Data-driven peakon and periodic peakon travelling wave solutions of some
nonlinear dispersive equations via deep learning

Li Wang and Zhenya Yan∗ ∗Email address<EMAIL_ADDRESS>(Corresponding
author)

Key Laboratory of Mathematics Mechanization, Academy of Mathematics and
Systems Science,
Chinese Academy of Sciences, Beijing 100190, China
School of Mathematical Sciences, University of Chinese Academy of Sciences,
Beijing 100049, China

###### Abstract

In the field of mathematical physics, there exist many physically interesting
nonlinear dispersive equations with peakon solutions, which are solitary waves
with discontinuous first-order derivative at the wave peak. In this paper, we
apply the multi-layer physics-informed neural networks (PINNs) deep learning
to successfully study the data-driven peakon and periodic peakon solutions of
some well-known nonlinear dispersion equations with initial-boundary value
conditions such as the Camassa-Holm (CH) equation, Degasperis-Procesi
equation, modified CH equation with cubic nonlinearity, Novikov equation with
cubic nonlinearity, mCH-Novikov equation, $b$-family equation with quartic
nonlinearity, generalized modified CH equation with quintic nonlinearity, and
etc. These results will be useful to further study the peakon solutions and
corresponding experimental design of nonlinear dispersive equations.

Key words: Nonlinear dispersive equation; initial-boundary value conditions;
physics-informed neural networks; deep learning; data-driven

##  1 Introduction

The well-known nonlinear dispersive Camassa-Holm (CH) equation can be used to
describe the propagation of shallow water waves, and deduced by using the
distinct approaches such as the symmetry method [1], an asymptotic expansion
directly in the Hamiltonian for Euler’s equations [2], and the tri-Hamiltonian
duality related to the Korteweg-de Vries equation [3]. The CH equation
admitted the peaked travelling wave solutions (alias peakons, i.e., solitary
waves with discontinuous first-order derivative at the wave peak) [2]. After
that, much attention has been paid to some physically interesting nonlinear
wave equations describing breaking waves and possessing the peakon solution in
the form $u_{t}-u_{xxt}+F(u,u_{x},u_{xx},u_{xxx})=0$ such as the Degasperis-
Procesi (DP) equation [4], $b$-family equation [5, 6], Fokas-Olver-Rosenau-
Qiao (FORQ) equation (alias modified CH (mCH) equation) [7, 8, 3], mCH-CH
equation [9], Novikov equation [10], mCH-Novikov equation [11], and etc. Up to
now, it is still an interesting subject to study these nonlinear peakon
equations via distinct approaches.

Recently, mechanics learning with deep neural networks plays a more and more
important role in many fields [12, 13]. Particularly, various deep neural
network learning approaches [14, 15, 16, 17, 18, 20, 19, 21, 22] have been
paid more and more attention in the study of linear and nonlinear partial
differential equations (PDEs). The physics-informed neural network (PINN)
approach [21, 22] were presented, and has been applied to many linear and
nonlinear PDEs [23, 24, 25, 26, 27], as well as other types of equations such
as the fractional PDEs [28], and stochastic differential equations [29]. More
recently, Shin, et al [30] theoretically showed the consistency of PINNs for
the linear second-order elliptic and parabolic type PDEs.

In this paper, we would like to use the PINN deep learning to consider the
Cauchy problem of the nonlinear dispersive equations with peakon solutions

$\displaystyle\left\\{\begin{array}[]{l}(1-\partial_{x}^{2})u_{t}+{\cal
N}(u,u_{x},u_{xx},u_{xxx},...)=0,\quad x\in(L_{1},L_{2}),\quad
t\in(0,T),\vspace{0.1in}\\\ u(x,0)=u_{0}(x),\quad
x\in[L_{1},L_{2}],\vspace{0.1in}\\\ u(L_{1},t)=u(L_{2},t),\quad
t\in[0,T],\end{array}\right.$ (4)

where the subscripts denote the partial derivatives (e.g., $u_{t}=\partial
u/\partial t$), $\partial_{x}=\partial/\partial x$, ${\cal
N}(u,u_{x},u_{xx},u_{xxx},...)$ is some nonlinear function of related
variables such that Eq. (4) contains many famous nonlinear peakon equations,
for example:

  * i)

The generalized $b$-family (gbf) equation with nonlinearities of degree
$(k+1)$ [31, 32, 33, 34]

$\displaystyle(1-\partial_{x}^{2})u_{t}+u^{k}\omega_{x}+bu^{k-1}u_{x}\omega=0,\quad\omega:=u-u_{xx},\,\,\,k\in\mathbb{N}^{+},\quad
b\in\mathbb{R},$ (5)

which contains the well-known models: ia) the integrable CH equation [2, 1]
for $k=1,b=2$

$\displaystyle(1-\partial_{x}^{2})u_{t}+3uu_{x}-2u_{x}u_{xx}-uu_{xxx}=0$ (6)

admitting the second-order Lax pair [2]; ib) the integrable DP equation [4]
for $k=1,b=3$

$\displaystyle(1-\partial_{x}^{2})u_{t}+4uu_{x}-3u_{x}u_{xx}-uu_{xxx}=0,$ (7)

which is related to a negative flow of the Kaup-Kupershmidt hierarchy by a
reciprocal transformation, and admits the third-order Lax pair [5]; ic) the
$b$-family equation [5, 6] for $k=1$

$\displaystyle(1-\partial_{x}^{2})u_{t}+(b+1)uu_{x}-bu_{x}u_{xx}-uu_{xxx}=0,$
(8)

id) the Novikov equation with cubic nonlinearity [10] for $k=2,\,b=3$

$\displaystyle(1-\partial_{x}^{2})u_{t}+4u^{2}u_{x}-3uu_{x}u_{xx}-u^{2}u_{xxx}=0,$
(9)

which is also completely integrable, and related to a negative flow in the
Sawada-Kotera hierarchy by a reciprocal transformation, as well as admits the
third-order Lax pair [35]; and ie) the modified Novikov equation [36] for
$k=2$

$\displaystyle(1-\partial_{x}^{2})u_{t}+(b+1)u^{2}u_{x}-buu_{x}u_{xx}-u^{2}u_{xxx}=0;$
(10)

  * ii)

The generalized modified Camassa-Holm (gmCH) equation with nonlinearities of
degree $(2k+1)$ [37, 38]

$\displaystyle(1-\partial_{x}^{2})u_{t}+[(u^{2}-u_{x}^{2})^{k}(u-u_{xx})]_{x}=0,\quad
k\in\mathbb{N}^{+},$ (11)

containing the integrable FORQ equation [7, 3, 8] (or the mCH equation) with
cubic nonlinearity for $k=1$

$\displaystyle(1-\partial_{x}^{2})u_{t}+[(u^{2}-u_{x}^{2})(u-u_{xx})]_{x}=0,$
(12)

which is also completely integrable, and admits the second-order Lax pair [8].

