# RAR-PINN algorithm for the data-driven vector-soliton solutions and
parameter discovery of coupled nonlinear equations

Shu-Mei Qina, Min Lia,, Tao Xub, Shao-Qun Dongb

a School of Mathematics and Physics,
North China Electric Power University, Beijing 102206, China
b College of Science, China University of Petroleum, Beijing 102249, China
Corresponding author, e-mail<EMAIL_ADDRESS>
###### Abstract

This work aims to provide an effective deep learning framework to predict the
vector-soliton solutions of the coupled nonlinear equations and their
interactions. The method we propose here is a physics-informed neural network
(PINN) combining with the residual-based adaptive refinement (RAR-PINN)
algorithm. Different from the traditional PINN algorithm which takes points
randomly, the RAR-PINN algorithm uses an adaptive point-fetching approach to
improve the training efficiency for the solutions with steep gradients. A
series of experiment comparisons between the RAR-PINN and traditional PINN
algorithms are implemented to a coupled generalized nonlinear Schrödinger
(CGNLS) equation as an example. The results indicate that the RAR-PINN
algorithm has faster convergence rate and better approximation ability,
especially in modeling the shape-changing vector-soliton interactions in the
coupled systems. Finally, the RAR-PINN method is applied to perform the data-
driven discovery of the CGNLS equation, which shows the dispersion and
nonlinear coefficients can be well approximated.

Keywords: Deep learning; $N$-soliton solutions; $N$-coupled nonlinear
equation; RAR-PINN; Parameters discovery

##  1 Introduction

In recent decades, with the explosion of computing resources and available
data, deep learning has been successfully applied in diverse fields, such as
language understanding [1], face recognition [2], risk assessment [3],
mathematical physics [4, 5, 6] and so on. Particularly, the idea of using
continuous dynamical systems to model high-dimensional nonlinear functions was
proposed in Ref.[7], i.e., the relationship between differential equations and
deep learning. Furthermore, Ref.[4] proposed the physics-informed neural
network (PINN) algorithm to solve partial differential equations (PDEs).
Compared with the traditional mesh-based methods, such as the finite element
method and the finite difference method, PINN method is mesh-free by taking
advantage of the automatic differentiation technique [8]. The original PINN
method has been successfully applied in solving nonlinear partial differential
equations with a small number of data sets [9, 10], and could break the curse
of dimensionality [11] in classical numerical methods. Recently, a lot of
high-quality data-driven solutions for classical integrable systems were
obtained by using the PINN method [12, 13]. The breathers and rogue wave
solutions of the nonlinear Schrödinger (NLS) equation were recovered with the
aid of the PINN model [14]. What’s more, a variety of femtosecond optical
soliton solutions of the high-order NLS equation were investigated with the
PINN method [15].

In order to improve the accuracy and convergence rate of the traditional PINN
(TPINN) algorithm, many modified algorithms have been proposed. For instance,
Ref. [16] introduced the gradient pathologies into PINN algorithm based on the
idea of adaptive loss function weights, Ref. [14] presented neuron-wise
locally adaptive activation function to derive localized wave solutions and
Ref. [17] proposed a generalized space-time domain decomposition approach to
solve PDEs on arbitrary complex-geometry domains. Each of these approaches
makes change to the neural network itself. In this paper, we will propose a
new RAR-PINN method, which combines a residual-based adaptive refinement
method to improve the training efficiency. The idea is that, solutions with
steep gradients can be learned better by putting more points near the sharp
front. Moreover, we will compare the TPINN and RAR-PINN algorithms by
comparing the residuals, loss function curves, absolute errors, and relative
errors in detail.

Most of the previous works investigated the data-driven scalar solitons. In
this study, we would like to explore the data-driven vector solutions of
integrable coupled nonlinear equations. Specially, the coupled generalized
nonlinear Schrödinger (CGNLS) equation [18] is used as an example to
illustrate, namely

$\left\\{\begin{aligned}
&ih_{1t}+h_{1xx}+2(\alpha|h_{1}|^{2}+\beta|h_{2}|^{2}+\gamma
h_{1}h_{2}^{*}+\gamma^{*}h_{2}h_{1}^{*})h_{1}=0,x\in[-L,L],t\in[-T,T]\\\ \
&ih_{2t}+h_{2xx}+2(\alpha|h_{1}|^{2}+\beta|h_{2}|^{2}+\gamma
h_{1}h_{2}^{*}+\gamma^{*}h_{2}h_{1}^{*})h_{2}=0,\end{aligned}\right.$ (1)

where $h_{1}$ and $h_{2}$ are slowly varying pulse envelopes, $\alpha$ and
$\beta$ are real constants corresponding to the self-phase modulation and
cross-phase modulation effects, respectively. Here $\gamma$ is a complex
constant corresponding to four-wave mixing, the subscripts $x$ and $t$ are the
partial derivatives about the scaled distance and retarded time, and $*$
represents complex conjugation. When $\alpha=\beta$ and $\gamma=0$, the above
equation reduces to the Manakov system, which is a well known two-coupled NLS
equation in the literature [19]. What’s more, when $\alpha=-\beta$ and
$\gamma=0$, it reduces to the mixed coupled NLS equation [20, 21]. The
nonlinear system (1) is proved to be completely integrable for arbitrary
values of the system parameters $\alpha,\gamma$ and $\beta$ through the Weiss-
Tabor-Carnevale algorithm [22]. Multihump solitons were demonstrated via
experiments when the self-trapped incoherent wave packets propagate in a
dispersive nonlinear medium [23, 24]. The non-degenerate vector-soliton
solutions can be applied in the fields of Bose-Einstein condensates [25, 26]
and optics communication [12]. Up until now, the Manakov system has been
solved by a pre-fixed multi-stage training algorithm [27]. In this paper, we
will apply the RAR-PINN method to model various types of vector solitons and
their interactions of Eq. (1) which is general than the Manakov system.

The structure of this paper is as follows. In Section 2, the algorithm of the
RAR-PINN is provided for the $N$-component coupled nonlinear equations. In
Section 3, the vector one-, two- and three-soliton solutions of Eq. (1) are
predicted by using the TPINN and RAR-PINN methods, respectively. The relative
errors, loss functions, and absolute errors for those two methods are also
compared. In Section 4, we discuss how to apply the RAR-PINN method to the
data-driven parameters discovery for Eq. (1). Finally, conclusions are given
in Section 5.

##  2 Method

Multi-layer feedforward neural networks have strong expressive power.
Theoretically, as long as the number of neurons is large enough, the
feedforward neural networks can simultaneously approximate any function and
their partial derivatives [28]. TPINN [4] is a class of deep learning
algorithm where the PDE is satisfied through the loss function of the multi-
layer neural networks. The TPINN algorithm is simple, which may not be
efficient for some PDEs that exhibit solutions with steep gradients [29]. In
order to improve the training efficiency of TPINN, the RAR-PINN method is
proposed. The improved algorithm has great potential to be applied to various
fields where adaptivity is needed [30].

In this paper, we consider the following form of $(1+1)$-dimensional
$N$-coupled nonlinear time-dependent system in complex space,

$\displaystyle
i\bm{H}_{t}+\bm{\chi}(\bm{\lambda},\bm{H},\bm{H}_{xx},\bm{H}_{xxx},...)=0,x\in\Omega,t\in[-T,T]$
(2) $\displaystyle\mathcal{B}(\bm{H},x,t)=0,x\in\partial\Omega,$

and solve their soliton solutions
$\bm{H}=[h_{1}(x,t),h_{2}(x,t),...,h_{n}(x,t)]$ is a complex-valued vector.
$\bm{\chi}=[\chi_{1},\chi_{2},...,\chi_{n}]$ are nonlinear differential
operators, specially, they can be taxonomized into a hyperbolic [31],
parabolic [32] or elliptic differential operator [33].
$\bm{\lambda}=[\lambda_{1},\lambda_{2},...]$ are parameters of the equation.
$\mathcal{B}(\bm{H},x,t)$ could be hyperbolic Neumann boundary conditions
[34], Dirichlet boundary conditions [35], periodic boundary conditions [36] or
mixed boundary conditions [37]. Besides, the initial condition can be simply
treated as a special type of Dirichlet boundary condition on the spatia-
temporal domain [13]. The subscripts $t$ and $x$ are the partial derivatives
and correspond to the retarded time and scaled distance. Then, Eq. (2) can be
converted into

