# Data driven localized wave solution of the Fokas-Lenells equation using
modified PINN

Gautam Kumar Saharia, Sagardeep Talukdar, Riki Dutta and
Sudipta Nandy a email<EMAIL_ADDRESS>
Abstract

We investigate data driven localized wave solutions of the Fokas-Lenells
equation by using physics informed neural network(PINN). We improve basic PINN
by incorporating control parameters into the residual loss function. We also
add conserve quantity as another loss term to modify the PINN. Using modified
PINN we obtain the data driven bright soliton and dark soliton solutions of
Fokas-Lenells equation. Conserved quantities informed loss function achieve
more accuracy in terms of relative $L2$ error between predicted and exact
soliton solutions. We hope that the present investigation would be useful to
study the applications of deep learning in nonlinear optics and other branches
of nonlinear physics. Source codes are available at
https://github.com/gautamksaharia/Fokas-Lenells

##  1 Introduction

Fokas-Lenells equation(FLE) [1, 2] is one of the four integrable equations
which describes stable propagation of ultrashort pulses in nonlinear mediums.
Significance of this equation lies on the presence of spatio-temporal term in
addition to group velocity dispersion term. This equation is used to model the
propagation of ultra-fast light pulses in nonlinear mediums such as pulse
propagation in monomode optical fiber. The other three integrable equations
are nonlinear Schrödinger equation (NLSE) [3, 4], derivative NLSE (DNLSE) [5],
higher order NLSE (HNLSE) [6, 7]. All four equations play a significant role
in the study of localized waves in nonlinear optical mediums [8].

Nonlinear equations do not always yield stable solutions or even a solution.
There are only a few analytical techniques available to deal with such
solutions. While the analytical methods, namely inverse scattering
transformation [9], direct bi-linear method [10, 11] are elegant but problem
specific and are applicable to mostly integrable cases. Due to the
unavailability of analytical solution in many cases, numerical methods and
other approximation methods[12, 13] are used as an alternative technique [14].

However, the accuracy of the solution depends on a number of parameters,
namely the number of iterations, calculation of higher order differential
terms, step size etc. Moreover, some complex numerical methods are found to be
very time consuming and computationally expensive. Such computationally
intensive resources are not available to everyone. There is an urgent need for
more general approaches to overcome limitations of analytical and numerical
methods.

Deep neural network (DNN) is one of the important discoveries in the 20th
century. After a few initial successes it failed to generate much interest but
it returned in the 21st century and showed promising results when applied to a
variety of fields across arts and science, namely in language processing [15],
image recognition [16] and many others [17]. DNN generated results have shown
enough potential to be a good alternative to the analytical and numerical
methods specially in solving complex nonlinear differential equations. This is
because of the most recent improvements in the computation powers and
availability of abundance data. The neural network also has an obvious
advantage over the analytical and numerical method in that the neural network
avoids complex calculations and formulas used in the conventional methods.

Recently Raissi et al. introduced PINN, which is computationally more
efficient than traditional numerical methods, especially when the physical
laws are highly nonlinear or when the geometrical domain is large and complex.
After the discovery of PINN, many modifications and improvements of this
method have been developed and applied to numerous fields [18, 19, 20, 21, 22,
23]. While PINNs are becoming more popular, but up to this point PINNs were
not capable of accurately simulating many nonlinear dynamical systems. Some
researchers addressed this drawback by modifying the loss function to increase
accuracy of PINN. This resulted in overcoming some of the earlier problems
seen with basic PINN. To mention a few notable approaches are, using a self
adaptive loss function through the adaptive weights for each loss term [24],
using a soft attention mechanism where the adaptation weights are fully
trainable and applied to each training point individually [25], least squares
weighted residual (LSWR) method [26], re-formulation of PINN loss functions
that can explicitly account for physical laws during model training [27],
gradient optimization algorithm which balances the interaction between
different terms in the loss function during model training by means of
gradient statistics [28].