  * iii)

The generalized mCH and $b$-family equation with nonlinearities of degree
${\rm max}\\{2k+1,s+1\\}$

$\displaystyle(1-\partial_{x}^{2})u_{t}+k_{1}[(u^{2}-u_{x}^{2})^{k}\omega]_{x}+k_{2}(u^{s}\omega_{x}+bu^{s-1}u_{x}\omega)=0,\quad
k,\,s\in\mathbb{N}^{+},$ (13)

containing iiia) the mCH-CH equation for $k=s=1,\,b=2$ [9]

$\displaystyle(1-\partial_{x}^{2})u_{t}+k_{1}[(u^{2}-u_{x}^{2})\omega]_{x}+k_{2}(u\omega_{x}+2u_{x}\omega)=0,\quad
k_{1},\,k_{2}\in\mathbb{R},$ (14)

and iiib) the mCH-Novikov equation with cubic nonlinearity [11]

$\displaystyle(1-\partial_{x}^{2})u_{t}+k_{1}[(u^{2}-u_{x}^{2})\omega]_{x}+k_{2}(u^{2}\omega_{x}+3uu_{x}\omega)=0,\quad
k_{1},\,k_{2}\in\mathbb{R},$ (15)

  * iv)

The combination of generalized mCH and $b$-family equations

$\displaystyle(1-\partial_{x}^{2})u_{t}+\sum_{k=1}^{N}k_{j}[(u^{2}-u_{x}^{2})^{k}\omega]_{x}+\sum_{s=1}^{M}k_{s}(u^{s}\omega_{x}+bu^{s-1}u_{x}\omega)=0,$
(16)

  * v)

The general family of nonlinear peakon equations [38]

$\displaystyle(1-\partial_{x}^{2})u_{t}+f_{1}(u,u_{x})\omega+(f_{2}(u,u_{x})\omega)_{x}=0,$
(17)

where $f_{j}$’s are some functions of $u$ nd $u_{x}$.

Notice that i) The generalized $b$-family equation (5) containing the CH
equation (6), DP equation (7), $b$-family equation (8), and Novikov equation
(9), possesses the peakon solution $u(x,t)=c^{1/k}e^{-|x-ct|}$, and periodic
peakon solution $u(x,t)=c^{1/k}\,{\rm sech}\pi\,p(x)$ with
$p(x)=\cosh(x-ct-2\pi[(x-ct)/(2\pi)]-\pi)$ and $\lfloor\cdot\rfloor$ denoting
the floor function or the greatest integer function [31, 32, 33, 34]; ii) The
mCH equation (12) has the peakon solution [39]
$u(x,t)=\sqrt{3c/2}e^{-|x-ct|},\,c>0$ and periodic peakon solution [40]
$u(x,t)=\sqrt{3c/(2\cosh^{2}\pi+1)}\,p(x),\,c>0$; iii) The mCH-Novikov
equation (15) has the peakon solution [11]
$u(x,t)=\sqrt{3c/(2k_{1}+3k_{2})}e^{-|x-ct|}$ and periodic peakon solution
$u(x,t)=\sqrt{3c/[k_{1}(2\cosh^{2}\pi+1)+3k_{2}\cosh^{2}\pi]}\,p(x);$ iv) The
generalized mCH equation (11) with $k=2$ has the peakon solution [41]
$u(x,t)=\sqrt[4]{15c/8}e^{-|x-ct|}$ and periodic peakon solution
$u(x,t)=\sqrt[4]{15c/[\sinh\pi(8\cosh^{4}\pi+4\cosh^{2}\pi+3)]}\,p(x)$.

In this paper, we would like to consider the data-driven peakon and periodic
peak solutions of the above-mentioned some physical interesting nonlinear PDEs
via deep learning. The rest of this paper is arranged as follows: In Sec. 2,
we introduce the PINN deep leaning approach for the Cauchy problem of
nonlinear PDEs (4). In Sec. 3, we use the PINN deep learning scheme to
investigate the data-driven peakon and periodic peakon solutions of some
famous nonlinear PDEs such as the CH equation (6), DP equation (7), Novikov
equation (9), $b$-family equation (5) with $b=k=3$, mCH equation (12),
generalized mCH equation (11) with $k=2$, mCH-CH equation (14), and the mCH-
Novikov equation (15). In Sec. 4, we give some conclusions and discussions.

##  2 The PINN deep learning scheme of Eq. (4)

For the given Cauchy problem (4) we consider the residual PINN $f(x,t)$ as

$\displaystyle f(x,t):=(1-\partial_{x}^{2})\widehat{u}_{t}+{\cal
N}(\widehat{u},\widehat{u}_{x},\widehat{u}_{xx},\widehat{u}_{xxx},...),$ (18)

where ${\cal
N}(\widehat{u},\widehat{u}_{x},\widehat{u}_{xx},\widehat{u}_{xxx},...)$ can be
chosen as the some physical models mentioned in Sec. 1, a deep neural network
$\widehat{u}(x,t;w,b)$ denotes the continuous latent function with two
families of network parameters: the weights $w$ and biases $b$, and can be
used to approximate $u(x,t)$ such that one can consider its partial
derivatives of arbitrary order with respect to its input with the aid of
automatic differentiation (e.g., Tensorflow ) [42, 43], which differs from the
numerical or symbolic differentiation. And then the residual PINN $f(x,t)$
given by Eq. (18) can also be found.