$\left\\{\begin{aligned}
&ih_{1t}+\chi_{1}(\lambda,h_{1},h_{2},h_{1x},h_{2x},...)=0,\\\
&ih_{2t}+\chi_{2}(\lambda,h_{1},h_{2},h_{1x},h_{2x},...)=0,\\\ &\vdots\\\
&ih_{nt}+\chi_{n}(\lambda,h_{1},h_{2},h_{1x},h_{2x},...)=0.\end{aligned}\right.$
(3)

Next, we decompose $h_{i}(x,t)$ into the real part $u_{i}(x,t)$ and the
imaginary part $v_{i}(x,t)(i=1,2,...,n)$, i.e., $h_{1}(x,t)=u_{1}+v_{1}i$,
$h_{2}(x,t)=u_{2}+v_{2}i,...,h_{n}(x,t)=u_{n}+v_{n}i$. Then Eq. (3) reduces to

$\left\\{\begin{aligned}
&u_{1t}+\chi_{1u}(\lambda,u_{1},v_{1},u_{2},v_{2},...)=0,\\\
&v_{1t}+\chi_{1v}(\lambda,u_{1},v_{1},u_{2},v_{2},...)=0,\\\
&u_{2t}+\chi_{2u}(\lambda,u_{1},v_{1},u_{2},v_{2},...)=0,\\\
&v_{2t}+\chi_{2v}(\lambda,u_{1},v_{1},u_{2},v_{2},...)=0,\\\ &\vdots\\\
&u_{nt}+\chi_{nu}(\lambda,u_{1},v_{1},u_{2},v_{2},...)=0,\\\
&v_{nt}+\chi_{nv}(\lambda,u_{1},v_{1},u_{2},v_{2},...)=0.\end{aligned}\right.$
(4)

Specifically, $\chi_{1u},\chi_{1v},\chi_{2u},...,\chi_{nv}$ are nonlinear
functions of the soliton solutions and their spatial derivatives
$u_{1},v_{1},u_{2},v_{2},u_{1x},v_{1x},u_{2x},v_{2x}...$. We define the
residuals $f_{1u}(x,t),f_{1v}(x,t),f_{2u}(x,t),...$ and $f_{nv}(x,t)$,
respectively

$\left\\{\begin{aligned}
&f_{1u}:=\hat{u}_{1t}+\chi_{1u}(\lambda,\hat{u}_{1},\hat{v}_{1},\hat{u}_{2},\hat{v}_{2},...),\\\
&f_{1v}:=\hat{v}_{1t}+\chi_{1v}(\lambda,\hat{u}_{1},\hat{v}_{1},\hat{u}_{2},\hat{v}_{2},...),\\\
&f_{2u}:=\hat{u}_{2t}+\chi_{2u}(\lambda,\hat{u}_{1},\hat{v}_{1},\hat{u}_{2},\hat{v}_{2},...),\\\
&f_{2v}:=\hat{v}_{2t}+\chi_{2v}(\lambda,\hat{u}_{1},\hat{v}_{1},\hat{u}_{2},\hat{v}_{2},...),\\\
&\vdots\\\
&f_{nu}:=\hat{u}_{nt}+\chi_{nu}(\lambda,\hat{u}_{1},\hat{v}_{1},\hat{u}_{2},\hat{v}_{2},...),\\\
&f_{nv}:=\hat{v}_{nt}+\chi_{nv}(\lambda,\hat{u}_{1},\hat{v}_{1},\hat{u}_{2},\hat{v}_{2},...).\end{aligned}\right.$
(5)

Considering governing equations, initial condition and boundary condition, the
loss function for solving Eq. (3) can be expressed as

$\begin{split}\begin{aligned}
Loss=loss_{0}+loss_{b}+loss_{f},\end{aligned}\end{split}$ (6)

where $loss_{b}$ and $loss_{0}$ represent the mean square error of boundary
and initial condition, respectively, while $loss_{f}$ is the collocation
points error. Specifically,

$\begin{split}\begin{aligned}
loss_{0}=&\frac{1}{N_{0}}\sum_{i=1}^{N_{0}}(|\hat{u}_{1}(x^{i},0)-u_{10}^{i}|^{2}+|\hat{v}_{1}(x^{i},0)-v_{10}^{i}|^{2}\\\
&+|\hat{u}_{2}(x^{i},0)-u_{20}^{i}|^{2}+|\hat{v}_{2}(x^{i},0)-v_{20}^{i}|^{2}+...),\end{aligned}\end{split}$
(7) $\begin{split}\begin{aligned}
loss_{b}=&\frac{1}{N_{b}}\sum_{j=1}^{N_{b}}(|\hat{u}_{1}(x^{j},t^{j})-u_{1b}^{j}|^{2}+|\hat{v}_{1}(x^{i},t^{j})-v_{1b}^{j}|^{2}\\\
&+|\hat{u}_{2}(x^{j},t^{j})-u_{2b}^{j}|^{2}+|\hat{v}_{2}(x^{j},t^{j})-v_{2b}^{j}|^{2}+...),\end{aligned}\end{split}$
(8) $\begin{split}\begin{aligned}
loss_{f}=&\frac{1}{N_{f}}\sum_{p=1}^{N_{f}}(|f_{1u}(x_{f}^{p},t_{f}^{p})|^{2}+|f_{1v}(x_{f}^{p},t_{f}^{p})|^{2}+|(f_{2u}(x_{f}^{p},t_{f}^{p})|^{2}+|f_{2v}(x_{f}^{p},t_{f}^{p})|^{2}+...),\end{aligned}\end{split}$
(9)

here
$\\{(x^{i},0),u_{10}^{i},v_{10}^{i},u_{20}^{i},v_{20}^{i},...\\}_{i=1}^{N_{0}}$
denotes the initial training data $\tau_{0}$ and the boundary training data
$\tau_{b}$ is
$\\{(x^{j},t^{j}),u_{1b}^{j},v_{1b}^{j},u_{2b}^{j},v_{2b}^{j},...\\}_{j=1}^{N_{b}}$
of Eq. (3). What’s more, $\\{(x_{f}^{p},t_{f}^{p})\\}_{p=1}^{N_{f}}$ is the
collocation points $\tau_{f}$ in space and time for the residuals
$f_{1u}(x_{f}^{p},t_{f}^{p}),f_{1v}(x_{f}^{p},t_{f}^{p}),...,f_{nv}(x_{f}^{p},t_{f}^{p})$.
$N_{0},N_{b}$ and $N_{f}$ are the numbers of auxiliary coordinates required to
calculate the $Loss$ terms in the initial conditions, boundary conditions and
governing equations, respectively. $\hat{u}_{1},\hat{v}_{1},...,\hat{v}_{n}$
are the predicted output data through multi-layer neural networks.

As we discussed above, the collocation points for residuals are usually
randomly distributed in the solution domain. Furthermore, all randomly sampled
point locations are generated using a space filling Latin Hypercube Sampling
strategy [38]. This sampled method works well for many equations, however, it
may not be efficient for some PDEs that exhibit solutions with steep gradients
[29]. For better results, we use a RAR-PINN algorithm to improve the
distribution of residual points during process. The modified algorithm
structure for solving Eq. (3) is shown in Figure 1, which also applies to
other coupled nonlinear equations.

Figure 1: Schematic of RAR-PINN for solving the $N$-coupled nonlinear
equation.

We can divide this structure into three parts, including deep neural networks
(DNNs), physical information and loss function. The part of DNNs is the
traditional fully connected feedforward neural networks. In detail, the input
layer consists of two neurons $x$ and $t$, the number of hidden layers is $k$,
and the output layer consists of $2n$ neurons
$\hat{u}_{1},\hat{v}_{1},\hat{u}_{2},...,\hat{v}_{n}$. We can compute the
derivatives of the neural network outputs with respect to the neural network
inputs time $t$ and space $x$ by AD, and incorporate them into residual as a
part of the loss function. AD, also known as algorithmic differentiation or
simply “autodiff”, is a technique similar to but more general than
backpropagation for efficiently and accurately evaluating the derivative of a
function expressed as a computer program [39]. TensorFlow [40] or PyTorch [8]
is used to implement AD of the parameters in a neural network. In the physical
information section, $loss_{0}$ is determined by the initial condition, the
governing equation controls the numerical value of $loss_{f}$, and the
numerical value of $loss_{b}$ is determined by the boundary condition. In
addition, the value of the loss function can be obtained from Eq. (6).
Furthermore, in order to improve the efficiency of the neural networks, we
judge whether the mean residual of the PDE is less than a threshold
$\varepsilon_{0}$. If it is less than $\varepsilon_{0}$, the training ends.
Otherwise, we add new points to continue the training until the mean residual
is less than $\varepsilon_{0}$, as shown in algorithm 1. RAR-PINN method adds
more degrees of freedom when the solution PDE residual is high.