PINN is applied to many nonlinear equations in optics with high degree of
success, namely Raissi et. al. predicted optical soliton solution of NLSE[29],
fang et al. [30] predicted femtosecond optical soliton of the high-order NLSE,
peng et al. obtained rogue periodic wave of Chen–Lee–Liu equation[31]. Data-
driven solutions and parameter discovery of other nonlinear systems such as
defocusing NLSE with the time dependent potential [32], Manakov system [33],
generalized Gross–Pitaevskii (GP) equation with PT symmetric potentials [34],
the Sasa–Satsuma [35], Yajima–Oikawa (YO) system [36], dark soliton of multi-
component Manakov model [37], LakshmananPorsezian-Daniel [38] have been
reported.

In all the above mentioned problems only dynamical equations along with
initial-boundary conditions are used as physical information. However, the
benefits of using the conserved quantity of the integrable system to the PINN
method have not received enough attention. Conserved quantities are crucial in
studying various optical dynamics, namely the stability of solitons, soliton
collisions and testing the stability of numerical methods. Therefore,
incorporating the information of conserved quantities should improve the
performance of neural networks by improving the convergence as well as
generalization of neural networks. Wu et. al. [39] has used this concept to
predict optical solitons of NLSE and other notable contributions to mention
are [40, 41]. To our knowledge, the data-driven solution of FLE using
conservation law in the PINN is not reported earlier. During our investigation
we noticed that the basic PINN algorithm converges to the minimum but cannot
learn the complex dynamics of FLE efficiently. Therefore, it becomes important
to study complex solutions of FLE by modifying existing PINN. In this paper,
we improve the PINN by modifying the loss function by incorporating a few of
the conserved quantities along with other physical information into the loss
function. Here, we aim to generate data driven bright soliton and dark soliton
solution of FLE using modified PINN. We have shown conserved quantities
informed loss function achieve more accuracy in terms of relative $L2$ error
between predicted and exact solution.

This paper is organized as follows. We present a review of FLE along with its
analytical bright and dark soliton solutions in section 2. In section 3 we
describe the basic PINN structure and present the modified PINN structure by
adding conserved quantities in the loss function. We show our results in
section 4 and section 5 concludes the paper.

##  2 FLE and bright dark soliton solutions

We consider the Fokas-Lenells equation

$u_{xt}=u-2i\sigma|u|^{2}u_{x}$ (1)

where $u=u(x,t)$ is a complex valued function, where subscripts x and t denote
partial differentiation with respect to $x$ and $t$. Here, $|u|^{2}u_{x}$
accounts for the nonlinearity and $u_{xt}$ is the spatio-temporal dispersion
term.

Under the vanishing background condition $|u|\rightarrow$ 0 as
$x\rightarrow\pm\infty$ we derive a bright soliton solution using Hirota
bilinear method [42].

$u=\frac{g_{1}}{f_{0}+f_{2}}$ (2)

where

$g_{1}=\alpha e^{\theta(x,t)}$ $f_{0}=\beta$
$f_{2}=\beta_{2}e^{\theta(x,t)+\theta^{*}(x,t)}$

$\theta=px+\frac{t}{p}+\theta_{0}$

$\beta_{2}=i\frac{|\alpha|^{2}|p|^{2}p}{\beta^{*}(p+p^{*})}$

$\alpha$, $\beta$, $p$ and $\theta_{0}$ are arbitrary complex constants,
$\alpha$ represent the polarization state, $\beta$ represents the initial
central position, $p$ corresponds to the spectral parameter obtained in the
Inverse scattering method and $\theta_{0}$ represents the initial phase. For
$p=a+ib$ where $a$ and $b$ are real constant, we get the following form of
bright 1-SS which we used in this paper.

$u=-\frac{2a}{a+ib}\frac{e^{\theta+i\chi}}{e^{2\theta}-b-ia}$ (3)

where

$\theta=a(x+vt)$ $\chi=b(x-vt)$ $v=\frac{1}{a^{2}+b^{2}}$

Again, considering FLE with non-vanishing boundary, we obtain following [43],
the dark 1-SS on the background of a plane wave, that is, when

$u\rightarrow\rho e^{i(\kappa x-\omega t)},\quad x\rightarrow\pm\infty$

The dark 1-SS is

$u=\rho e^{i(\kappa x-\omega
t)}\frac{1-\frac{k+b+ia}{2a}\frac{a+ib}{a-ib}e^{2\theta}}{1+\frac{k+b-ia}{2a}e^{2\theta}}$
(4)

where the amplitude $\rho$, frequency $\omega$ and wave number $\kappa$ of the
plane wave are real constants and are constrained with the relation, namely
$\omega=\frac{1}{\kappa}+2\rho^{2}$. $a$ and $b$ are real constant given by
$a=\sqrt{k^{3}\rho^{2}(1+k\rho^{2})}sin(\phi)$ and
$b=k\rho^{2}+\sqrt{k^{3}\rho^{2}(1+k\rho^{2})}cos(\phi)$ and $0<\phi<2\pi$