Therefore, with the aid of L-BFGS optimization method [44], the common
parameters in the latent function $\widehat{u}(x,t)$ and residual PINN
$f(x,t)$ can be trained by using the multi-hidden-layer deep NN with some
neurons per layer combined with a hyperbolic tangent activation function and
minimizing the whole mean squared error (MSE) loss in the form

$MSE=MSE_{ie}+MSE_{b}+MSE_{f},$ (19)

where

$\displaystyle MSE_{ie}$
$\displaystyle=\frac{1}{N_{int}}\sum_{j=1}^{N_{int}}\left|\widehat{u}(x_{0}^{j},0)-u(x_{0}^{j},0)\right|^{2}+\frac{1}{N_{end}}\sum_{j=1}^{N_{end}}\left|\widehat{u}(x_{T}^{j},T)-u(x_{T}^{j},T)\right|^{2},$
(20) $\displaystyle MSE_{b}$
$\displaystyle=\frac{1}{N_{b}}\sum_{j=1}^{N_{b}}\left(\left|\widehat{u}(L_{1},t_{b}^{j})-u(L_{1},t_{b}^{j})\right|^{2}+\left|\widehat{u}(L_{2},t_{b}^{j})-u(L_{2},t_{b}^{j})\right|^{2}+\left|\widehat{u}(L_{1},t_{b}^{j})-\widehat{u}(L_{2},t_{b}^{j})\right|^{2}\right),$
(21) $\displaystyle MSE_{f}$
$\displaystyle=\frac{1}{N_{f}}\sum_{j=1}^{N_{f}}\left|f(x_{f}^{j},t_{f}^{j})\right|^{2},$
(22)

the observed measurements $\\{\widehat{u}(x_{0}^{j},0)\\}_{1}^{N_{int}}$ and
$\\{\widehat{u}(x_{T}^{j},T)\\}_{1}^{N_{end}}$ of the hidden field
$\widehat{u}(x,t)$ are linked with the sampled initial and end training data
$\\{x_{0}^{j},\,u(x_{0}^{j},0)\\}_{1}^{N_{int}}$ and
$\\{x_{T}^{j},\,u(x_{T}^{j},T)\\}_{1}^{N_{end}}$, respectively.
$\\{\widehat{u}(L_{1,2},t_{b}^{j})\\}_{1}^{N_{b}}$ connect the selected
boundary training data $\\{t_{b}^{j},\,u(L_{1,2},t_{b}^{j}\\}_{1}^{N_{b}}$,
and $\\{x_{f}^{j},t_{f}^{j}\\}^{N_{f}}_{1}$ are connected with the marked
points for the PINN $f(x,t)$. As a result, for the randomly chosen points,
$MSE_{ie}$, and $MSE_{b}$ represent the MSE losses of initial-end and periodic
boundary data, respectively, and $MSE_{f}$ is associated with the MSE loss of
the PINN (18). The aims of $MSE_{ie}$ and $MSE_{b}$ try to match the learning
solution to exact one for the initial and end data, and boundary data,
respectively. The aim of $MSE_{f}$ is to make the hidden $\widehat{u}(x,t)$
satisfy the considered physical equation (4).

In what follows we would like to use the PINN deep learning scheme to
investigate the data-driven peakon and periodic peakon solutions of some
famous nonlinear PDEs such as the CH equation (6), DP equation (7), Novikov
equation (9), $b$-family equation (5) with $b=k=3$, mCH equation (12),
generalized mCH equation (11) with $k=2$, mCH-CH equation (14), and the mCHN
equation (15).

##  3 The data-driven peakon and periodic peakon solutions

###  3.1 The CH equation

In the subsection, we use the above-mentioned PINN deep learning approach to
consider the data-driven peakon and periodic peakon solutions of the initial-
boundary value problem of the CH equation given by Eqs. (4) and (6).

Case 1. For the peakon solution [2] $u(x,t)=ce^{-|x-ct|}$ of the CH equation
(6), it is easy to see that when $c>0$, the peakon solution is a right-going
travelling wave solution, and bright peakon solution, whereas $c<0$, it is a
left-going travelling wave solution, and dark peakon solution. We here
consider the stationary peakon solution at $t=0$ as the initial condition

$\displaystyle u(x,0)=ce^{-|x|},\quad x\in[-L,L],$ (23)

and the periodic boundary condition $u(-L,t)=u(L,t)$. The considered residual
PINN $f_{ch}(x,t)$ is written as

$\displaystyle f_{ch}(x,t):=u_{t}-u_{xxt}+3uu_{x}-2u_{x}u_{xx}-uu_{xxx}.$ (24)

Here, the hidden neural network $\widehat{u}(x,t)$ in Python can be defined as

⬇

def u(x, t):

u = neural_net(tf.concat([x,t],1), weights, biases)

return u

such that the residual PINN $f_{ch}(x,t)$ in Python is written as

⬇

def f_ch(x, t):

u = u(x, t)

u_t = tf.gradients(u, t)[0]

u_x = tf.gradients(u, x)[0]

u_xx = tf.gradients(u_x, x)[0]

u_xxt = tf.gradients(u_xx, t)[0]

u_xxx = tf.gradients(u_xx, x)[0]

f_ch = u_t - u_xxt + 3*u*u_x - 2*u_x*u_xx - u*u_xxx

return f_ch

i) For the case $c>0$ in the initial condition (23), i.e., the bright peakon
solution, we take $c=0.8$ and use the Fourier pseudo-spectral method [45]
(i.e., one can take Fourier transform in space, and choose the explicit
fourth-order Runge-Kutta method in time) to simulate the CH equation (6) with
the initial value condition (23) and periodic boundary condition. The spatial
region is $x\in[-6,\,6]$ with 512 Fourier modes, and the temporal region is
$t\in[0,2]$ with time-step $\Delta t=0.004$. As a consequence, we produce the
data-set for the PINN pertaining to the CH equation. The training data-set
used in the PINN $f_{ch}(x,t)$ consists of randomly chosen $N_{int}=10$ points
from the initial data $u(x,0)$, $N_{end}=10$ points from the end data
$u(x,2)$, $N_{b}=10$ points pertaining to the periodic boundary data, and
$N_{f}=1000$ points from the spatial-temporal solution zone.

Figures 1(a1) and (c1) exhibit the two-dimensional (2D) and three-dimensional
(3D) profiles of the latent bright-peakon solution $\widehat{u}(x,t)$ learned
by a 5-hidden-layer deep PINN $f_{ch}(x,t)$ with 20 neurons per layer combined
with a hyperbolic tangent activation function, and minimizing the MSE loss
(19). The comparison of learning solution (red dashed line), numerical
solution (blue solid line) and exact solution (green plus) is shown at three
different times $t=0.4,\,1.0$, and $1.4$ (see Fig. 1(b1)). The
$\mathbb{L}_{2}$-norm error between learning solution $\widehat{u}(x,t)$ and
numerical solution is 2.73e-02.

Figure 1: The CH equation. (a1,a2) The data-driven peakon solutions resulted
from the PINN; (b1,b2) The comparisons between the learning, numerical, and
exact peakon solutions at the distinct times $t=0.4,\,1$, and $1.4$; The
$\mathbb{L}_{2}$-norm errors $\widehat{u}(x,t)$ between learning and numerical
peakon solutions are (b1) 2.73e-02 and (b2) 4.62e-02; (c1,c2) The 3D profiles
of the learning bright and dark peakon solutions. $c=0.8$ for (a1, b1, c1) and
$c=-0.9$ for (a2, b2, c2).