Input: Initial training data set $\tau_{0},\tau_{b}$ and $\tau_{f}$, number of
new points added $m$, number of iterations $n$ and a tolerance
$\varepsilon_{0}$;

while  _$err <\varepsilon_{0}$_ do

– Calculate the mean coupled nonlinear differential partial equation residual
$err$, specifically,

$err=\dfrac{1}{N}(|f_{1u}|+|f_{1v}|+|f_{2u}|+|f_{2v}|+...);$

– Select the $m$ points with the largest residual;

– Add the $m$ new points to the training data set;

end while

Output: Predictive solutions of Eq. (2).

Algorithm 1 RAR-PINN for solving Eq. (3)

##  3 Data-driven vector $N$-soliton solutions of the CGNLS equation

###  3.1 Vector one-soliton solutions

In the following, we will take Eq. (1) as an example. Vector one-soliton
solutions of Eq. (1) was derived in [41] via the Hirota bilinear method.
However, the data-driven vector one-soliton solutions of Eq. (1) have not been
investigated so far. In order to assess the accuracy of our method, here, we
give the exact vector one-soliton solution for System (1) as [41]

$\left\\{\begin{aligned}
&h_{1}=\frac{ae^{\theta}}{1+e^{\theta+\theta^{*}+c}}=\frac{a}{2}e^{-\frac{c}{2}}e^{i\theta_{I}}{\rm
sech}(\theta_{R}+\frac{c}{2}),\\\
&h_{2}=\frac{be^{\theta}}{1+e^{\theta+\theta^{*}+c}}=\dfrac{b}{2}e^{-\frac{c}{2}}e^{i\theta_{I}}{\rm
sech}(\theta_{R}+\frac{c}{2}),\end{aligned}\right.$ (10)

with $\theta=kx+ik^{2}t$ and
$e^{c}=\frac{|a|^{2}\alpha+|b|^{2}\beta+a_{1}b^{*}\gamma+a^{*}b\gamma^{*}}{(k+k^{*})^{2}}$,
where $\frac{c}{2}$ represents the localization position of one-solitons; $a$,
$b$ and $k$ are complex constants. Here and elsewhere, the subscripts $R$ and
$I$ represent the real and imaginary parts, respectively.

As an example, by choosing $a=1$, $b=2$, $k=1.5+i$ and $\alpha=\beta=\gamma=1$
in Solution (10) the corresponding initial condition is obtained as

$\left\\{\begin{aligned}
&h_{10}(x)=h_{1}(x,-2)=\frac{e^{(1.5+i)x+6-2.5i}}{1+e^{3x+12}}=\frac{1}{2}e^{(1.5i-1)x+6i+2.5}{\rm
sech}(1.5x+6),\\\
&h_{20}(x)=h_{2}(x,-2)=\frac{2e^{(1.5+i)x+6-2.5i}}{1+e^{3x+12}}=e^{(1.5i-1)x+6i+2.5}{\rm
sech}(1.5x+6),\end{aligned}\right.$ (11)

and the Dirichlet-Neumann periodic boundary condition is expressed as

$\left\\{\begin{aligned}
&h_{1}(-10,t)=h_{1}(10,t),h_{2}(-10,t)=h_{2}(10,t),t\in[-2,2],\\\
&h_{1x}(-10,t)=h_{1x}(10,t),h_{2x}(-10,t)=h_{2x}(10,t).\end{aligned}\right.$
(12)

The training data set can be obtained by using random sampling. More
specifically, the computation domain $[-10,10]\times[-2,2]$ is divided into
$[300\times 201]$ data points, vector one-soliton solutions $h_{1}$ and
$h_{2}$ are discretized into 201 snapshots on regular space-time grid with
$\bigtriangleup t=0.02$. A small training data set containing initial data is
generated by randomly extracting $N_{0}=50$ from original data set. Similarly,
the number of boundary points is $N_{b}=50$. Initially, 4000 points are
selected as the residual points, and then we add 10 more new points adaptively
via RAR with $m=5$ and $\varepsilon_{0}=0.01$. To compare the RAR-PINN
solutions with TPINN solutions, we obtain the TPINN solutions based on
$N_{f}=4010$ points by the Latin Hypercube Sampling method [38]. After giving
a data set of initial points, boundary points and residual points, the latent
vector one-solitons $h_{1}(x,t)$ and $h_{2}(x,t)$ can be successfully learned
by minimizing the loss function Eq. (6). In order to minimiz the loss
function, we use 6 hidden layers of 32 neurons in each layer to numerically
simulate soliton solutions. The results are displayed in Figure 2, which
demonstrates the effectiveness of the RAR-PINN method.

(a) Exact solution $|h_{1}(x,t)|$ (b) Exact solution $|h_{2}(x,t)|$
(c) Trational PINN solution $|h_{1}(x,t)|$ (d) Trational PINN solution
$|h_{2}(x,t)|$
(e) RAR-PINN solution $|h_{1}(x,t)|$ (f) RAR-PINN solution $|h_{2}(x,t)|$

Figure 2: The vector one-soliton solution $h_{1}(x,t)$ and $h_{2}(x,t)$:
Comparison among the exact solutions (panels (a) and (b)), predictive
solutions via TPINN (panels (c) and (d)) and predictive results by RAR-PINN
(panels (e) and (f)) at different propagation times $t=-1.70,0.0$ and 1.70.
The left and right graphs represent the density diagrams and the profiles at
different times for the one-soliton solutions $h_{1}(x,t)$ and $h_{2}(x,t)$,
respectively.

From Figure 2, we can clearly observe the reconstructed vector one-soliton
space-time dynamics. The density plots with the corresponding peak scale for
diverse dynamics include exact dynamics, learned dynamics with TPINN and
learned dynamics with RAR-PINN, respectively. Specifically, Figure 2(a) and
Figure 2(b) indicate the magnitude of the exact vector one-soliton solutions
$|h_{1}(x,t)|=\sqrt{u_{1}^{2}(x,t)+v_{1}^{2}(x,t)}$ and
$|h_{2}(x,t)|=\sqrt{u_{2}^{2}(x,t)+v_{2}^{2}(x,t)}$; Figure 2(c) and Figure
2(d) show the the predicted vector one-soliton solutions via TPINN; and Figure
2(e) and Figure 2(f) demonstrate the results via RAR-PINN. To make a more
detailed comparison, the relative $\mathbb{L}_{2}$ errors of the TPINN model
are $9.189407\times 10^{-2}$ for $h_{1}(x,t)$ and $8.453297\times 10^{-2}$ for
$h_{2}(x,t)$; while the relative $\mathbb{L}_{2}$ errors of the RAR-PINN model
are $4.979434\times 10^{-2}$ for $h_{1}(x,t)$ and $4.661114\times 10^{-2}$ for
$h_{2}(x,t)$. Besides, Figure 2 demonstrates the comparison of the exact
vector one-soliton solution (blue curve) and the predicted one (red curve) at
different times.