##  3 PINN Deep learning method

###  3.1 PINN

In order to write PINN for FLE, we first simplify the complex structure of
FLE. We convert $u(x,t)$ into real and imaginary parts, namely
$u(x,t)$=$r(x,t)$ $+$ $i$ $m(x,t)$, where $r(x,t)$ and $m(x,t)$ are real
valued functions. Writing PINN for real and imaginary part of FLE as $f_{r}$
and $f_{m}$, we have:

$f_{r}:=\hat{r}_{xt}-\hat{r}-2(\hat{r}^{2}+\hat{m}^{2})\hat{m}_{x}$ (5)

$f_{m}:=\hat{m}_{xt}-\hat{m}+2(\hat{r}^{2}+\hat{m}^{2})\hat{r}_{x}$ (6)

where $\hat{r}(x,t;w,b)$ and $\hat{m}(x,t;w,b)$ are the latent solutions of
Neural Netowrks(NN) with the weight $W$ and bias parameters $b$, which have to
be optimized to learn the exact solution $u(x,t)$.

We construct a feed-forward NN, which consists of an input layer, M number of
hidden layers and an output layer as shown in Figure 1. The input layer takes
the coordinates $(x,t)$ as input, multiplies them with weight $W$ and adds
bias $b$ to it. Before sending them as an input to the next layer, we apply an
activation function $\sigma$, namely tanh to add non-linearity in the output.
The network is called feed-forward because each hidden layer of the NN
receives input from the previous layer. We use Glorot Normal initialization to
randomly initialize the network weight $W$ and bias term $b$. The final NN
representation is given by,

$u(X,\Theta)=\sigma(\sigma(\sigma(W_{0}X+b_{0})W_{1}+b_{1})W_{2}+b_{2})......$
(7)

Figure 1: Schematic diagram of PINN. A NN consists of an input and an output
layer and some number of hidden layers. $\sigma$ represents activation
functions in all layer. The input of the NN goes through all the hidden
layers. Outputs of the NN, namely $r$ and $m$ are considered inputs for the
initial and boundary condition loss functions, the physics equation loss
function, and the conserve quantity loss function. The total loss function is
the combination of all these loss functions, which are minimized by the ADAM
optimizer.

$u(X,\Theta)=[\hat{r}(x,t;w,b),\hat{m}(x,t;w,b)]$ is the output of the NN, and
$X=(x,t)$ is the input to the NN, $\Theta$ =
$\left\\{W^{k},b^{k}\right\\}^{M}_{k=1}$ represent trainable parameters in the
NN and $\sigma$ represents activation function. Our goal is to optimize these
trainable parameters so that $\hat{u}(x,t)$ satisfy the FLE and PINN $f_{r}$
and $f_{m}$ become minimum, such that the output of the NN approximates the
solution of FLE, i.e. $\hat{u}(x,t;w,b)\approx u(x,t)$.

PINN $f_{r}$ and $f_{m}$ share same parameters with the NN $u(X,\Theta)$
therefore, these common parameters are trained by minimizing the following
loss function of the network:

$Loss(\Theta)=Loss_{r_{0}}+Loss_{m_{0}}+Loss_{r_{b}}+Loss_{m_{b}}+Loss_{f_{r}}+Loss_{f_{m}}$
(8)

where, $Loss_{r_{0}}$, $Loss_{m_{0}}$, $Loss_{r_{b}}$, $Loss_{m_{b}}$,
$Loss_{f_{r}}$ and $Loss_{f_{m}}$ are defined by

$Loss_{r_{0}}=\frac{1}{N_{0}}\sum_{i=1}^{N_{0}}|\hat{r}(x^{i}_{0},t^{i}_{0})-r^{i}_{0}|^{2},\quad
Loss_{m_{0}}=\frac{1}{N_{0}}\sum_{i=1}^{N_{0}}|\hat{m}(x^{i}_{0},t^{i}_{0})-m^{i}_{0}|^{2}$
(9)

