DATA-DRIVEN FORWARD-INVERSE PROBLEMS FOR THE VARIABLE COEFFICIENTS HIROTA EQUATION USING DEEP LEARNING METHOD HUIJUAN ZHOU JUNCAI PU AND YONG CHEN

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DATA-DRIVEN FORWARD-INVERSE PROBLEMS FOR THE VARIABLE COEFFICIENTS

HIROTA EQUATION USING DEEP LEARNING METHOD

HUIJUAN ZHOU, JUNCAI PU, AND YONG CHEN

Abstract. Data-driven forward-inverse problems for the variable coeﬃcients Hirota (VC-Hirota) equation are discussed

in this paper. An improved physics-informed neural networks (IPINN) algorithm is used to recover the data-driven

solitons, high-order soliton, as well as the data-driven parameters discovery for the VC-Hirota equation. We propose

a PINN algorithm with sub-neural networks to learn the data-driven functions discovery of VC-Hirota equation with

unknown functions under noise of diﬀerent intensity. Numerical results are shown to demonstrate the facts: (i) data-

driven soliton solutions and parameters discovery of the VC-Hirota equation are successfully learned by adjusting

the network layers, neurons, the original training data, spatiotemporal regions and other parameters of the IPINN

algorithm; (ii) the data-driven functions discovery of VC-Hirota equation can be trained stably and accurately via the

PINN algorithm with sub-neural networks. The results achieved in this work verify that the forward-inverse problems

including the data-driven function discovery of the variable coeﬃcients equation can be solved based on deep learning

method.

1. Introduction

As an important branch of nonlinear systems, solitons have been widely studied in many ﬁelds, such as plasma

physics, ﬂuid mechanics, optical ﬁber communication, condensed matter physics and so on [1–6]. In the ﬁeld of optical

ﬁber communication, it is well known that a modiﬁed NLS equation, which is called Hirota equation, can be used

to describe the propagation of subpicosecond or femtosecond optical pulse in ﬁbers thanks to it taking into account

higher-order dispersion and time-delay corrections to the cubic nonlinearity. Though there are extensive studies about

Hirota equation and some types of exact soliton solutions in optical have been obtained, it is worth noting that these

investigations of optical solitons have revolved mainly around homogeneous ﬁbers. However, considering long-distance

communication and manufacturing problems in realistic ﬁber transmission lines, the inhomogeneous variable coeﬃcient

model needs to be considered.

Taking into account the higher-order eﬀects inﬂuenced by spatial variations of the ﬁber parameters, the inhomo-

geneous variable coeﬃcient Hirota (VC-Hirota) equation which can be used to describe the certain ultrashort optical

pulses propagating in a nonlinear inhomogeneous ﬁber is as follows [7].

iqz+α1(z)qtt −1

3δiα1(z)qttt +δα2(z)q|q|2−iα2(z)|q|2qt−iα3(z)q= 0,(1.1)

where α3(z) = α1,z α2−α1α2,z

2α1α2,q=q(t, z) is complex-valued solution about the space tand time z,δis a real number,

α1(z) and α2(z) are dispersion and nonlinear eﬀects, respectively. In order to understand the nonlinear phenomena

of optical pulse propagation in inhomogeneous ﬁber media, it is necessary to analyze the analytical and numerical

solutions of the VC-Hirota model. Due to the important application of the VC-Hirota model, there has been much

research on this model, such as the exact bright and dark solitary wave solutions near the zero dispersion point is

derived in [8]. Three combined solitary wave solutions are given in the same expression, and the properties of bright

and dark solitary waves are described respectively. Furthermore, the features of the solutions are analyzed, and

numerically discuss the stability of these solitary waves under slight violations of the parameter conditions and ﬁnite

initial perturbations [9]. It has also been extensively studied by many other authors and some types of localized waves

solution have been obtained [10–15]. Recently, we derive the exact form of N-soliton and high-order soliton solutions

for the VC-Hirota equation by utilizing the Riemann-Hilbert approach [16].