Specifically, to obtain the optimal parameters to minimize the loss function,
we first use the Adam optimizer [42] for 10,000 iterations, followed by the
L-BFGS optimizer [43]. Importantly, the L-BFGS method has global convergence
on uniformly convex problems and it is very competitive due to its low
iteration cost [43]. The specific loss function curve is shown in Figure 3.
Figure 3(a) and (b) represent the loss function curves via TPINN, in detail,
Figure 3(a) shows the loss function descent curve for 10,000 iterations using
the Adam optimizer and Figure 3(b) demonstrates the loss function curve using
the L-BFGS optimizer. From Figure 3(b) the initial condition error is
$loss_{0}=2.2284491\times 10^{-6}$, the boundary condition error is
$loss_{b}=4.350938\times 10^{-7}$, the collocation points error is
$loss_{f}=1.6587395\times 10^{-5}$, and the overall error is
$Loss=1.9250938\times 10^{-5}$. Figure 3(c) and Figure 3(d) illustrate the
loss function curves with RAR-PINN, specifically, Figure 3(c) indicates the
loss function curve for 10,000 iterations using the Adam optimization and
Figure 3(d) shows the loss function curve using the L-BFGS optimization. From
Figure 3(d) the initial condition error is $loss_{0}=6.016333\times 10^{-7}$,
the boundary condition error is $loss_{b}=1.3800212\times 10^{-7}$, the
collocation points error is $loss_{f}=7.81987\times 10^{-6}$ and the overall
error is $Loss=8.559506\times 10^{-6}$. By comparing the loss functions of the
two methods, the RAR-PINN method is more efficient for solving the vector one-
soliton solutions of Eq. (1).

(a) (b)
(c) (d)

Figure 3: The loss function curve figures of one-soliton solutions
$h_{1}(x,t)$ and $h_{2}(x,t)$ arising from (a) TPINN algorithm with the 10000
iterations Adam optimization; (b) TPINN algorithm with the 9864 iterations
L-BFGS optimization; similarly, (c) RAR-PINN algorithm with the 10000
iterations Adam optimization; (d) RAR-PINN algorithm with the 13506 iterations
L-BFGS optimization.

The three-dimensional diagram of the residual error is shown in Figure 4. From
Figure 4(a), we can see that the residuals of some positions before the
adaptive pointing are relatively large up to 3.520, and from Figure 4(b), the
residuals of all positions after the adaptive pointing can be controlled
within 0.08. Combining the above numerical simulation results, we can find
that RAR-PINN is more accurate than TPNN for solving the vector one-soliton
solutions of Eq. (1). %ͨ ͼ4 ǿ Կ Ӧȡ ̵֮ǰ Щλ õĲв Ƚϴ Դﵽ Ӧȡ ̵֮ λ õĲв Կ ڡ

(a) (b)

Figure 4: Three-dimensional stereogram of residual error: (a) three-
dimensional stereogram residual error without adding adaptive points; (b)
three-dimensional stereogram residual error after adding adaptive points.

###  3.2 Vector two-soliton solutions

Through the Hirota bilinear method, the vector two-soliton solutions can be
derived via symbolic computation. It is found that the vector two-soliton
solutions admit the inelastic and elastic collisions depending on the choice
of solitonic parameters. Simplifying the resultant expressions we have the
exact vector two-soliton solutions of the form [44]

$\left\\{\begin{aligned}
&h_{1}=\frac{\xi_{1}^{(1)}e^{\eta_{1}}+\xi_{2}^{(1)}e^{\eta_{2}}+e^{\eta_{1}+\eta_{1}^{*}+\eta_{2}+\delta_{11}}+e^{\eta_{1}+\eta_{2}^{*}+\eta{2}+\delta_{21}}}{D},\\\
&h_{2}=\frac{\xi_{1}^{(2)}e^{\eta_{1}}+\xi_{2}^{(2)}e^{\eta_{2}}+e^{\eta_{1}+\eta_{1}^{*}+\eta_{2}+\delta_{12}}+e^{\eta_{1}+\eta_{2}^{*}+\eta{2}+\delta_{22}}}{D},\end{aligned}\right.$
(13)

and

$\begin{split}\begin{aligned}
D=1+e^{\eta_{1}+\eta_{1}^{*}+R_{1}}+e^{\eta_{1}+\eta_{2}^{*}+\delta_{0}}+e^{\eta_{1}^{*}+\eta_{2}+\delta_{0}^{*}}+e^{\eta_{2}+\eta_{2}^{*}+R_{2}}+e^{\eta_{1}+\eta_{1}^{*}+\eta_{2}+\eta_{2}^{*}+R_{3}},\end{aligned}\end{split}$
(14)

where we have defined the constants as

$\begin{split}&\eta_{1}=k_{1}(x+ik_{1}t),\eta_{2}=k_{2}(x+ik_{2}t),e^{\delta_{0}}=\frac{\varphi_{12}}{k_{1}+k_{2}^{*}},e^{\delta_{R_{1}}}=\frac{\varphi_{11}}{k_{1}+k_{1}^{*}},\\\
&e^{\delta_{R_{2}}}=\frac{\varphi_{22}}{k_{2}+k_{2}^{*}},e^{\delta_{R_{3}}}=\frac{|k_{1}-k_{2}|^{2}(\varphi_{11}\varphi_{22}-\varphi_{12}\varphi_{21})}{(k_{1}+k_{1}^{*})(k_{2}+k_{2}^{*})|k_{1}+k_{2}^{*}|^{2}},\\\
&e^{\delta_{11}}=\frac{(k_{1}-k_{2})(\xi_{1}^{(1)}\varphi_{21}-\xi_{2}^{(1)}\varphi_{11})}{(k_{1}+k_{1}^{*})(k_{2}+k_{1}^{*})},e^{\delta_{21}}=\frac{(k_{2}-k_{1})(\xi_{2}^{(2)}\varphi_{12}-\xi_{1}^{(1)}\varphi_{22})}{(k_{2}+k_{2}^{*})(k_{1}+k_{2}^{*})},\\\
&e^{\delta_{12}}=\frac{(k_{1}-k_{2})(\xi_{1}^{(2)}\varphi_{21}-\xi_{2}^{(2)}\varphi_{11})}{(k_{1}+k_{1}^{*})(k_{2}+k_{1}^{*})},e^{\delta_{22}}=\frac{(k_{2}-k_{1})(\xi_{2}^{(2)}\varphi_{12}-\xi_{1}^{(2)}\varphi_{22})}{(k_{2}+k_{2}^{*})(k_{1}+k_{2}^{*})},\end{split}$
(15)

and

$\varphi_{mn}=\dfrac{\alpha\xi_{m}^{(1)*}+\beta\xi_{m}^{(2)}\xi_{n}^{(2)*}+\gamma\xi_{m}^{(1)}\xi_{n}^{(2)*}+\gamma\xi_{m}^{(1)*}\xi_{n}^{(2)}}{k_{m}+k_{n}^{*}},m,n=1,2.$
(16)

The two-soliton solutions $h_{1}(x,t)$ and $h_{2}(x,t)$ are characterized by
arbitrary complex parameters $k_{1}$, $k_{2}$, $\xi_{1}^{(1)}$,
$\xi_{1}^{(2)}$, $\xi_{2}^{(1)}$ and $\xi_{2}^{(2)}$.

####  3.2.1 Elastic collision of vector two-soliton solution

To illustrate the collision behavior of vector two-soliton solutions, we take
$[-L,L]$ and $[-T,T]$ in Eq. (1) as $[-7,7]$ and $[-1,1]$, respectively. By
choosing $k_{1}=1+i$, $k_{2}=2-i$,
$\xi_{1}^{(1)}=\xi_{1}^{(2)}=\xi_{2}^{(1)}=\xi_{2}^{(2)}=1$, $\alpha=\beta=2$
and $\gamma=0.5+0.5i$ in Solution (13) the corresponding initial condition is
driven as shown in the following formula: $h_{10}(x)=h_{1}(x,-1)$ and
$h_{20}(x)=h_{2}(x,-1)$. What’s more, the boundary condition is expressed as

$\left\\{\begin{aligned}
&h_{1}(-7,t)=h_{1}(7,t),h_{2}(-7,t)=h_{2}(7,t),t\in[-1,1],\\\
&h_{1x}(-7,t)=h_{1x}(7,t),h_{2x}(-7,t)=h_{2x}(7,t).\end{aligned}\right.$ (17)

The computation domain $[-7,7]\times[-1,1]$ is divided into $[300\times 201]$
data points, and then vector two-soliton solutions $h_{1}$ and $h_{2}$ are
discretized into 201 snapshots on regular space-time grid with $\bigtriangleup
t=0.01$. A training data set containing initial data and boundary data are
generated by randomly sampling [45]. Specifically, the number of boundary
points is $N_{b}=100$, and the number of initial points is $N_{0}=100$.
Firstly, we choose 4000 points selected as the residual points and then add 15
more new points adaptively via RAR with $m=3$ and $\varepsilon_{0}=0.055$. In
order to compare these two algorithms, we select $N_{0}=100$, $N_{b}=100$ and
$N_{f}=4015$ to obtain the vector two-soliton solutions using the TPINN
algorithm. The predicted vector two-soliton solutions $h_{1}(x,t)$, and
$h_{2}(x,t)$ can be obtained by minimizing the loss function. Here a fully
connected neural network with 6 hidden layers and 32 neurons per hidden layer
is used to learn the parameters of the neural network. Besides, Figure 5
specifically demonstrates the effectiveness of RAR-PINN in solving the elastic
collision vector two-soliton solutions.