Figure 2: The CH equation: (a, c) The data-driven periodic peakon solution
resulted from the PINN; (b) The comparisons between the learning, numerical,
and exact periodic peakon solutions at the distinct times $t=0.2$, $0.9$, and
$1.5$. The $\mathbb{L}_{2}$-norm error between the learning and numerical
periodic peakon solutions is 3.56e-02.

ii) For the case $c<0$ in the initial condition (23), i.e., dark peakon
solution, we take $c=-0.9$ and use the same pseudo-spectral approach to
produce the data-set for PINN pertaining to the CH equation in the spatio-
temporal region $(x,t)\in[-6,6]\times[0,2]$ with the 512 Fourier modes in the
$x$ direction and time-step $\Delta t=0.004$. The training data-set used in
the PINN consists of randomly chosen $N_{int}=20$ points from the initial data
$u(x,0)$, $N_{end}=20$ points from the end data $u(x,2)$, $N_{b}=20$ points
pertaining to the periodic boundary data, and $N_{f}=1000$ points from the
spatio-temporal solution zone. Figures 1(a2) and (c2) exhibit the 2D and 3D
profiles of the latent dark-peakon solution $\widehat{u}(x,t)$ learned by a
6-hidden-layer deep PINN with 20 neurons per layer combined with a hyperbolic
tangent activation function. The comparison of learning solution (red dashed
line), numerical solution (blue solid line) and exact solution (green plus) is
shown at three different times $t=0.4,\,1.0$, and $1.4$ (see Fig. 1(b2)). The
$\mathbb{L}_{2}$-norm error between learning solution $\widehat{u}(x,t)$ and
numerical solution is 4.62e-02.

Case 2. For the periodic peakon solution [32] $u(x,t)=c\,{\rm
sech}\pi\cosh(x-ct-2\pi\lfloor(x-ct)/(2\pi)\rfloor-\pi)$ of the CH equation
(6), we consider the stationary periodic peakon solution at $t=0$ as the
initial value condition

$\displaystyle u(x,0)=c\,{\rm
sech}\pi\cosh\left(x-2\pi\left\lfloor\frac{x}{2\pi}\right\rfloor-\pi\right),\quad
x\in[L_{1},L_{2}]$ (25)

and the periodic boundary $u(L_{1},t)=u(L_{2},t)$, as well as $c=1$.
Similarly, we use the pseudo-spectral approach to generate the data-set for
the PINN pertaining to the CH equation in the spatio-temporal region
$(x,t)\in[-8.64,10.21]\times[0,2]$ with the 512 Fourier modes in the $x$
direction and time-step $\Delta t=0.004$. The training data-set used in the
PINN consists of randomly chosen $N_{int}=60$ points from the initial data
$u(x,0)$, $N_{end}=10$ points from the end data $u(x,2)$, $N_{b}=10$ points
pertaining to the periodic boundary data, and $N_{f}=1100$ points from the
spatio-temporal solution zone. The hidden solution $\widehat{u}(x,t)$ is
learned by a 6-hidden-layer deep PINN $f_{ch}(x,t)$ with 40 neurons pere layer
combined with a hyperbolic tangent activation function.

Figures 2(a) and (c) exhibit the 2D and 3D profiles of the latent periodic
peakon solution $\widehat{u}(x,t)$. The comparison of learning solution (red
dashed line), numerical solution (blue solid line) and exact solution (green
plus) is shown at three different times $t=0.2,\,0.9$, and $1.5$ (see Fig.
2(b)). The $\mathbb{L}_{2}$-norm error between learning solution
$\widehat{u}(x,t)$ and numerical solution is 3.56e-02.

Similarly, the data-driven peakon and periodic peakon solutions of the DP
equation (7) are also studied in Appendix A.

###  3.2 The Novikov equation with cubic nonlinearity

In the subsection, we use the PINN to consider the data-driven peakon
solutions of the initial-boundary value problem of the Novikov equation with
cubic nonlinearity (9).

Case 1. For the peakon solution [35] $u(x,t)=\sqrt{c}e^{-|x-ct|}$ with $c>0$
of the Novikov equation (9), which is a right-going travelling wave solution,
and bright peakon solution. It follows from the peakon solutions of CH and
Novikov equations that for the right-going travelling wave with the same
velocity, i) when $c>1$, the amplitude of the CH peakon solution is larger
than one of the Novikov peakon solution; when $0<c<1$, the amplitude of the CH
peakon solution is smaller than one of the Novikov peakon solution; iii) when
$c=1$, the amplitudes of the CH and Novikov peakon solutions are same. We here
consider the the stationary peakon solution at $t=0$ as the initial condition

$\displaystyle u(x,0)=\sqrt{c}\,e^{-|x|},\quad x\in[-L,L]$ (26)

and the periodic boundary $u(-L,t)=u(L,t)$. The considered PINN $f_{nov}(x,t)$
is written as

$\displaystyle
f_{nov}(x,t):=u_{t}-u_{xxt}+4u^{2}u_{x}-3uu_{x}u_{xx}-u^{2}u_{xxx}.$ (27)

We here choose $c=0.36$ in the initial condition (26), and use the Fourier
pseudo-spectral method (i.e., one can take Fourier transform in space, and
choose the explicit fourth-order Runge-Kutta method in time) to simulate the
Novikov equation with the initial value condition (26) and periodic boundary
condition. The spatial region is $x\in[-10,\,10]$ with 256 Fourier modes, and
the temporal region is $t\in[0,2]$ with time-step $\Delta t=0.004$. As a
result, we produce the data-set for the PINN pertaining to the Novikov
equation. The training data-set used in the PINN $f_{nov}(x,t)$ consists of
randomly chosen $N_{int}=20$ points from the initial data $u(x,0)$,
$N_{end}=20$ points from the end data $u(x,2)$, $N_{b}=20$ points pertaining
to the periodic boundary data, and $N_{f}=1000$ points from the spatio-
temporal solution zone. Figures 3(a1) and (c1) exhibit the 2D and 3D profiles
of the latent bright peakon solution $\widehat{u}(x,t)$ learned by a 6-hidden-
layer deep PINN $f_{nov}(x,t)$ with 20 neurons per layer combined with a
hyperbolic tangent activation function. The comparison of learning solution
(red dashed line), numerical solution (blue solid line) and exact solution
(green plus) is shown at three different times $t=0.2,1.0$ and $1.8$ (see Fig.
3(b1)). The $\mathbb{L}_{2}$-norm error between learning solution
$\widehat{u}(x,t)$ and numerical solution is 3.82e-02.