$Loss_{r_{b}}=\frac{1}{N_{b}}\sum_{i=1}^{N_{b}}|\hat{r}(x^{i}_{b},t^{i}_{b})-r^{i}_{b}|^{2},\quad
Loss_{m_{b}}=\frac{1}{N_{b}}\sum_{i=1}^{N_{b}}|\hat{m}(x^{i}_{b},t^{i}_{b})-m^{i}_{b}|^{2}$
(10)

$Loss_{f_{r}}=\frac{1}{N_{f}}\sum_{i=1}^{N_{f}}|f_{r}(x^{i}_{f},t^{i}_{f})|^{2},\quad
Loss_{f_{m}}=\frac{1}{N_{f}}\sum_{i=1}^{N_{f}}|f_{m}(x^{i}_{f},t^{i}_{f})|^{2}$
(11)

Here, $\left\\{x^{i}_{0},t^{i}_{0};r^{i}_{0},m^{i}_{0}\right\\}^{N_{0}}_{i=1}$
denotes initial data points,
$\left\\{x^{i}_{b},t^{i}_{b};r^{i}_{b},m^{i}_{b}\right\\}^{N_{b}}_{i=1}$
corresponds to collocation points on the boundary data, and
$\left\\{x^{i}_{f},t^{i}_{f}\right\\}^{N_{f}}_{i=1}$ represent the collocation
points on $f_{r}$ and $f_{m}$. $Loss_{r_{0}}$ and $Loss_{m_{0}}$ corresponds
to the loss on the initial data, $Loss_{r_{b}}$ and $Loss_{m_{b}}$ enforce the
vanishing boundary conditions, and $f_{r}$ and $f_{m}$ penalizes the FLE for
not being satisfied on the collocation points. These training data were
obtained from an exact solution, considering their initial and boundary
conditions. Collocation points are randomly chosen from a uniform distribution
in the computational domain i.e. $x\in[-L,L]$ and $t\in[t_{0},t_{1}]$.

With the help of the ADAM optimizer [44], we minimize the loss function by
optimizing the trainable parameters $\Theta$ =
$\left\\{W^{k},b^{k}\right\\}^{M}_{k=1}$. Thus, solving the FLE is now
transformed into an optimization problem. The global minimum of the loss
function corresponds to the solution of the FLE subject to particular initial
and boundary conditions.

###  3.2  Loss function with conserve quantity

Although the basic PINN method converges to a minimum value, it cannot learn
the exact soliton solution of the FLE. To address this problem, we modify the
loss function by multiplying some real coefficient $\gamma$ in the physics
informed loss function, i.e $f_{r}$ and $f_{m}$. Physics informed loss refers
to imposing FLE residuals in the total loss functions to regularize NN
training. The modified loss function is given as follows:

$Loss(\Theta)=Loss_{r_{0}}+Loss_{m_{0}}+Loss_{r_{b}}+Loss_{m_{b}}+\gamma(Loss_{f_{r}}+Loss_{f_{m}})$
(12)

where $\gamma$ determines how much of a physics informed loss function is to
be contributed to the total loss function.

Embedding the knowledge of physical information, namely energy conservation
law, momentum conservation law etc into a NN as a loss function improves the
PINN to capture the right solution and generalize better result even with a
small number of training data. Therefore, we further improve the loss function
by introducing additional physical information from the FLE. Being an
integrable equation, FLE has many conserved quantities. We use conserve
quantities of FLE in the PINN method to improve the loss function by adding
them as additional loss terms which have to be minimized during training.

The first few conserved quantities for FLE are given below, including both
positive and negative hierarchy, which are obtained by solving the methods
described in [42].