As shown in Figure 5, we can clearly see the reconstructed vector two-soliton
space-time dynamics. The density plots with the corresponding peak scale for
diverse dynamics include exact dynamics and two dynamics learned by different
methods. More specifically, Figure 5 (a-b) are the exact interactions of two
solitons, while Figure 5 (c-d) and Figure 5 (e-f) show the results given by
TPINN and RAR-PINN methods, respectively. The blue and red curves represent
the exact and predicted solutions, respectively. The predicted solutions
obtained by the two methods are compared with the exact solutions,
respectively. The relative $\mathbb{L}_{2}$ errors of the TPINN model are
$7.381569\times 10^{-2}$ for $h_{1}(x,t)$ and $7.395515\times 10^{-2}$ for
$h_{2}(x,t)$; while the relative $\mathbb{L}_{2}$ errors of the RAR-PINN model
are $3.764622\times 10^{-2}$ for $h_{1}(x,t)$ and $3.756750\times 10^{-2}$ for
$h_{2}(x,t)$.

(a) Exact solution $|h_{1}(x,t)|$ (b) Exact solution $|h_{2}(x,t)|$
(c) Tractional PINN solution $|h_{1}(x,t)|$ (d) Tractional PINN solution
$|h_{2}(x,t)|$
(e) RAR-PINN solution $|h_{1}(x,t)|$ (f) RAR-PINN solution $|h_{2}(x,t)|$

Figure 5: The elastic collision two-soliton solutions $h_{1}(x,t)$ and
$h_{2}(x,t)$: Comparison among the exact solutions (panels (a) and (b)),
predictive solutions via TPINN (panels (c) and (d)) and predictive results by
RAR-PINN (panels (e) and (f)) at different propagation times $t=-0.85,0.0$ and
0.85. The left and right graphs represent the density diagrams and the
profiles at different times for the vector two-soliton solutions $h_{1}(x,t)$
and $h_{2}(x,t)$, respectively.

During the process of neural network training, we first use Adam optimizer
[42] to train 10,000 times, then use L-BFGS optimizer [43] to train. Adam is
an algorithm for optimizing stochastic objective functions based on first-
order gradients. The main advantages of the Adam method are: simple and easy
adaptive estimation based on low order moments on the gradient;
computationally efficient and low memory requirements well suited for problems
with large data and/or parameters and suitable for non-smooth objectives and
problems with very noisy and/or sparse gradients [42]. Figure 6(a) shows the
loss function curve using Adam’s algorithm trained 10,000 iterations and
Figure 6(b) represents the loss function descent curve using L-BFGS’s
algorithm trained 9423 iterations. More specially, initial condition error is
$loss_{0}=1.401873\times 10^{-5}$; boundary condition error is
$loss_{b}=2.5388058\times 10^{-6}$; the collocation points error is
$loss_{f}=5.429575\times 10^{-5}$ and overall error is $Loss=7.085329\times
10^{-5}$. Figure 6(c) and (d) represent the loss function curves with RAR-PINN
using Adam optimizer and L-BFGS optimizer, respectively. After training,
initial condition error is $loss_{0}=6.1637432\times 10^{-6}$; boundary
condition error is $loss_{b}=1.3867134\times 10^{-6}$; the collocation points
error is $loss_{f}=2.861435\times 10^{-5}$ and overall error is
$Loss=3.616529\times 10^{-5}$.

(a) (b)
(c) (d)

Figure 6: The loss function curve figures of vector two-soliton solutions
$h_{1}(x,t)$ and $h_{2}(x,t)$ arising from (a) TPINN algorithm with the 10000
iterations Adam optimization; (b) TPINN algorithm with the 9423 iterations
L-BFGS optimization; (c) RAR-PINN algorithm with the 10000 iterations Adam
optimization; (d) RAR-PINN algorithm with the 13272 iterations L-BFGS
optimization.

The three-dimensional diagram of the residual error is shown in Figure 7.
Figure 7(a) is three-dimensional stereogram residual error without adding
adaptive points, from which we can see that the residuals of some positions
before the adaptive pointing are relatively large up to 2.270. Figure 7(b)
denotes the denotes the three-dimensional stereogram residual error after
adding adaptive points, which indicates that the residuals of all positions
after the adaptive pointing can be controlled within 1.30. From these
experimental results, we can find that the RAR-PINN method is more efficient
than TPINN for solving the elastic collision of vector two-soliton solutions.

(a) (b)

Figure 7: Three-dimensional stereogram of residual error: (a) three-
dimensional stereogram residual error without adding adaptive points; (b)
three-dimensional stereogram residual error after adding adaptive points.

####  3.2.2 Inelastic collision of vector two-soliton solution

In this section, by choosing the parameters $k_{1}=1+i$, $k_{2}=2-i$,
$\xi_{1}^{(1)}=\xi_{1}^{(2)}=\xi_{2}^{(2)}=1$,
$\xi_{2}^{(1)}=\dfrac{39+80i}{89}$, $\alpha=\beta=2$ and $\gamma=0.5+0.5i$, we
get the soliton collision behaviors. We take $[-L,L]$ and $[-T,T]$ in System
(1) as $[-4,4]$ and $[-0.3,0.3]$, respectively. The initial values can be
computed by Eq. (13) at $t=-0.3$ and the boundary values are periodic boundary
condition. We divide the domain into $[400\times 301]$ data points. The vector
two-soliton solutions $h_{1}$ and $h_{2}$ are discretized into 301 snapshots.
We predict the data-driven solution based on the neural networks architecture
with 6 layers and 32 neurons per hidden layer. The training set of the TPINN
model consists of initial training points $N_{0}=150$, boundary training
points $N_{b}=150$ and collocation points $N_{f}=15040$. As for the RAR-PINN
model, we first choose $N_{0}=150$, $N_{b}=150$ and $N_{f}=15000$, then add
100 more collocation points adaptivly. In particular, we choose the threshold
$\varepsilon_{0}=0.13$ and the number of points added each time $m=10$.

(a) Exact solution $|h_{1}(x,t)|$ (b) Exact solution $|h_{2}(x,t)|$
(c) Tractional PINN solution $|h_{1}(x,t)|$ (d) Tractional PINN solution
$|h_{2}(x,t)|$
(e) RAR-PINN solution $|h_{1}(x,t)|$ (f) RAR-PINN solution $|h_{2}(x,t)|$

Figure 8: The shape-changing vector two-soliton solutions $h_{1}(x,t)$ and
$h_{2}(x,t)$: Comparison among the exact solutions (panels (a) and (b)),
predictive solutions via TPINN (panels (c) and (d)) and predictive results by
RAR-PINN (panels (e) and (f)) at different propagation times $t=-0.1,0.0$ and
0.1. The left and right graphs represent the density diagrams and the profiles
at different times for the shape-changing vector two-soliton solutions
$h_{1}(x,t)$ and $h_{2}(x,t)$, respectively.

(a) (b)
(c) (d)

Figure 9: The loss function curve figures of shape-changing vector two-soliton
solutions $h_{1}(x,t)$ and $h_{2}(x,t)$ arising from (a) TPINN algorithm with
the 20000 iterations Adam optimization; (b) TPINN algorithm with the 11438
iterations L-BFGS optimization; similarly, (c)RAR-PINN algorithm with the
20000 iterations Adam optimization; (d) RAR-PINN algorithm with the 7117
iterations L-BFGS optimization.

(a) (b)

Figure 10: Three-dimensional stereogram of residual error: (a) three-
dimensional stereogram residual error without adding adaptive points; (b)
three-dimensional stereogram residual error after adding adaptive points.