Figure 3: The Novikov equation. The data-driven peakon (a1,c1) and periodic
peakon (a2, c2) solutions resulted from the PINN; (b1, b2) The comparisons
between the learning, numerical, and exact peakon or periodic peakon solutions
at the distinct times (b1) $t=0.2,\,1$, and $1.8$, or (b2) $t=0.5,\,0.8$, and
$1.6$. The $\mathbb{L}_{2}$-norm errors between the learning and numerical
solutions are (b2) 3.82e-02 and (b2) 4.56e-02. $c=0.36$ for (a1,b1,c1), and
$c=0.25$ for (a2,b2,c2).

Case 2. For the periodic peakon solution [32, 46] $u(x,t)=\sqrt{c}\,{\rm
sech}\pi\cosh(x-ct-2\pi\lfloor(x-ct)/(2\pi)\rfloor-\pi)$ of the Novikov
equation (9), we consider the stationary periodic peakon solution at $t=0$ as
the initial value condition

$\displaystyle u(x,0)=\sqrt{c}\,{\rm
sech}\pi\cosh\left(x-2\pi\left\lfloor\frac{x}{2\pi}\right\rfloor-\pi\right),$
(28)

and the periodic boundary $u(-L,t)=u(-L,t)$, as well as $c=0.25$. Similarly,
we use the pseudo-spectral approach to generate the data-set for the PINN
pertaining to the Novikov equation in the spatio-temporal region
$(x,t)\in[-3\pi,3\pi]\times[0,2]$ with the 512 Fourier modes in the $x$
direction and time-step $\Delta t=0.004$. The training data-set used in the
PINN consists of randomly chosen $N_{int}=70$ points from the initial data
$u(x,0)$, $N_{end}=0$ point from the end data $u(x,2)$, $N_{b}=10$ points
pertaining to the periodic boundary data, and $N_{f}=2000$ points from the
spatio-temporal solution zone. The hidden solution $\widehat{u}(x,t)$ is
learned by a 6-hidden-layer deep PINN $f_{nov}(x,t)$ with 20 neurons pere
layer combined with a hyperbolic tangent activation function.

Figures 3(a2) and (c2) exhibit the 2D and 3D profiles of the latent solution
$\widehat{u}(x,t)$ learned by a 6-hidden-layer deep PINN $f_{nov}(x,t)$ with
20 neurons per layer combined with a hyperbolic tangent activation function.
The comparison of learning solution (red dashed line), numerical solution
(blue solid line) and exact solution (green plus) is shown at three different
times $t=0.5,0.8$ and $1.6$ (see Fig. 3(b2)). The $\mathbb{L}_{2}$-norm error
between learning solution $\widehat{u}(x,t)$ and numerical solution is
4.56e-02.

###  3.3 The generalized $b$-family equation with quartic nonlinearity

In the subsection, we use the PINN to consider the data-driven peakon
solutions of the initial-boundary value problem of the generalized $b$-family
equation (5) with quartic nonlinearity for $b=k=3$.

Figure 4: The generalized b-family equation (5) with $b=k=3$. The data-driven
peakon (a1,c1) and periodic peakon (a2, c2) solutions resulted from the PINN;
(b1, b2) The comparisons between the learning, numerical, and exact peakon or
periodic peakon solutions at the distinct times (b1) $t=0.2,\,1.1$, and $1.7$,
or (b2) $t=0.3,\,0.9$, and $1.7$. The $\mathbb{L}_{2}$-norm errors between the
learning and numerical solutions are (b2) 4.62e-02 and (b2) 6.2e-02. $c=0.125$
for (a1,b1,c1), and $c=-0.216$ for (a2,b2,c2).

Case 1. The generalized $b$-family equation (5) with $b=k=3$ has the peakon
solution [32] $u(x,t)=c^{1/3}e^{-|x-ct|}$. When $c>0$, the peakon solution is
a right-going travelling wave solution, and bright peakon solution, whereas
$c<0$, it is a left-going travelling wave solution, and dark peakon solution.
We here consider the stationary peakon solution at $t=0$ as the initial
condition

$\displaystyle u(x,0)=c^{1/3}\,e^{-|x|},$ (29)

and the periodic boundary $u(-L,t)=u(L,t)$. The considered PINN $f_{gbf}(x,t)$
is written as

$\displaystyle
f_{gbf}(x,t):=u_{t}-u_{xxt}+4u^{3}u_{x}-3u^{2}u_{x}u_{xx}-u^{3}u_{xxx}.$ (30)

We here consider $c=0.125$ in the initial condition (29), and use the Fourier
pseudo-spectral method (i.e., one can take Fourier transform in space, and
choose the explicit fourth-order Runge-Kutta method in time) to simulate the
generalized $b$-family equation (5) with $b=k=3$, the initial value condition
(29), and periodic boundary condition. The spatial region is $x\in[-10,\,10]$
with 256 Fourier modes, and the temporal region is $t\in[0,2]$ with time-step
$\Delta t=0.004$. As a result, we produce the data-set for the PINN pertaining
to the generalized $b$-family equation (5) with quartic nonlinearity for
$b=k=3$. The training data-set used in the PINN $f_{gbf}(x,t)$ consists of
randomly chosen $N_{int}=40$ points from the initial data $u(x,0)$,
$N_{end}=10$ points from the end data $u(x,2)$, $N_{b}=10$ points pertaining
to the periodic boundary data, and $N_{f}=3000$ points from the spatio-
temporal solution zone. Figures 4(a1) and (c1) exhibit the 2D and 3D profiles
of the latent solution $\widehat{u}(x,t)$ learned by a 5-hidden-layer deep
PINN $f_{gbf}(x,t)$ with 20 neurons per layer combined with a hyperbolic
tangent activation function. The comparison of learning solution (red dashed
line), numerical solution (blue solid line) and exact solution (green plus) is
shown at three different times $t=0.2,1.1$ and $1.7$ (see Fig. 4(b1)). The
$\mathbb{L}_{2}$-norm error between learning solution $\widehat{u}(x,t)$ and
numerical solution is 4.62e-02.