$H_{1}=\int^{+\infty}_{-\infty}|u_{x}|^{2}dx$ (13)

$H_{3}=\int^{+\infty}_{-\infty}(|u_{x}|^{4}-iu^{*}_{x}u_{xx})dx$ (14)

$H_{5}=\int^{+\infty}_{-\infty}(-u^{*}_{x}u_{xxx}+2|u_{x}|^{6}-3i|u_{x}|^{2}u^{*}_{x}u_{xx})dx$
(15)

$H_{-1}=\int^{+\infty}_{-\infty}-iu^{*}_{x}udx$ (16)

$H_{-3}=\int^{+\infty}_{-\infty}-(|u|^{2}+i|u|^{2}uu^{*}_{x})dx$ (17)

Since the temporal derivative of conservative quantity must be minimum, we
write

$\frac{\partial}{\partial t}H_{i}=0,\quad i=1,3,5,-1,-3$ (18)

$L_{conserve}=\sum_{i=1,3,5,-1,-3}||\frac{\partial}{\partial
t}\hat{H}_{i}||^{2}$ (19)

Where $L_{conserve}$ corresponds to loss on conserve quantities of FLE. Thus,
improved loss function with conserved quantity:

$Loss(\Theta)=Loss_{r_{0}}+Loss_{m_{0}}+Loss_{r_{b}}+Loss_{m_{b}}+\gamma_{1}(Loss_{f_{r}}+Loss_{f_{m}})+\gamma_{2}L_{cnsr}$
(20)

where $\gamma_{1}$ and $\gamma_{2}$ are two real-valued coefficients that
determine the contribution from the physics loss function and conserved
quantity loss function respectively. Improved loss functions present more
precise soliton behaviors with fewer sample points.

###  3.3 Algorithm

Step 1: Specification of training data, initial training data:
$(x^{i},t^{i},r^{i},m^{i})^{N_{0}}_{i=1}$; boundary training data:
$(x^{j},t^{j},r^{j},m^{j})^{N_{b}}_{j=1}$; residual training points:
$(x^{l}_{f},t^{l}_{f})^{N_{f}}_{l=1}$

Step 2: Construct the NN $u_{NN}(\Theta)$ with random initialization of
trainable parameters $\Theta\in\left\\{W,b\right\\}$

Step 3: Construct the PINN by substituting surrogate $u_{NN}$ into the
governing equation.

Step 4: Specification of the loss function, that include the weighted loss
function and conserved quantity.

Step 5 : Minimize the loss function to find best parameters $\Theta$ using
ADAM Optimizer

##  4 Results

###  4.1 Bright Soliton

We consider the initial condition $u(x,-1)=g(x)$ for $a=1$ and $b=1$

$g(x)=-\frac{2}{1+i}\frac{{e^{(x-\frac{1}{2})+i(x+\frac{1}{2})}}}{e^{2(x-\frac{1}{2})}-1-i}\quad
x\in[-5,5]$ (21)

and the vanishing boundary condition, namely

$u(x=-5,t)=u(x=5,t)=0,\quad t\in[-1,1]$

Here, we chose $x\in[-5,5]$ and $t\in[-1,1]$ as the spatial and temporal
interval.

Training data $\left\\{x^{i}_{0},-1\right\\}^{N_{0}}_{i=1}$ for initial
condition consists of $N_{0}=100$ data points, randomly drawn from uniform
distribution over the half open interval $[-5,5)$, similarly, $N_{b}=100$ data
points $\left\\{\pm 5,t^{i}_{b}\right\\}^{N_{b}}_{i=1}$ are drawn from uniform
distribution over the half open interval $[-1,1)$ to enforce the vanishing
boundary condition. Moreover, we have selected $N_{f}=1000$ number of randomly
sampled collocation points to enforce the eq. 1 inside the computational
domain. All these randomly sampled points are drawn from the uniform
distribution over the spatial and temporal intervals, $x\in[-5,5]$ and
$t\in[-1,1]$.

To obtain a data-driven bright soliton, we construct a feed-forward NN with
eight hidden layers with 40 neurons in each layer and hyperbolic tan as the
activation function. Minimizing the loss function by optimizing all the
learnable parameters using the ADAM optimizer with a learning rate of 0.001,
we approximate the bright 1-SS.

Considering $\gamma=0.001$ in eq. 12 we obtain $L_{2}$ error, where total
loss= 0.00085254 and physics loss = 0.79478168 after 30,000 iterations in time
1187.6953 seconds. Again considering $\gamma_{1}=0.001$ and
$\gamma_{2}=0.00001$ multiplying with $H_{-1}$ conserved loss function in eq.
20 we get $L2$ error where total loss = 0.00072432 and physics loss = 0.647380
after 30,000 iterations in 4393.4230 seconds. Thus, we find that inclusion of
conserved quantities to the loss function enables us to minimization of loss
more effectively than when it is excluded.