Figure 8 demonstrates the inelastic collision of vector two-soliton solution
predicted by the TPINN model and the RAR-PINN model, respectively. The
relative $\mathbb{L}_{2}$ errors of the shape-changing vector two-soliton
solutions $h_{1}(x,t)$ and $h_{2}(x,t)$ via the TPINN model are
$1.196291\times 10^{-1}$ and $1.205316\times 10^{-1}$, respectively; while the
relative $\mathbb{L}_{2}$ errors via the RAR-PINN model are $8.64846\times
10^{-2}$ for $h_{1}(x,t)$ and $8.71967\times 10^{-2}$ for $h_{2}(x,t)$. Adam
optimizer and L-BFGS optimizer are used to minimize the loss function. Figure
9(a) demonstrates the loss function descent curve of the TPINN model with the
Adam optimizer for 20,000 iterations. Figure 9(b) represents the loss function
curve of the TPINN model with the L-BFGS optimizer for 11438 iterations. We
can find the initial error $loss_{0}=5.6866372\times 10^{-5}$, the boundary
error $loss_{b}=9.3172368\times 10^{-6}$, the residual error
$loss_{f}=2.0258679\times 10^{-5}$ and the total error $Loss=8.644289\times
10^{-5}$. Figure 9(c) indicates the loss function curve of the RAR-PINN model
via the Adam optimizer for 20,000 iterations. Figure 9(d) illustrates the loss
function curve of the RAR-PINN model with the L-BFGS optimizer for 7117
iterations. Furthermore, we can observe the initial error
$loss_{0}=2.923557\times 10^{-5}$, the boundary error $loss_{b}=4.950383\times
10^{-6}$, the residual $loss_{f}=6.765904\times 10^{-6}$ and the total error
$Loss=4.095181\times 10^{-5}$. On the whole, the RAR-PINN model is more
efficient than the TPINN model in both relative and absolute errors. Figure
10(a) and (b) demonstrate the residual errors of the three-dimensional
stereogram without the addition of adaptive points and the residual errors of
the three-dimensional stereogram after the addition of adaptive points,
respectively. Through the above comparison, we can see that the RAR-PINN model
is more efficient for the shape-changing soliton collisions of Eq. (1).

###  3.3 Vector three-soliton solutions

The exact vector three-soliton solution of the CGNLS equation has been studied
by various methods, that is given as[44]

$\displaystyle h_{1}=$
$\displaystyle\frac{\xi_{1}^{(1)}e^{\eta_{1}}+\xi_{2}^{(1)}e^{\eta_{2}}+\xi_{3}^{(1)}e^{\eta_{3}}+e^{\eta_{1}+\eta_{1}^{*}+\eta_{2}+\delta_{11}}+e^{\eta_{1}+\eta_{1}^{*}+\eta_{3}+\delta_{21}}+e^{\eta_{1}+\eta_{2}^{*}+\eta_{2}+\delta_{21}}}{D}$
(18)
$\displaystyle+\frac{e^{\eta_{2}+\eta_{2}^{*}+\eta_{3}+\delta_{41}}+e^{\eta_{3}+\eta_{3}^{*}+\eta_{1}+\delta_{51}}+e^{\eta_{3}+\eta_{3}^{*}+\eta_{2}+\delta_{61}}+e^{\eta_{1}^{*}+\eta_{2}+\eta_{3}+\delta_{71}+}e^{\eta_{1}+\eta_{2}^{*}+\eta_{3}+\delta_{81}}}{D}$
$\displaystyle+\frac{e^{\eta_{1}+\eta_{2}+\eta_{3}^{*}+\delta_{91}}+e^{\eta_{1}+\eta_{1}^{*}+\eta_{2}+\eta_{2}^{*}+\eta_{3}+\tau_{11}}+e^{\eta_{1}+\eta_{1}^{*}+\eta_{3}+\eta_{3}^{*}+\eta_{2}+\tau_{21}}+e^{\eta_{2}+\eta_{2}^{*}+\eta_{3}+\eta_{3}^{*}+\eta_{1}+\tau_{31}}}{D},$
$\displaystyle h_{2}=$
$\displaystyle\frac{\xi_{1}^{(2)}e^{\eta_{1}}+\xi_{2}^{(2)}e^{\eta_{2}}+\xi_{3}^{(2)}e^{\eta_{3}}+e^{\eta_{1}+\eta_{1}^{*}+\eta_{2}+\delta_{12}}+e^{\eta_{1}+\eta_{1}^{*}+\eta_{3}+\delta_{22}}+e^{\eta_{1}+\eta_{2}^{*}+\eta_{2}+\delta_{32}}}{D}$
$\displaystyle+\frac{e^{\eta_{2}+\eta_{2}^{*}+\eta_{3}+\delta_{42}}+e^{\eta_{3}+\eta_{3}^{*}+\eta_{1}+\delta_{52}}+e^{\eta_{3}+\eta_{3}^{*}+\eta_{2}+\delta_{62}}+e^{\eta_{1}^{*}+\eta_{2}+\eta_{3}+\delta_{72}+}e^{\eta_{1}+\eta_{2}^{*}+\eta_{3}+\delta_{82}}}{D}$
$\displaystyle+\frac{e^{\eta_{1}+\eta_{2}+\eta_{3}^{*}+\delta_{92}}+e^{\eta_{1}+\eta_{1}^{*}+\eta_{2}+\eta_{2}^{*}+\eta_{3}+\tau_{12}}+e^{\eta_{1}+\eta_{1}^{*}+\eta_{3}+\eta_{3}^{*}+\eta_{2}+\tau_{22}}+e^{\eta_{2}+\eta_{2}^{*}+\eta_{3}+\eta_{3}^{*}+\eta_{1}+\tau_{32}}}{D},$

where

$\begin{split}\begin{aligned}
D=&1+e^{\eta_{1}+\eta_{1}^{*}+R_{1}}+e^{\eta_{2}+\eta_{2}^{*}+R_{2}}+e^{\eta_{3}+\eta_{3}^{*}+R_{3}}+e^{\eta_{1}+\eta_{2}^{*}+\delta_{10}}+e^{\eta_{1}^{*}+\eta_{2}+\delta_{10}^{*}}\\\
&+e^{\eta_{1}+\eta_{3}^{*}+\delta_{20}}+e^{\eta_{1}^{*}+\eta_{3}+\delta_{20}^{*}}+e^{\eta_{2}+\eta_{3}^{*}+\delta_{30}}+e^{\eta_{2}^{*}+\eta_{3}+\delta_{30}^{*}}+e^{\eta_{1}+\eta_{1}^{*}+\eta_{2}+\eta_{2}^{*}+R_{4}}\\\
&+e^{\eta_{1}+\eta_{1}^{*}+\eta_{3}+\eta_{3}^{*}+R_{5}}+e^{\eta_{2}+\eta_{2}^{*}+\eta_{3}+\eta_{3}^{*}+R_{6}}+e^{\eta_{1}+\eta_{1}^{*}+\eta_{2}+\eta_{3}^{*}+\tau_{10}}+e^{\eta_{1}+\eta_{1}^{*}+\eta_{3}+\eta_{2}^{*}+\tau_{10}^{*}}\\\
&+e^{\eta_{1}+\eta_{1}^{*}+\eta_{1}+\eta_{1}^{*}+\eta_{3}+\eta_{3}^{*}+R_{7}},\\\
\eta_{l}=&k_{l}(x+ik_{l}t),l=1,2,3.\end{aligned}\end{split}$ (19)

The expressions for the other parameters of the vector three-soliton solutions
have been given in detail in the Ref.[44]. We make choose the vector three-
soliton solutions $h_{1}(x,t)$ and $h_{2}(x,t)$ with the parameter choices
$k_{1}=1+i$, $k_{2}=2-i$,
$\xi_{1}^{(1)}=\xi_{1}^{(2)}=\xi_{2}^{(1)}=\xi_{2}^{(2)}=1$, $\alpha=\beta=2$
and $\gamma=0.5+0.5i$. Besides, we take $[-L,L]$ and $[-T,T]$ in Eq. (1) as
$[-6,6]$ and $[-0.8,0.8]$, respectively. The corresponding initial condition
is driven as $h_{10}=h_{1}(x,-1)$ and $h_{20}=h_{2}(x,-1)$. Similarly, here we
still use periodic boundary condition to numerically simulate the vector
three-soliton solutions of Eq. (1).The computation domain
$[-6,6]\times[-0.8,0.8]$ is divided into $[400\times 301]$ data points, three-
soliton solutions $h_{1}$ and $h_{2}$ are discretized into 301 snapshots on
regular space-time grid with $\bigtriangleup t=0.005$. A training data set
containing initial data and boundary data are generated by randomly sampling
[45]. Specifically, the number of boundary points is $N_{b}=100$ and the
number of initial points is $N_{0}=120$. Firstly, we choose 6000 points
selected as the residual points and then add 18 more new points adaptively via
RAR with $m=3$ and $\varepsilon_{0}=0.07$. In order to compare these two
algorithms, we select $N_{0}=120$, $N_{b}=100$ and $N_{f}=6018$ to obtain the
vector three-soliton solutions using the TPINN algorithm. The predicted vector
three-soliton solutions $h_{1}(x,t)$ and $h_{2}(x,t)$ can be obtained by
minimizing the loss function.