Case 2. The generalized $b$-family equation (5) with $b=k=3$ has the periodic
peakon solution [32] $u(x,t)=c^{1/3}\,{\rm
sech}\pi\cosh(x-ct-2\pi\lfloor(x-ct)/(2\pi)\rfloor-\pi)$, we consider the
stationary periodic peakon solution at $t=0$ as the initial value condition

$\displaystyle u(x,0)=c^{1/3}\,{\rm
sech}\,\pi\cosh\left(x-2\pi\left\lfloor\frac{x}{2\pi}\right\rfloor-\pi\right),$
(31)

and the periodic boundary $u(L_{1},t)=u(L_{2},t)$, as well as $c=-0.216$.
Similarly, we use the pseudo-spectral approach to generate the data-set for
the PINN pertaining to the generalized $b$-family equation (5) with $b=k=3$ in
the spatio-temporal region $(x,t)\in[-3\pi,3\pi]\times[0,2]$ with the 512
Fourier modes in the $x$ direction and time-step $\Delta t=0.004$. The
training data-set used in the PINN consists of randomly chosen $N_{int}=50$
points from the initial data $u(x,0)$, $N_{end}=0$ point from the end data
$u(x,2)$, $N_{b}=10$ points pertaining to the periodic boundary data, and
$N_{f}=2000$ points from the spatio-temporal solution zone. The hidden
solution $\widehat{u}(x,t)$ is learned by a 6-hidden-layer deep PINN
$f_{gbf}(x,t)$ with 20 neurons per layer combined with a hyperbolic tangent
activation function.

Figures 4(a2) and (c2) exhibit the 2D and 3D profiles of the latent solution
$\widehat{u}(x,t)$ learned by a 6-hidden-layer deep PINN $f_{gbf}(x,t)$ with
20 neurons per layer combined with a hyperbolic tangent activation function.
The comparison of learning solution (red dashed line), numerical solution
(blue solid line) and exact solution (green plus) is shown at three different
times $t=0.3,0.9$ and $1.7$ (see Fig. 4(b2)). The $\mathbb{L}_{2}$-norm error
between learning solution $\widehat{u}(x,t)$ and numerical solution is
6.2e-02.

###  3.4 The mCH equation with cubic nonlinearity

In the subsection, we use the PINN to consider the data-driven peakon
solutions of the initial-boundary value problem of the mCH equation (12).

Figure 5: The mCH equation (12). The data-driven peakon (a1,c1) and periodic
peakon (a2, c2) solutions resulted from the PINN; (b1, b2) The comparisons
between the learning, numerical, and exact peakon or periodic peakon solutions
at the distinct times (b1) $t=0.2,\,1$, and $1.8$, or (b2) $t=0.1,\,0.5$, and
$0.9$. The $\mathbb{L}_{2}$-norm errors between the learning and numerical
solutions are (b2) 4.74e-02 and (b2) 4.09e-02. $c=0.04$ for (a1,b1,c1), and
$c=0.06$ for (a2,b2,c2).

Case 1. For the peakon solution [39] $u(x,t)=\sqrt{3c/2}\,e^{-|x-ct|}$ with
$c>0$ of the mCH equation (12), which is a right-going travelling wave
solution, and bright peakon solution. We here consider the the stationary
peakon solution at $t=0$ as the initial condition

$\displaystyle u(x,0)=\sqrt{3c/2}\,e^{-|x|},\quad x\in[-L,L],$ (32)

and the periodic boundary $u(-L,t)=u(L,t)$. The considered PINN $f_{mch}(x,t)$
is written as

$\displaystyle f_{mch}(x,t):=u_{t}-u_{xxt}+[(u^{2}-u_{x}^{2})(u-u_{xx})]_{x}.$
(33)

We here consider $c=0.04$ in the initial condition (32), and use the Fourier
pseudo-spectral method (i.e., one can take Fourier transform in space, and
choose the explicit fourth-order Runge-Kutta method in time) to simulate the
mCH equation (12) with the initial value condition (32) and periodic boundary
condition. The spatial region is $x\in[-10,\,10]$ with 256 Fourier modes, and
the temporal region is $t\in[0,2]$ with time-step $\Delta t=0.005$. As a
result, we produce the data-set for the PINN pertaining to the mCH equation.
The training data-set used in the PINN $f_{mch}(x,t)$ consists of randomly
chosen $N_{int}=40$ points from the initial data $u(x,0)$, $N_{end}=20$ points
from the end data $u(x,2)$, $N_{b}=20$ points pertaining to the periodic
boundary data, and $N_{f}=2000$ points from the spatio-temporal solution zone.
Figures 5(a1) and (c1) exhibit the 2D and 3D profiles of the latent solution
$\widehat{u}(x,t)$ learned by a 7-hidden-layer deep PINN $f_{mch}(x,t)$ with
20 neurons per layer combined with a hyperbolic tangent activation function.
The comparison of learning solution (red dashed line), numerical solution
(blue solid line) and exact solution (green plus) is shown at three different
times $t=0.2,1$, and $1.8$ (see Fig. 5(b1)). The $\mathbb{L}_{2}$-norm error
between learning solution $\widehat{u}(x,t)$ and numerical solution is
4.74e-02.

Case 2. The mCH equation has the periodic peakon solution [40]
$u(x,t)=\sqrt{3c/(2\cosh^{2}\pi+1)}\cosh(x-ct-2\pi\lfloor(x-ct)/(2\pi)\rfloor-\pi$
with $c>0$. We consider the stationary periodic peakon solution at $t=0$ as
the initial value condition

$\displaystyle
u(x,0)=\sqrt{\frac{3c}{2\cosh^{2}\pi+1}}\cosh\left(x-2\pi\left\lfloor\frac{x}{2\pi}\right\rfloor-\pi\right),\quad
c>0$ (34)

and the periodic boundary $u(L_{1},t)=u(L_{2},t)$, as well as $c=0.06$.
Similarly, we use the pseudo-spectral approach to generate the data-set for
the PINN pertaining to the mCH equation (12) in the spatio-temporal region
$(x,t)\in[-3\pi,3\pi]\times[0,1]$ with the 512 Fourier modes in the $x$
direction and time-step $\Delta t=0.002$. The training data-set used in the
PINN consists of randomly chosen $N_{int}=60$ points from the initial data
$u(x,0)$, $N_{end}=0$ point from the end data $u(x,1)$, $N_{b}=10$ points
pertaining to the periodic boundary data, and $N_{f}=2000$ points from the
spatio-temporal solution zone. The hidden solution $\widehat{u}(x,t)$ is
learned by a 6-hidden-layer deep PINN $f_{mch}(x,t)$ with 20 neurons per layer
combined with a hyperbolic tangent activation function.

Figures 5(a2) and (c2) exhibit the 2D and 3D profiles of the latent solution
$\widehat{u}(x,t)$ learned by a 6-hidden-layer deep PINN $f_{mch}(x,t)$ with
20 neurons per layer combined with a hyperbolic tangent activation function.
The comparison of learning solution (red dashed line), numerical solution
(blue solid line) and exact solution (green plus) is shown at three different
times $t=0.1,0.5$ and $0.9$ (see Fig. 5(b2)). The $\mathbb{L}_{2}$-norm error
between learning solution $\widehat{u}(x,t)$ and numerical solution is
4.09e-02.