The results are demonstrated in Figure 2 and Figure 3. Figure. 2 (a) shows
density plot of exact bright 1-SS, Figure.2(b) shows density plot of data-
driven bright 1-SS and Figure 2(c) shows density plot of error between exact
and data-driven bright 1-SS. Figure 3 shows a comparison of the soliton at
different time instance (i) at t = -0.70, (ii) at t = -0.29 (iii) at t = 0.52.

(a) Exact

(b) Data-driven

(c) Error

Figure 2: Exact and data-driven bright 1-SS and error between them

(a) at t = -0.70

(b) at t = -0.29

(c) at t = 0.52

Figure 3: Comparison between exact and data driven bright 1-SS at three
different time instance (a)at t = -0.70, (b) at t = -0.29, (c) at t = 0.72

###  4.2 Dark Soliton

Initial value condition for dark soliton $u(x,-1)=g(x)$, considering
$a=\sqrt{2}$, $b=1$, $\kappa=1$, $\rho=1$ and $\omega=3$.

$g(x)=e^{i(x+3)}\frac{1-\frac{2+i}{2}\frac{1+i}{1-i}e^{2x-1}}{1+\frac{2-i}{2}e^{2x-1}}\quad,x\in[-5,5]$

and the boundary conditions

$u(-5,t)=u(5,t)=1,\quad\text{t}\in[-1,1]$

Here , we chose $x\in[-5,5]$ and $t\in[-1,1]$ as the spatial and temporal
interval.

The training data set is obtained considering the exact solution at the
initial boundary data i.e. at $x=-5$, $x=5$, and $t=-1$ dividing the spatial
region [-5, 5] into $N_{b}=200$ data points and temporal region [-1,1] into
$N_{0}=100$ points. We have also selected a $N_{f}=2000$ number of randomly
sampled collocation points to enforce the eq. 4 inside the computational
domain. All these randomly sampled points are drawn from the uniform
distribution over the spatial and temporal intervals $x\in[-5,5]$ and
$t\in[-1,1]$.

We use the 9 hidden layer deep PINN with 60 neurons per layer and a hyperbolic
tan activation function to obtain data driven dark 1-SS. To minimizing the
loss function, we use the ADAM optimizer with a learning rate 0.001. We obtain
dark 1-SS with total loss= 0.00221427 and physics loss = 1.88655448 after
30,000 iterations with coefficient value of $\gamma_{1}=0.001$ to physics loss
function and $\gamma_{2}=0.001$ to the $H_{-3}$ conserved quantities loss
function.

The results are demonstrated in Figure. 4 and 5. Figure. 2 (a) shows density
plot of exact dark 1-SS, Figure. 2 (b) shows density plot of data-driven dark
1-SS and Figure. 2 (c) shows density plot of error between exact and data-
driven dark 1-SS. We present a comparison between the exact and data-driven
dark soliton solution at three different time instants (i) $t=-0.29$, (ii)
$t=-0.70$ and (ii) $t=0.72$ in Figure 5.

(a) data driven

(b) exact

(c) error

Figure 4: Data-driven and exact dark 1-SS and error between them

(a) at t = -0.70

(b) at t = -0.29

(c) at t =0.72

Figure 5: Comparison between exact and data driven dark soliton solution at
three different time instance (a)at t = -0.70, (b) at t = -0.29, (c) at t =
0.72

##  5 Conclusion

We improve basic PINN by incorporating control parameters into the residual
loss function. We add the conserved quantities of FLE into the loss function
to modify the PINN. We have obtained both bright and dark 1-SS of FLE using an
improved PINN. Although PINN can predict bright soliton solutions with great
accuracy, for dark soliton, there is still scope for improving its accuracy.
We find that using conserved quantities of FLE as another loss term helps us
to obtain a data-driven soliton solution with greater accuracy. Therefore,
incorporating the information of conserved quantities should improve the
performance of the NN by improving its convergence as well as generalization.

## Acknowledgement

G.K. Saharia acknowledges Google Colab and Kaggle for their free GPU services.
S. Talukdar and R. Dutta acknowledges DST, Govt. of India for Inspire
fellowship, grant nos. DST Inspire Fellowship 2020/IF200278; 2020/IF200303.

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