In the past few decades, many methods and techniques have been presented to solve the nonlinear evolution

equations, including Hirota bilinear method, Darboux transformation, Riemann-Hilbert method, physics-informed

neural networks (PINN) deep learning method, etc. In scientiﬁc computing, the neural network (NN) method provides

an ideal representation for the solution of diﬀerential equations due to its universal approximation properties. Recently,

a PINN method which is controlled by mathematical physical systems based on the multi-layer NNs has been proposed

and proved to be particularly suitable for dealing with both the forward problems and highly ill-posed inverse problems

by obtaining the approximate solutions of governing equations and discovering parameters involved in the governing

equation are inferred from the training data. Numerical results show that the PINN architecture is able to obtain

remarkably accurate solutions with remarkably little data [17]. At the same time, this method also provides a better

physical explanation for predicted solutions because of the underlying physical constraints, which are usually explicitly

described by diﬀerential equations. Later on, using the PINN method to obtain data-driven solutions, parameters

discovery and reveal the dynamic behavior of nonlinear partial diﬀerential equations with physical constraints has

attracted extensive attention and raised a hot wave of research. Recently, PINN has played an important role in many

arXiv:2210.09656v2 [nlin.PS] 22 Dec 2022

2 HUIJUAN ZHOU, JUNCAI PU, AND YONG CHEN

physical applications [18]. Afterwards, global adaptive activation functions and locally adaptive activation functions

are proposed to approximate smooth and discontinuous functions as well as solutions of linear and nonlinear partial

diﬀerential equations by introducing a scalable parameters in the activation function and adding a slope recovery term

based on activation slope to the loss function of locally adaptive activation functions. It demonstrates the locally

adaptive activation functions further improve the training speed, performance and speed up the training process of

NNs. [19, 20].

Due to the good properties of integrable systems, it is possible to solve the data-drive forward and inverse problems

for abundant classical integrable nonlinear evolution equations via the PINN method. Since Chen’s team ﬁrst proposed

the concept of integrable deep learning in 2020, a large number related literature has been published. In particular, Li

and Chen construct abundant numerical solutions of second-order and third-order nonlinear integrable equations with

diﬀerent initial and boundary conditions by deep learning method based on the PINN model [21, 22]. Pu, Peng et al.

recover the solitons, breathers, rogue wave solutions and rogue wave on the periodic background of the nonlinear partial

diﬀerential equation with the aid of the PINN model [23,24]. Miao and Chen use the PINN method to high-dimensional

system to solve the (N+1)-dimensional initial boundary value problem with 2N+1 hyperplane boundaries [25]. Lin

and Chen devise a two-stage PINN method which is tailored to the nature of equations by introducing features of

physical systems into NNs based on conserved quantities [26]. In addition, many important work on data-driven

solutions and parameter discovery of diﬀerent nonlinear systems have been doned by other scholars [27–32].

In the previous literature, many soliton solutions of constant coeﬃcients integrable systems have been accurately

simulated numerically. The soliton solutions of variable coeﬃcients integrable systems studied in this paper are quite

diﬀerent from the soliton solutions of constant coeﬃcients equations. From the perspective of the solution dynamics,

the center trajectory equation of the one-soliton solution of the constant coeﬃcient equation is often a straight line,

while the one-soliton solution of the variable coeﬃcient equation presents a richer and more complex shape. For

example, in this paper, we learned the case of the center trajectory equations of the one-solitons are parabola, ”S-

shape” and cosine wave type. For the two-soliton solution of variable coeﬃcients equation, it can take on a more

complex shape. To our knowledge, there is rare study about the data-driven soliton solutions, parameter and function

discovery for the variable coeﬃcient equation by the IPINN method. To ﬁll this gap, the main purpose of this paper is

to recover various solitons, high-order soliton, as well as parameters and functions discovery for the variable coeﬃcient

equation when only know the initial-boundary value conditions. It is note that we use PINN algorithm with sub-neural

networks to achieve data-driven function discovery of variable coeﬃcient equations.

The rest of this paper is built up as follows. In Section 2, the IPINN method for the VC-Hirota equation is

introduced. To provide a direct and precise description of the IPINN method, the algorithm ﬂow schematic and

algorithm steps are given. Section 3 provides the data-driven multi-soliton and high-order soliton solutions of the VC-

Hirota equation using the IPINN approach, and related plots and dynamic analyses are revealed in detail. Section 4

presents experimental results of learning data-driven parameter discovery of VC-Hirota system. Furthermore, we learn

the unknown function of the VC-Hirota via a PINN algorithm with sub-neural networks successfully. The conclusions

and discussions are given in Section 5.

2. The improved physics-informed neural networks method for the variable coefficient Hirota

equation

An improved PINN (IPINN) approach with neuron-wise locally adaptive activation function was presented to derive

data-driven localized waves and learn unknown parameters of integrable systems in complex space, and numerical

results demonstrated the improved approach has faster convergence and better simulation eﬀect than the classical

PINN method [33,34]. In the following, we will introduce the IPINN method in detail and extend it to the application

of variable coeﬃcient Hirota equations.

