Give me a list of 50 INC
leaders, they should have been prominent during 2014 and 2019 elections. Also,
include variations in their names. For example Chidambaram, and Chidhambaram.”

## Acknowledgements

ChatGPT was utilized to generate sections of this Work, including text, code,
and data.

## References

  * Aalberg, De Vreese, and Strömbäck (2016) Aalberg, T.; De Vreese, C.; and Strömbäck, J. 2016.  Strategy and game framing.  In _Comparing political journalism_ , 33–49. Routledge.
  * Aalberg, Strömbäck, and De Vreese (2012) Aalberg, T.; Strömbäck, J.; and De Vreese, C. H. 2012.  The framing of politics as strategy and game: A review of concepts, operationalizations and key findings.  _Journalism_ , 13(2): 162–178.
  * Aalberg, Strömbäck, and de Vreese (2012) Aalberg, T.; Strömbäck, J.; and de Vreese, C. 2012.  The Framing of Politics as Strategy and Game: A Review of Concepts, Operationalizations and Key Findings.  _Journalism_ , 13: 162–178.
  * Akyürek et al. (2020) Akyürek, A. F.; Guo, L.; Elanwar, R.; Ishwar, P.; Betke, M.; and Wijaya, D. T. 2020.  Multi-Label and Multilingual News Framing Analysis.  In _Proceedings of the 58th Annual Meeting of the Association for Computational Linguistics_ , 8614–8624. Online: Association for Computational Linguistics.
  * Ali and Hassan (2022) Ali, M.; and Hassan, N. 2022.  A Survey of Computational Framing Analysis Approaches.  In _Proceedings of the 2022 Conference on Empirical Methods in Natural Language Processing_ , 9335–9348. Abu Dhabi, United Arab Emirates: Association for Computational Linguistics.
  * Bedi et al. (2020) Bedi, K.; Bjp; Gautam; and Gambhir. 2020.  Twitter in the 2019 Indian General Elections: Trends of Use Across States and Parties.
  * Bjarnøe, de Vreese, and Albæk (2020) Bjarnøe, C.; de Vreese, C. H.; and Albæk, E. 2020.  The effect of being conflict non-avoidant: linking conflict framing and political participation.  _West European Politics_ , 43(1): 102–128.
  * Burscher, Vliegenthart, and Vreese (2016) Burscher, B.; Vliegenthart, R.; and Vreese, C. H. d. 2016.  Frames beyond words: Applying cluster and sentiment analysis to news coverage of the nuclear power issue.  _Social Science Computer Review_ , 34(5): 530–545.
  * Cabot et al. (2020) Cabot, P.-L. H.; Dankers, V.; Abadi, D.; Fischer, A.; and Shutova, E. 2020.  The pragmatics behind politics: Modelling metaphor, framing and emotion in political discourse.  In _Findings of the association for computational linguistics: emnlp 2020_ , 4479–4488.
  * Cappella and Jamieson (1997) Cappella, J. N.; and Jamieson, K. H. 1997.  _Spiral of cynicism: The press and the public good_.  Oxford University Press.
  * Card et al. (2015) Card, D.; Boydstun, A.; Gross, J.; Resnik, P.; and Smith, N. 2015.  The Media Frames Corpus: Annotations of Frames Across Issues.  2: 438–444.
  * Dekavalla (2018) Dekavalla, M. 2018.  Issue and game frames in the news: Frame-building factors in television coverage of the 2014 Scottish independence referendum.  _Journalism_ , 19(11): 1588–1607.
  * Dimitrova and Strömbäck (2012) Dimitrova, D. V.; and Strömbäck, J. 2012.  Election news in Sweden and the United States: A comparative study of sources and media frames.  _Journalism_ , 13(5): 604–619.
  * Dunaway and Lawrence (2015) Dunaway, J.; and Lawrence, R. G. 2015.  What predicts the game frame? Media ownership, electoral context, and campaign news.  _Political Communication_ , 32(1): 43–60.
  * Entman (2006) Entman, R. M. 2006.  Framing: Toward Clarification of a Fractured Paradigm.  _Journal of Communication_ , 43(4): 51–58.
  * FORCE11 (2020) FORCE11. 2020.  The FAIR Data principles.  https://force11.org/info/the-fair-data-principles/.
  * Gabielkov et al. (2016) Gabielkov, M.; Ramachandran, A.; Chaintreau, A.; and Legout, A. 2016.  Social Clicks: What and Who Gets Read on Twitter?  In _ACM SIGMETRICS / IFIP Performance 2016_. Antibes Juan-les-Pins, France.
  * Gamson (1992) Gamson, W. A. 1992.  _Talking politics_.  Cambridge university press.
  * Gebru et al. (2021) Gebru, T.; Morgenstern, J.; Vecchione, B.; Vaughan, J. W.; Wallach, H.; Iii, H. D.; and Crawford, K. 2021.  Datasheets for datasets.  _Communications of the ACM_ , 64(12): 86–92.
  * Hänggli and Kriesi (2010) Hänggli, R.; and Kriesi, H. 2010.  Political framing strategies and their impact on media framing in a Swiss direct-democratic campaign.  _Political communication_ , 27(2): 141–157.
  * Hibbing and Theiss-Morse (2002) Hibbing, J. R.; and Theiss-Morse, E. 2002.  _Stealth democracy: Americans’ beliefs about how government should work_.  Cambridge University Press.
  * Higgins (1996) Higgins, E. T. 1996.  Activation: Accessibility, and salience.  _Social psychology: Handbook of basic principles_ , 133–168.