###  3.5 The mCH-CH equation with both quadratic and cubic nonlinearities

Case 1. The mCH-CH equation (14) has the peakon solution [9]
$u(x,t)=(-3k_{2}\pm\sqrt{9k_{2}^{2}+24ck_{1}})/(4k_{1})e^{-|x-ct|}$, which is
a right-going ($c>0$ and $9k_{2}^{2}+24ck_{1}>0$) or left-going ($c<0$ and
$9k_{2}^{2}+24ck_{1}>0$) peaked travelling wave solution. We here consider the
the stationary peakon solution at $t=0$ as the initial condition

$\displaystyle
u(x,0)=\frac{-3k_{2}+\sqrt{9k_{2}^{2}+24ck_{1}}}{4k_{1}}e^{-|x|},$ (35)

and the periodic boundary $u(-L,t)=u(L,t)$. The considered PINN
$f_{mch2}(x,t)$ is written as

$\displaystyle
f_{mch2}(x,t):=u_{t}-u_{xxt}+k_{1}[(u^{2}-u_{x}^{2})(u-u_{xx})]_{x}+k_{2}(3uu_{x}-2u_{x}u_{xx}-uu_{xxx}).$
(36)

i) For $c=0.5,\,k_{1}=-0.2$ and $k_{2}=2$ in the initial condition (35), which
is a bright peakon solution, we use the Fourier pseudo-spectral method (i.e.,
one can take Fourier transform in space, and choose the explicit fourth-order
Runge-Kutta method in time) to simulate the mCH-CH equation (14) with the
initial value condition (35) and periodic boundary condition. The spatial
region is $x\in[-10,\,10]$ with 640 Fourier modes, and the temporal region is
$t\in[0,2]$ with time-step $\Delta t=0.004$. As a result, we produce the data-
set for the PINN pertaining to the mCH-CH equation. The training data-set used
in the PINN $f_{mch2}(x,t)$ consists of randomly chosen $N_{int}=40$ points
from the initial data $u(x,0)$, $N_{end}=10$ points from the end data
$u(x,2)$, $N_{b}=10$ points pertaining to the periodic boundary data, and
$N_{f}=5000$ points from the spatio-temporal solution zone. Figures 6(a1) and
(c1) exhibit the 2D and 3D profiles of the latent solution $\widehat{u}(x,t)$
learned by a 5-hidden-layer deep PINN $f_{mch2}(x,t)$ with 10 neurons per
layer combined with a hyperbolic tangent activation function. The comparison
of learning solution (red dashed line), numerical solution (with blue solid
line) and exact solution (green plus) is shown at three different times
$t=0.2,0.7$, and $1.4$ (see Fig. 6(b1)). The $\mathbb{L}_{2}$-norm error
between learning solution $\widehat{u}(x,t)$ and numerical solution is
6.02e-02.

ii) For $c=-2,\,k_{1}=-0.6$ and $k_{2}=4$ in the initial condition (35), which
is a dark peakon solution, we similarly choose the spatial region
$x\in[-10,\,10]$ with 640 Fourier modes, and the temporal region is
$t\in[0,2]$ with time-step $\Delta t=0.004$. As a result, we produce the data-
set for the PINN pertaining to the mCH-CH equation. The training data-set used
in the PINN $f_{mch2}(x,t)$ consists of randomly chosen $N_{int}=40$ points
from the initial data $u(x,0)$, $N_{end}=10$ points from the end data
$u(x,2)$, $N_{b}=10$ points pertaining to the periodic boundary data, and
$N_{f}=6000$ points from the spatio-temporal solution zone. Figures 6(a2) and
(c2) exhibit the 2D and 3D profiles of the latent solution $\widehat{u}(x,t)$
learned by a 6-hidden-layer deep PINN $f_{mch2}(x,t)$ with 10 neurons per
layer combined with a hyperbolic tangent activation function. The comparison
of learning solution (red dashed line), numerical solution (blue solid line)
and exact solution (green plus) is shown at three different times $t=0.1,0.6$,
and $1.3$ (see Fig. 6(b2)). The $\mathbb{L}_{2}$-norm error between learning
solution $\widehat{u}(x,t)$ and numerical solution is 8.47e-02.

Figure 6: The mCH-CH equation with peakon solutions: (a1,a2) The data-driven
peakon solution resulted from the PINN; (b1,b2) The comparisons between the
learning, numerical, and exact peakon solutions at the distinct times (b1)
$t=0.2,\,0.7$, and $1.4$, (b2) $t=0.1,\,0.6$, and $1.3$. The
$\mathbb{L}_{2}$-norm errors $\widehat{u}(x,t)$ between learning and numerical
peakon solutions are (b1) 6.02e-02 and (b2) 8.47e-02; (c1,c2) The 3D profiles
of the learning bright and dark peakon solutions.
$c=0.5,\,k_{1}=-0.2,\,k_{2}=2$ for (a1, b1, c1) and
$c=-2,\,k_{1}=-0.6,\,k_{2}=4$ for (a2, b2, c2).

Case 2. The mCH-CH equation has the periodic peakon solution
$u(x,t)=(-3k_{2}\cosh\pi\pm
a)/[2k_{1}(2\cosh^{2}\pi+1)]\cosh(x-ct-2\pi\lfloor(x-ct)/(2\pi)\rfloor-\pi)$
with $a=\sqrt{9k_{2}^{2}\cosh^{2}\pi+12ck_{1}(2\cosh^{2}\pi+1)}$. We here
consider the the stationary periodic peakon solution at $t=0$ as the initial
condition

$\displaystyle
u(x,0)=\frac{-3k_{2}\cosh\pi\\!+\\!\mu\sqrt{9k_{2}^{2}\cosh^{2}\\!\pi\\!+\\!12ck_{1}(2\cosh^{2}\\!\pi\\!+\\!1)}}{2k_{1}(2\cosh^{2}\\!\pi+1)}\cosh\left(x\\!-\\!2\pi\left\lfloor\frac{x}{2\pi}\right\rfloor\\!-\\!\pi\right),\,\,\,\,\mu=\pm
1$ (37)

and the periodic boundary condition $u(L_{1},t)=u(L_{2},t)$

i) We consider $\mu=1,\,c=1.5,\,k_{1}=-0.5$, and $k_{2}=2$ in the initial
condition (37), and use the pseudo-spectral approach to generate the data-set
for the PINN pertaining to the mCH-CH equation in the spatio-temporal region
$(x,t)\in[-75\pi/32,117\pi/32]\times[0,2]$ with the 640 Fourier modes in the
$x$ direction and time-step $\Delta t=0.004$. The training data-set used in
the PINN consists of randomly chosen $N_{int}=90$ points from the initial data
$u(x,0)$, $N_{end}=0$ point from the end data $u(x,2)$, $N_{b}=10$ points
pertaining to the periodic boundary data, and $N_{f}=6000$ points from the
spatio-temporal solution zone. The hidden solution $\widehat{u}(x,t)$ is
learned by a 6-hidden-layer deep PINN $f_{mch2}(x,t)$ with 40 neurons per
layer combined with a hyperbolic tangent activation function.