Considering the (1+1)-dimensional nonlinear time-dependent systems with unknown constant parameter δand

variable coeﬃcients α1(z), α2(z) and α3(z) in complex space as below.

qz+N[q;δ, α1(z), α2(z), α3(z)] = 0,(2.1)

where N[·;δ, α1(z), α2(z), α3(z)] is nonlinear diﬀerential operators in space. In order to simplify the structure of the

complex-valued solutions q(t, z) in Eq. (2.1), we decompose q(t, z) into q(t, z) = u(t, z) + iv(t, z), where u(t, z) and

v(t, z) are real-valued functions. Then substituting q(t, z) = u(t, z) + iv(t, z) into Eq. (2.1), and letting the real and

imaginary parts be equal to 0, we have

uz+Nu[u, v;δ, α1(z), α2(z), α3(z)] = 0,

vz+Nv[u, v;δ, α1(z), α2(z), α3(z)] = 0,(2.2)

where the Nuand Nvare nonlinear diﬀerential operators in space. Deﬁning the physics-informed part fu(x, t) and

fv(x, t) as

fu:= uz+Nu[u, v;δ, α1(z), α2(z), α3(z)],

fv:= vz+Nv[u, v;δ, α1(z), α2(z), α3(z)],(2.3)

which play a role of regularization.

DATA-DRIVEN FORWARD-INVERSE PROBLEMS FOR THE VARIABLE COEFFICIENTS HIROTA EQUATION USING DEEP LEARNING METHOD3

The IPINN of depth Dcorresponding to the NN with an input layer, D−1 hidden-layers and an output layer.

Ndpresents the number of neurons in the dth hidden-layer, and each hidden-layer of the IPINN receives an output

xd−1∈RNd−1from the previous layer. Denote an aﬃne transformation as follows:

Ld(xd−1),Wdxd−1+bd,(2.4)

where the network weights Wd∈RNd×Nd−1and bias term bd∈RNdare associated with the dth layer. Then deﬁne

neuron-wise locally adaptive activation function as

σnad

iLdxd−1i, n > 1,···d= 1,2,··· , D −1, i = 1,2,··· , Nd,

where σis the activation function, nis a scaling factor and {ad

i}is additional

D−1

d=1

Ndparameter to be optimized.

When ngreater than or equal to critical scaling factor ncin each problem set, the optimization algorithm will become

sensitive. The neuron activation function acts as a vector activation function in each hidden layer, and each neuron

has its own activation function slope.

The IPINN method with neuron-wise locally adaptive activation function can be expressed as:

q(x;¯

Θ) = (LD)i0◦σ◦naD−1

i(LD−1)i◦ ··· ◦ σ◦na1

i(L1)i(x), i0= 1,2,(2.5)

where xrepresent the two inputs and q(x;¯

Θ) represent the two outputs in the IPINN. The trainable parameters

set ¯

Θ∈¯

Pconsists of Wd,bdD

d=1 and ad

iD−1

d=1 ,∀i= 1,2,··· , Nd, where the ¯

Pis a parameter space. In order to

minimize the loss function below certain tolerance εuntil a prescribed maximum number of iterations, seek a optimal

values of weights W, biases band scalable parameter ad

iis necessary. Here, we initialize the scalable parameters in

the case that nad

i= 1,∀n>1 (we ﬁxed n=10 in this paper).

Deﬁne the loss function as follows:

L(¯

Θ) = Loss =Lossu+Lossv+Lossfu+Lossfv+Lossa,(2.6)

where Lossu, Lossv, Lossfu,Lossfvand Lossaare deﬁned as below:

Lossu=1

j=1 ˆu(tj, zj)−uj2,

Lossv=1

j=1 ˆv(tj, zj)−vj2,

Lossfu=1

l=1 fu(tl

f, zl

f)2,

Lossfv=1

l=1 fv(tl

f, zl

f)2,

Lossa=Na

D−1

d=1

exp Nd

i=1

Nd!.