  * Hopmann, Shehata, and Strömbäck (2015) Hopmann, D. N.; Shehata, A.; and Strömbäck, J. 2015.  Contagious media effects: How media use and exposure to game-framed news influence media trust.  _Mass Communication and Society_ , 18(6): 776–798.
  * Indian Readership Survey (2019) Indian Readership Survey, R. S. C. I. 2019.  Indian readership survey, 2019.
  * Iyengar (1994) Iyengar, S. 1994.  _Is anyone responsible?: How television frames political issues_.  University of Chicago Press.
  * Khanehzar, Turpin, and Mikolajczak (2019) Khanehzar, S.; Turpin, A.; and Mikolajczak, G. 2019.  Modeling political framing across policy issues and contexts.  In _Proceedings of the The 17th Annual Workshop of the Australasian Language Technology Association_ , 61–66.
  * Kim and Zhou (2020) Kim, Y.; and Zhou, S. 2020.  The effects of political conflict news frame on political polarization: A social identity approach.  _International Journal of Communication_ , 14: 22.
  * Lee et al. (2021) Lee, K.; Guu, K.; He, L.; Dozat, T.; and Chung, H. W. 2021.  Neural Data Augmentation via Example Extrapolation.  arXiv:2102.01335.
  * Mccombs and Shaw (2017) Mccombs, M.; and Shaw, D. 2017.  The Agenda-Setting function of mass media.  _The Agenda Setting Journal_ , 1.
  * Mendelsohn, Budak, and Jurgens (2021) Mendelsohn, J.; Budak, C.; and Jurgens, D. 2021.  Modeling Framing in Immigration Discourse on Social Media.  In _Proceedings of the 2021 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies_ , 2219–2263. Online: Association for Computational Linguistics.
  * Mudgal (2015) Mudgal, V. 2015.  Framing the 2014 Elections: The Curious Absence of Development.  _Television & New Media_, 16(4): 354–360.
  * Naderi and Hirst (2017) Naderi, N.; and Hirst, G. 2017.  Classifying Frames at the Sentence Level in News Articles.  536–542.
  * Neuman, Just, and Crigler (1992) Neuman, W. R.; Just, M. R.; and Crigler, A. N. 1992.  _Common knowledge: News and the construction of political meaning_.  University of Chicago Press.
  * Nguyen (2015) Nguyen, V.-A. 2015.  _Guided probabilistic topic models for agenda-setting and framing_.  Ph.D. thesis, University of Maryland, College Park.
  * OpenAI (2023) OpenAI. 2023.  ChatGPT (August 3 Version).
  * Price and Zaller (1993) Price, V.; and Zaller, J. 1993.  Who gets the news? Alternative measures of news reception and their implications for research.  _Public opinion quarterly_ , 57(2): 133–164.
  * Rhee (2006) Rhee, J. W. 2006.  Strategy and Issue Frames in Election Campaign Coverage: A Social Cognitive Account of Framing Effects.  _Journal of Communication_ , 47(3): 26–48.
  * Roberts et al. (2014) Roberts, M.; Stewart, B.; Tingley, D.; Lucas, C.; Leder-Luis, J.; Gadarian, S.; and Albertson, B. 2014.  Structural Topic Models for Open-Ended Survey Responses.  _American Journal of Political Science_ , 58.
  * Sanderink (2020) Sanderink, L. 2020.  Shattered frames in global energy governance: Exploring fragmented interpretations among renewable energy institutions.  _Energy research & social science_, 61: 101355.
  * Sanh et al. (2019) Sanh, V.; Debut, L.; Chaumond, J.; and Wolf, T. 2019.  DistilBERT, a distilled version of BERT: smaller, faster, cheaper and lighter.  _CoRR_ , abs/1910.01108.
  * Schmuck et al. (2017) Schmuck, D.; Heiss, R.; Matthes, J.; Engesser, S.; and Esser, F. 2017.  Antecedents of strategic game framing in political news coverage.  _Journalism_ , 18(8): 937–955.
  * Schuck, Vliegenthart, and De Vreese (2016) Schuck, A. R.; Vliegenthart, R.; and De Vreese, C. H. 2016.  Who’s afraid of conflict? The mobilizing effect of conflict framing in campaign news.  _British Journal of Political Science_ , 46(1): 177–194.
  * Shehata (2014) Shehata, A. 2014.  Game frames, issue frames, and mobilization: Disentangling the effects of frame exposure and motivated news attention on political cynicism and engagement.  _International journal of public opinion research_ , 26(2): 157–177.
  * Shetty (2023) Shetty, S. 2023.  Cricket remains India’s most watched game – A look at other sports urban Indians follow.  https://www.cnbctv18.com/photos/sports/cricket-kabaddi-tennis-sports-indians-watch-and-follow-the-most-16057821.htm.
  * Trimble and Sampert (2004) Trimble, L.; and Sampert, S. 2004.  Who’s in the Game? The Framing of Election 2000 by the Globe and Mail and the National Post.  _Canadian Journal of Political Science/Revue canadienne de science politique_ , 37(1): 51–71.
  * Yadav (1996) Yadav, Y. 1996.  Reconfiguration in Indian politics: State assembly elections, 1993-95.  _Economic and Political Weekly_ , 95–104.