(a) Exact solution $|h_{1}(x,t)|$ (b) Exact solution $|h_{2}(x,t)|$
(c) Tractional PINN solution $|h_{1}(x,t)|$ (d) Tractional PINN solution
$|h_{2}(x,t)|$
(e) RAR-PINN solution $|h_{1}(x,t)|$ (f) RAR-PINN solution $|h_{2}(x,t)|$

Figure 11: The vector three-soliton solution $h_{1}(x,t)$ and $h_{2}(x,t)$:
Comparison among the exact solutions (panels (a) and (b)), predictive
solutions via TPINN (panels (c) and (d)) and predictive results by RAR-PINN
(panels (e) and (f)) at different propagation times $t=-0.53,0.0$ and 0.53.
The left and right graphs represent the density diagrams and the profiles at
different times for the vector three-soliton solutions $h_{1}(x,t)$ and
$h_{2}(x,t)$, respectively.

Figure 11 shows the vector three-soliton solution given by the exact
expression, TPINN algorithm and RAR-PINN algorithm, respectively. The relative
$\mathbb{L}_{2}$ errors of the vector three-soliton solutions via the TPINN
algorithm are $8.505647\times 10^{-2}$ of $h_{1}(x,t)$ and $8.591699\times
10^{-2}$ of $h_{2}(x,t)$, respectively; while the relative $\mathbb{L}_{2}$
errors via the RAR-PINN algorithm are $6.3181865\times 10^{-2}$ of
$h_{1}(x,t)$ and $6.3181865\times 10^{-2}$ of $h_{2}(x,t)$. Here, we use Adam
optimizer and L-BFGS optimizer to minimize the loss function. Figure 12(a)
demonstrates the loss function descent curve of the TPINN model with the Adam
optimizer for 20,000 iterations. Figure 12(b) indicates the loss function
curve of the TPINN model with the L-BFGS optimizer for 9893 iterations.
Besides, we can find the initial error $loss_{0}=1.9974363\times 10^{-5}$, the
boundary error $loss_{b}=5.9046424\times 10^{-6}$, the residual
$loss_{f}=5.5316887\times 10^{-5}$, the total error $Loss=8.119589\times
10^{-5}$. Figure 12(c) demonstrates the loss function curve of the RAR-PINN
model via the Adam optimizer for 20,000 iterations. Figure 12(d) illustrates
the loss function curve of the RAR-PINN model with the L-BFGS optimizer for
11836 iterations. Furthermore, we can observe the initial error
$loss_{0}=1.415507\times 10^{-5}$, the boundary error $loss_{b}=3.168202\times
10^{-6}$, the residual $loss_{f}=4.163423\times 10^{-5}$, and the total error
$Loss=5.895750\times 10^{-5}$. On the whole, the RAR-PINN model is more
efficient than the TPINN model in both relative and absolute errors.

(a) (b)
(c) (d)

Figure 12: The loss function curve figures of vector three-soliton solutions
$h_{1}(x,t)$ and $h_{2}(x,t)$ arising from (a) TPINN algorithm with the 20000
iterations Adam optimization; (b) TPINN algorithm with the 9893 iterations
L-BFGS optimization; similarly, (c) RAR-PINN algorithm with the 20000
iterations Adam optimization; (d) RAR-PINN algorithm with the 11836 iterations
L-BFGS optimization.

In order to further illustrate the effectiveness of RAR-PINN, three-
dimensional diagram of the residual error are shown. Figure 13 (a) and (b)
show the residual errors of the three-dimensional stereogram before and after
adding the adaptive points, respectively. By comparing Figure 13 (a) and
Figure 13 (b), we can see that the residual can reach 0.85 before adding the
adaptive points and can be reduced to less than 0.45 after adding the adaptive
points. Through the comparison of the above experimental results, the RAR-PINN
model is more efficient for solving vector three-soliton solutions of Eq. (1).

(a) (b)

Figure 13: Three-dimensional stereogram of residual error: (a) three-
dimensional stereogram residual error without adding adaptive points; (b)
three-dimensional stereogram residual error after adding adaptive points.

##  4 Data-driven parameters discovery of the CGNLS equation

The deep learning method can tackle the data-driven parameters discovery of
the $N$-coupled nonlinear equation. For convenience, we use the CGNLS equation
as an example to illustrate. In this part, there are some unknown parameters
in System (1), but some extra information is known on these points besides the
CGNLS equation. We will consider the problem of data-driven discovery of PDEs
[46, 47, 48]. The CGNLS equation with unknown parameters is as follows

$\left\\{\begin{aligned}
&ih_{1t}+\lambda_{1}h_{1xx}+\lambda_{2}(\alpha|h_{1}|^{2}+\beta|h_{2}|^{2}+\gamma
h_{1}h_{2}^{*}+\gamma^{*}h_{2}h_{1}^{*})h_{1}=0,\\\
&ih_{2t}+\lambda_{3}h_{2xx}+\lambda_{4}(\alpha|h_{1}|^{2}+\beta|h_{2}|^{2}+\gamma
h_{1}h_{2}^{*}+\gamma^{*}h_{2}h_{1}^{*})h_{2}=0,\end{aligned}\right.$ (20)

where $h_{k}=u_{k}+v_{k}i$ ($k=1,2$; $u_{k}$ and $v_{k}$ are the real and
imaginary parts, respectively) is a complex field. Besides, coefficients
$\lambda_{1},\lambda_{2},\lambda_{3}$ and $\lambda_{4}$ are unknown parameters
that need to be trained. With the parameter choices $\alpha=\beta=\gamma=1$,
let us consider the physical model given explicitly by

$\left\\{\begin{aligned}
&f_{1}:=ih_{1t}+\lambda_{1}h_{1xx}+\lambda_{2}(|h_{1}|^{2}+|h_{2}|^{2}+h_{1}h_{2}^{*})h_{1},\\\
&f_{2}:=ih_{2t}+\lambda_{3}h_{2xx}+\lambda_{4}(|h_{1}|^{2}+|h_{2}|^{2}+h_{1}h_{2}^{*})h_{2},\end{aligned}\right.$
(21)

which contain real and imaginary parts, thus

$\left\\{\begin{aligned}
&f_{1u}:=v_{1t}-\lambda_{1}u_{1xx}-\lambda_{2}(u_{1}^{2}+v_{1}^{2}+u_{2}^{2}+v_{2}^{2}+2(u_{1}u_{2}+v_{1}v_{2}))u_{1},\\\
&f_{1v}:=u_{1t}+\lambda_{1}v_{1xx}+\lambda_{2}(u_{1}^{2}+v_{1}^{2}+u_{2}^{2}+v_{2}^{2}+2(u_{1}u_{2}+v_{1}v_{2}))v_{1},\\\
&f_{2u}:=v_{2t}-\lambda_{3}u_{2xx}-\lambda_{4}(u_{1}^{2}+v_{1}^{2}+u_{2}^{2}+v_{2}^{2}+2(u_{1}u_{2}+v_{1}v_{2}))u_{2},\\\
&f_{2v}:=u_{2t}+\lambda_{3}v_{2xx}+\lambda_{4}(u_{1}^{2}+v_{1}^{2}+u_{2}^{2}+v_{2}^{2}+2(u_{1}u_{2}+v_{1}v_{2}))v_{2}.\end{aligned}\right.$
(22)

We determine the values of the unknown parameters
$\lambda_{1},\lambda_{2},\lambda_{3}$ and $\lambda_{4}$ by minimizing the MSE
loss function, which is defined as follows