(2.7)

{tj, zj, uj, vj}Nq

j=1 denote the inputs data of initial-boundary value on Eqs. (2.2) and (2.3). ˆu(tj, zj) and ˆv(tj, zj)

represent the optimal training outputs data through the IPINN. Furthermore, {tl

f, zl

f}Nf

l=1 represent the collocation

points on networks fu(t, z) and fv(t, z). Nais the hyper-parameter for slope recovery term Lossa, and we take

Na=1

100 for dominating the loss function to ensure that the ﬁnal loss value is not too large in this paper. Lossu

and Lossvcorrespond to the loss on the initial and boundary data. Lossfuand Lossfvpenalize the collocation

points which not satisfy the VC-Hirota equation. The Lossachanges the topology of Loss function forces the NN

to increase the activation slope value quickly, which ensures the non-vanishing gradient of the loss function and

improves the convergence speed and network optimization ability. Therefore, the loss function is evaluated using the

contribution from the NN part as well as the residual from the governing equation given by the physics-informed part.

The resulting optimization algorithm will attempt to ﬁnd the optimized parameters including the weights, biases and

additional coeﬃcients in the activation to minimize the new loss function. In addition, the L2norm error is introduced

to measure the training error better. L2norm error is deﬁned as follows:

Error = sN

k=1 qexact(xk)−qpredict(xk;¯

Θ)2

k=1 qexact(xk)2

4 HUIJUAN ZHOU, JUNCAI PU, AND YONG CHEN

where qpredict(xk;¯

Θ) represent the model training prediction solution and qexact(xk) represent the exact analytical

solution at point xk= (tk, zk).

The physics-informed parts of the IPINN for VC-Hirota equation (1.1) can be deﬁned as:

fu:= −vz+α1(z)utt +1

3δα1(z)vttt +δα2(z)(u3+uv2) + α2(z)(u2vt+v2vt) + α3(z)v,

fv:= uz+α1(z)vtt −1

3δα1(z)uttt +δα2(z)(vu2+v3)−α2(z)(u2ut+v2ut)−α3(z)u.

(2.8)

The physics-informed parts of the IPINN for VC-Hirota equation (1.1) is integrable due to the equation (1.1) have

the lax pair as follows:

U=





λqδα2(z)

2α1(z)q(z, t)

−q∗(z, t)qδα2(z)

2α1(z)−λ



,(2.9)

V=A B

C−A,(2.10)

where

A=4α1(z)

3δλ3+ 2iα1(z)λ2+δα2(z)|q(z, t)|2

3δλ+δα2(z)(q∗(z, t)qt(z, t)−q(z, t)q∗

t(z, t))

6δ+iδα2(z)q(z, t)q∗(z, t)

B=sδα2(z)

2α1(z)(4α1(z)q(z, t)

3δλ2+2(α1(z)qt(z, t)

3δ+iα1(z)q(z, t))λ+iα1(z)qt(z, t)+ α1(z)

3δ(qtt(z, t) + δα2(z)|q(z, t)|2q(z, t)

α1(z)))

and

C=sδα2(z)

2α1(z)(−4α1(z)q∗(z, t)

3δλ2+2(α1(z)q∗

t(z, t)

3δ−iα1(z)q∗(z, t))λ+iα1(z)q∗

t(z, t)−α1(z)

3δ(q∗

tt(z, t)+δα2(z)|q(z, t)|2q∗(z, t)

α1(z))).

In order to describe the IPINN algorithm for the VC-Hirota equation more intuitively, we give the ﬂow chart of

the algorithm in Fig. 1. Since the complex function qis decomposed into u+iv, one can see that the “NN” part

has two output functions {u, v}, and there are two nonlinear equation constraints in the physics-informed part. What

need to note is that in the physics-informed part, we introduce the constant parameter δand variable coeﬃcients

α1(z), α2(z), α3(z) into the governing equation (2.3). Note f(z)=α1(z), α2(z), α3(z) and α3(z) = α1,z α2−α1α2,z

2α1α2is a

necessary constraint for the integrability of the equation (2.3). In addition, we also show the corresponding procedure

steps of the IPINN algorithm for the VC-Hirota equation in Tab.1.

Figure 1. (Color online) Schematic of IPINN for the VC-Hirota equation. The left NN is the

universal approximation network while the right one induced by the governing equation is the physics-

informed network. The two NNs share hyper-parameters and they both contribute to the loss function.

DATA-DRIVEN FORWARD-INVERSE PROBLEMS FOR THE VARIABLE COEFFICIENTS HIROTA EQUATION USING DEEP LEARNING METHOD5

Table 1. IPINN algorithm of the VC-Hirota equation.

Step one: Introduce constant parameter δand variable coeﬃcients α1(z), α2(z), α3(z) into the

governing equation (2.3) and take α3(z) = α1,z α2−α1α2,z

2α1α2insure Eq. (2.3) completely integrable.