### Ethics Checklist

All the analysis performed in this paper use public online news archives.
Since we examine primarily political parties and their representatives, we see
little risk potential of misuse or negative impact.

  1. 1.

For most authors…

    1. (a)

Would answering this research question advance science without violating
social contracts, such as violating privacy norms, perpetuating unfair
profiling, exacerbating the socio-economic divide, or implying disrespect to
societies or cultures? Yes

    2. (b)

Do your main claims in the abstract and introduction accurately reflect the
paper’s contributions and scope? Yes

    3. (c)

Do you clarify how the proposed methodological approach is appropriate for the
claims made? Yes

    4. (d)

Do you clarify what are possible artifacts in the data used, given population-
specific distributions? NA, the dataset is from the online archives on news
websites.

    5. (e)

Did you describe the limitations of your work? Yes

    6. (f)

Did you discuss any potential negative societal impacts of your work? No

    7. (g)

Did you discuss any potential misuse of your work? No

    8. (h)

Did you describe steps taken to prevent or mitigate potential negative
outcomes of the research, such as data and model documentation, data
anonymization, responsible release, access control, and the reproducibility of
findings? Yes

    9. (i)

Have you read the ethics review guidelines and ensured that your paper
conforms to them? Yes

  2. 2.

Additionally, if your study involves hypotheses testing…

    1. (a)

Did you clearly state the assumptions underlying all theoretical results? NA

    2. (b)

Have you provided justifications for all theoretical results? NA

    3. (c)

Did you discuss competing hypotheses or theories that might challenge or
complement your theoretical results? NA

    4. (d)

Have you considered alternative mechanisms or explanations that might account
for the same outcomes observed in your study? NA

    5. (e)

Did you address potential biases or limitations in your theoretical framework?
NA

    6. (f)

Have you related your theoretical results to the existing literature in social
science? NA

    7. (g)

Did you discuss the implications of your theoretical results for policy,
practice, or further research in the social science domain? NA

  3. 3.

Additionally, if you are including theoretical proofs…

    1. (a)

Did you state the full set of assumptions of all theoretical results? NA

    2. (b)

Did you include complete proofs of all theoretical results? NA

  4. 4.

Additionally, if you ran machine learning experiments…

    1. (a)

Did you include the code, data, and instructions needed to reproduce the main
experimental results (either in the supplemental material or as a URL)? Yes,
link will be provided after peer review

    2. (b)

Did you specify all the training details (e.g., data splits, hyperparameters,
how they were chosen)? Yes

    3. (c)

Did you report error bars (e.g., with respect to the random seed after running
experiments multiple times)? Yes

    4. (d)

Did you include the total amount of compute and the type of resources used
(e.g., type of GPUs, internal cluster, or cloud provider)? No, all the
analysis was done on the free Google Colab account or on personal machines.

    5. (e)

Do you justify how the proposed evaluation is sufficient and appropriate to
the claims made? Yes

    6. (f)

Do you discuss what is “the cost“ of misclassification and fault
(in)tolerance? Yes

  5. 5.

Additionally, if you are using existing assets (e.g., code, data, models) or
curating/releasing new assets…

    1. (a)

If your work uses existing assets, did you cite the creators? Yes

    2. (b)

Did you mention the license of the assets? Yes

    3. (c)

Did you include any new assets in the supplemental material or as a URL? Yes,
All the data and code used for the analysis will be publicly available after
peer review.

    4. (d)

Did you discuss whether and how consent was obtained from people whose data
you’re using/curating? NA, the data are is from publicly available news
archives.

    5. (e)

Did you discuss whether the data you are using/curating contains personally
identifiable information or offensive content? NA, the data are is from
publicly available news archives.

    6. (f)

If you are curating or releasing new datasets, did you discuss how you intend
to make your datasets FAIR (see FORCE11 (2020))? Yes

    7. (g)

If you are curating or releasing new datasets, did you create a Datasheet for
the dataset (see Gebru et al. (2021))? The datasheet will be available after
the peer review.

  6. 6.

Additionally, if you used crowdsourcing or conducted research with human
subjects…

    1. (a)

Did you include the full text of instructions given to participants and
screenshots? NA

    2. (b)

Did you describe any potential participant risks, with mentions of
Institutional Review Board (IRB) approvals? NA

    3. (c)

Did you include the estimated hourly wage paid to participants and the total
amount spent on participant compensation? NA

    4. (d)

Did you discuss how data is stored, shared, and deidentified? NA