$MSE=MSE_{p}+MSE_{f},$ (23)

where

$\begin{split}\begin{aligned}
MSE_{p}=&\frac{1}{N_{u}}\sum_{i=1}^{N_{u}}({|\hat{u}_{1}}(x^{i},t^{i})-u_{1}^{i}|^{2}+|\hat{v}_{1}(x^{i},t^{i})-v_{1}^{i}|^{2}\\\
&+|\hat{u}_{2}(x^{i},t^{i})-u_{2}^{i}|^{2}+|\hat{v}_{2}(x^{i},t^{i})-v_{2}^{i}|^{2}),\\\
MSE_{f}=&\frac{1}{N_{u}}\sum_{i=1}^{N_{u}}(|f_{1u}(x^{i},t^{i})|^{2}+|f_{1v}(x^{i},t^{i})|^{2}+|f_{2u}(x^{i},t^{i})|^{2}+|f_{2v}(x^{i},t^{i})|^{2}).\end{aligned}\end{split}$
(24)

The vector-soliton solutions are used as the data set. Using the pseudo-
spectral method, the solutions of the Eq. (21) are discretized into
$[300\times 201]$ data points. The exact solutions for the Eq. (21) with
parameters $\lambda_{1}=1,\lambda_{2}=2,\lambda_{3}=1$ and $\lambda_{4}=2$ are
considered to make sampling data set with space-time region
$(x,t)\in[-10,10]\times[-2,2]$.

To illustrate the effectiveness of our algorithm, we have created a training
data-set by randomly generating $N_{u}=5,000$ points across the entire domain
from the exact solution corresponding to
$\lambda_{1}=1,\lambda_{2}=2,\lambda_{3}=1$ and $\lambda_{4}=2$. During the
training process, the RAR-PINN algorithm is used in order to add more points
where the error is large. We add 10 more new points adaptively via RAR with
$m=5$ and $\varepsilon_{0}=0.15$, and then the total number of data points is
$N=5,010$. Table 1 indicates that the RAR-PINN algorithm is able to correctly
identify the unknown parameters with very high accuracy even when the training
data are corrupted with noise. Specifically, The predictions remain robust
even when the training data are corrupted with $1\%$ and $3\%$ uncorrelated
Gaussian noise.

Table 1: Identified obtained by learning $\lambda_{1},\lambda_{2},\lambda_{3}$ and $\lambda_{4}$ Correct CGNLS equation | $ih_{1t}+h_{1xx}+2(|h_{1}|^{2}+|h_{2}|^{2}+h_{1}h_{2}^{*})h_{1}=0$
---|---
$ih_{2t}+h_{2xx}+2(|h_{1}|^{2}+|h_{2}|^{2}+h_{1}h_{2}^{*})h_{2}=0$
Identified CGNLS equation (clean data) | $ih_{1t}+1.00012h_{1xx}+2.00270(|h_{1}|^{2}+|h_{2}|^{2}+h_{1}h_{2}^{*})h_{1}=0$
$ih_{2t}+1.00014h_{2xx}+2.00359(|h_{1}|^{2}+|h_{2}|^{2}+h_{1}h_{2}^{*})h_{2}=0$
Identified CGNLS equation ( $1\%$ noise) | $ih_{1t}+0.99912h_{1xx}+2.00500(|h_{1}|^{2}+|h_{2}|^{2}+h_{1}h_{2}^{*})h_{1}=0$
$ih_{2t}+1.00057h_{2xx}+2.00436(|h_{1}|^{2}+|h_{2}|^{2}+h_{1}h_{2}^{*})h_{2}=0$
Identified CGNLS equation ( $3\%$ noise) | $ih_{1t}+1.01020h_{1xx}+2.01574(|h_{1}|^{2}+|h_{2}|^{2}+h_{1}h_{2}^{*})h_{1}=0$
$ih_{2t}+0.99996h_{2xx}+2.01110(|h_{1}|^{2}+|h_{2}|^{2}+h_{1}h_{2}^{*})h_{2}=0$

To further check the stability of our algorithm, we conduct a series of
experiments with different amounts of training data and noise corruption
levels. The results are summarized in Tables 2 and 3. The key point here is
that this method seems to be very robust to the level of noise in the data,
producing reasonable recognition accuracy even for noise corruptions of up to
$12\%$. We also observe from Table 2 that the error decreases as the total
number of training points increases, while the error is non-monotonic along
with the change of the noise levels as seen in Table 3. Different factors
related to the equations themselves as well as the particular neural network
setup may be responsible for this variation.

Table 2: Error in the identified parameters $\lambda_{1},\lambda_{2},\lambda_{3}$, and $\lambda_{4}$ for different number of training data Parameters $N$ | 1015 | 2010 | 3008 | 4009 | 5010
---|---|---|---|---|---
$\lambda_{1}$ | 1.143306 | 0.0043837 | 0.0021511 | 0.0016654 | 0.0001152
$\lambda_{2}$ | 0.9430671 | 0.0022939 | 0.0007388 | 0.0038375 | 0.0013522
$\lambda_{3}$ | 1.0143343 | 0.0017641 | 0.0016935 | 0.0016654 | 0.0001363
$\lambda_{4}$ | 0.9438857 | 0.0017554 | 0.0001838 | 0.0038375 | 0.0017950
Table 3: Error in the identified parameters $\lambda_{1},\lambda_{2},\lambda_{3}$, and $\lambda_{4}$ by different noise levels Parameters Noise | $1\%$ | $3\%$ | $6\%$ | $9\%$ | $12\%$
---|---|---|---|---|---
$\lambda_{1}$ | 0.0008779 | 0.0101990 | 0.0059060 | 0.0009057 | 0.0073389
$\lambda_{2}$ | 0.0025021 | 0.0078678 | 0.0042040 | 0.0010816 | 0.0051344
$\lambda_{3}$ | 0.0005662 | 0.0000419 | 0.0024133 | 0.0015854 | 0.0059356
$\lambda_{4}$ | 0.0021807 | 0.0055518 | 0.0020984 | 0.0016636 | 0.0051588

##  5 Discussion and conclusion

In conclusion, we have investigated the CGNLS equation via the RAR-PINN
method. By improving the distribution of residual points during the training
process, the RAR-PINN method increases the training efficiency. We have used
RAR-PINN and TPINN methods to predict the vector one-, two-, and three-soliton
solutions of the CGNLS equation, respectively, and then compared the
experiment results. It has been revealed that the RAR-PINN method can present
higher accuracy for the same number of training points. In particular, the
RAR-PINN method still achieves good results for shape-changing soliton
solutions of the CGNLS equation. After a series of experiments, we can
conclude that when solving the coupled nonlinear equations, the RAR-PINN
method can well simulate vector-soliton interactions for both elastic and
inelastic collisions. In addition, the data-driven parameters discovery of the
CGNLS equation has been performed. We have added the noise to judge the
stability of the neural networks. The prediction error increases as the number
of sampling points decreases. These changes of error are within the
controllable range, which is enough to prove the excellent performance of the
RAR-PINN method. Such method can be applied into other coupled nonlinear
systems to discover the dynamic properties of nonlinear waves.

## Acknowledgments

This work was partially supported by the Natural Science Foundation of Beijing
Municipality (Grant No. 1212007), by the National Natural Science Foundation
of China (Grant No. 11705284), by the Fundamental Research Funds of the
Central Universities (Grant No. 2020MS043), and by the Science Foundations of
China University of Petroleum, Beijing (Grant Nos. 2462020YXZZ004 and
2462020XKJS02).

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11institutetext: Name of First Author 22institutetext: Name, Address of
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44institutetext: Name, Address of Institute 44email<EMAIL_ADDRESS>
# Contribution Title

Name of First Author and Name of Second Author

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Livelihood and survival mobility are oftentimes coutcomes of uneven
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Classes |  Subclass |  Length |  Action Mechanism
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Translation |  mRNAa |  22 (19–25) |  Translation repression, mRNA cleavage
Translation |  mRNA cleavage |  21 |  mRNA cleavage
Translation |  mRNA |  21–22 |  mRNA cleavage
Translation |  mRNA |  24–26 |  Histone and DNA Modification

a Table foot note (with superscript)

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$a\times b=c$ (4)