Step two: Speciﬁcation of training set in computational domain:

Training data:{tj, zj, uj, vj}Nq

j=1,Residual training points:{tl

f, zl

f}Nf

l=1.

Step three: Construct NN q(x;¯

Θ) with random initialization of parameters ¯

Θ.

Step four: Construct the residual NN {fu, fv}by substituting surrogate q(x;¯

Θ) into the gov-

erning equations using automatic diﬀerentiation and other arithmetic operations.

Step ﬁve: Speciﬁcation of the loss function L(¯

Θ) that includes the slope recovery term.

Step six: Find the best parameters ¯

Θ∗using a suitable optimization method for minimizing

the loss function L(¯

Θ) as ¯

Θ∗= arg min

Θ∈¯

L(¯

Θ).

Adam and L-BFGS algorithms are used to optimize all loss functions in the IPINN method. The Adam optimiza-

tion algorithm is a variant of the traditional stochastic gradient descent algorithm, while the L-BFGS optimization

algorithm is a full-batch gradient descent optimization algorithm based on the quasi-Newton method [35, 36]. In

particular, unless otherwise stated, the scalable parameters in the adaptive activation function are typically initialized

as n= 10, ad

i= 0.1. In addition, we chose feedforward NNs with Xavier initialization and hyperbolic tangent (tanh)

as activation functions. All the code in this article is based on Python 3.7 and Tensorﬂow 1.15. All the numerical

experiments reported in this article were run on a DELL Precision 7920 Tower computer. The computer is equipped

with a 2.10 GHz 8-core Xeon Silver 4110 processor, 64 GB memory and an 11 GB NVIDIA GeForce GTX 1080 Ti

video card.

3. Data-driven forward problems of the VC-Hirota equation

In this section, we will focus on the data-driven forward for the VC-Hirota equation with Dirichlet boundary

conditions and initial conditions by means of IPINN method. The VC-Hirota equation with initial–boundary value

conditions is as follows.











iqz+α1(z)qtt −1

3δiα1(z)qttt +δα2(z)q|q|2−iα2(z)|q|2qt−iα3(z)q= 0,

q(L0, z) = qlb(z), q(L1, z) = qub(z),

q(t, T0) = q0(t), t ∈[L0, L1], z ∈[T0, T1],

(3.1)

where “i” is an imaginary number, the subscripts denote the partial derivatives of the complex ﬁelds q(t, z) with

respect to the space tand time z, while the L0and L1represent the lower and upper boundaries of trespectively.

Similarly, T0and T1represent the initial and ﬁnal times of zrespectively. Moreover, the q0(t) represents initial value

of the q(t, z) at z=T0, the qlb(z) and qub(z) are the lower and upper boundaries of the q(t, z) corresponding to t=L0

and t=L1respectively.

The N-solitons of the VC-Hirota have been derived by the Riemann-Hilbert method in [16], which can be expressed

as follows:

q=−2i|F|

|M|f(z)eig(t,z),(3.2)

where Fis the following (N+ 1) ×(N+ 1) matrix







0e−θ∗

1... e−θ∗

c1eθ1M11 ... MN1

. . . .

cNeθNM1N... MNN







,(3.3)

the elements of the N×Nmatrix Mare given by

Mjk =e−(θk+θ∗

j)+c∗

jckeθk+θ∗

ζ∗

j−ζk

θk=−iζkT−(4iβζ3

k+ 2iγζ2

k)Z,

f(z) = sα1(z)

α2(z), Z =−√2δ

12βZα1(z)dz, T =√2δ

2(t−(γ2

36β2−δ)Zα1(z)dz),

and g(t, z) = −6βδ +γ√2δ

6βt−216δ2β3+ 54γβ2δ√2δ−γ3√2δ

324β3Zα1(z)dz,

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DATA-DRIVENFORWARD-INVERSEPROBLEMSFORTHEVARIABLECOEFFICIENTSHIROTAEQUATIONUSINGDEEPLEARNINGMETHODHUIJUANZHOU,JUNCAIPU,ANDYONGCHENAbstract.Data-drivenforward-inverseproblemsforthevariablecoecientsHirota(VC-Hirota)equationarediscussedinthispaper.Animprovedphysics-informedneuralnetworks(IPINN)algorith...

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DATA-DRIVEN FORWARD-INVERSE PROBLEMS FOR THE VARIABLE COEFFICIENTS HIROTA EQUATION USING DEEP LEARNING METHOD HUIJUAN ZHOU JUNCAI PU AND YONG CHEN

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