Figures 7(a1) and (c1) exhibit the 2D and 3D profiles of the latent solution
$\widehat{u}(x,t)$ learned by a 6-hidden-layer deep PINN $f_{mch2}(x,t)$ with
40 neurons per layer combined with a hyperbolic tangent activation function.
The comparison of learning solution (red dashed line), numerical solution
(blue solid line) and exact solution (green plus) is shown at three different
times $t=0.3,0.8$ and $1.7$ (see Fig. 7(b1)). The $\mathbb{L}_{2}$-norm error
between learning solution $\widehat{u}(x,t)$ and numerical solution is
7.35e-02.

Figure 7: The mCH-CH equation. (a1,a2) The data-driven periodic peakon
solution resulted from the PINN; (b1,b2) The comparisons between the learning,
numerical, and exact periodic peakon solutions at the distinct times (b1)
$t=0.3,\,0.8$, and $1.7$, (b2) $t=0.2,\,0.7$, and $1.5$. The
$\mathbb{L}_{2}$-norm errors $\widehat{u}(x,t)$ between learning and numerical
peakon solutions are (b1) 7.35e-02 and (b2) 7.99e-02; (c1,c2) The 3D profiles
of the learning bright and dark peakon solutions.
$\mu=1,\,c=1.5,\,k_{1}=-0.5,\,k_{2}=2$ for (a1, b1, c1) and
$\mu=-1,\,c=1,\,k_{1}=-1,\,k_{2}=-2$ for (a2, b2, c2).

ii) Similarly, we consider $\mu=-1,\,c=1,\,k_{1}=-1$, and $k_{2}=-2$ in the
initial condition (37), use the pseudo-spectral approach to generate the data-
set for the PINN pertaining to the mCH-CH equation in the spatio-temporal
region $(x,t)\in[-2.6\pi,3.4\pi]\times[0,2]$ with the 640 Fourier modes in the
$x$ direction and time-step $\Delta t=0.004$. The training data-set used in
the PINN consists of randomly chosen $N_{int}=90$ points from the initial data
$u(x,0)$, $N_{end}=0$ points from the end data $u(x,2)$, $N_{b}=20$ points
pertaining to the periodic boundary data, and $N_{f}=6000$ points from the
spatio-temporal solution zone. The hidden solution $\widehat{u}(x,t)$ is
learned by a 6-hidden-layer deep PINN $f_{mch2}(x,t)$ with 40 neurons per
layer combined with a hyperbolic tangent activation function.

Figures 7(a2) and (c2) exhibit the 2D and 3D profiles of the latent solution
$\widehat{u}(x,t)$ learned by a 6-hidden-layer deep PINN $f_{mch2}(x,t)$ with
40 neurons per layer combined with a hyperbolic tangent activation function.
The comparison of learning solution (red dashed line), numerical solution
(blue solid line) and exact solution (green plus) is shown at three different
times $t=0.2,\,0.7$ and $1.5$ (see Fig. 7(b2)). The $\mathbb{L}_{2}$-norm
error between learning solution $\widehat{u}(x,t)$ and numerical solution is
7.99e-02.

Figure 8: The mCH-Novikov equation. (a1,a2) The data-driven peakon solution
resulted from the PINN; (b1,b2) The comparisons between the learning,
numerical, and exact peakon solutions at the distinct times (b1)
$t=0.1,\,0.8$, and $1.5$, (b2) $t=0.1,\,0.8$, and $1.3$. The
$\mathbb{L}_{2}$-norm errors $\widehat{u}(x,t)$ between learning and numerical
peakon solutions are (b1) 6.02e-02 and (b2) 1.31e-02; (c1,c2) The 3D profiles
of the learning bright peakon solutions. $c=0.1,\,k_{1}=-0.25,\,k_{2}=1/3$ for
(a1, b1, c1) and $c=-0.7,\,k_{1}=-0.15,\,k_{2}=-1$ for (a2, b2, c2).

###  3.6 The mCH-Novikov equation with cubic nonlinearity

In the subsection, we use the PINN to consider the data-driven peakon and
periodic peakon solutions of the initial-boundary value problem of the mCH-
Novikov equation (15).

Case 1. The mCH-Novikov equation (15) has the peakon solution [11]
$u(x,t)=\sqrt{3c/(2k_{1}+3k_{2})}e^{-|x-ct|}$, which is a right-going for
$c>0,\,2k_{1}+3k_{2}>0$ or left-going for $c<0,\,2k_{1}+3k_{2}<0$ peaked
travelling wave solution. We here consider the the stationary peakon solution
at $t=0$ as the initial condition

$\displaystyle u(x,0)=\sqrt{3c/(2k_{1}+3k_{2})}e^{-|x|},\quad c>0,$ (38)

and the periodic boundary $u(-L,t)=u(L,t)$. The considered PINN
$f_{mchn}(x,t)$ is written as

$\displaystyle
f_{mchn}(x,t):=u_{t}-u_{xxt}+k_{1}[(u^{2}-u_{x}^{2})(u-u_{xx})]_{x}+k_{2}(4u^{2}u_{x}-3uu_{x}u_{xx}-u^{2}u_{xxx}).$
(39)

i) For $c=0.1,\,k_{1}=-0.25$ and $k_{2}=1/3$ in the initial condition (38),
which is a bright peakon solution, we use the Fourier pseudo-spectral method
(i.e., one can take Fourier transform in space, and choose the explicit
fourth-order Runge-Kutta method in time) to simulate the mCH-Novikov equation
(15) with the initial value condition (38) and periodic boundary condition.
The spatial region is $x\in[-10,\,10]$ with 256 Fourier modes, and the
temporal region is $t\in[0,2]$ with time-step $\Delta t=0.004$. As a result,
we produce the data-set for the PINN pertaining to the mCH-Novikov equation.
The training data-set used in the PINN $f_{mchn}(x,t)$ consists of randomly
chosen $N_{int}=30$ points from the initial data $u(x,0)$, $N_{end}=20$ points
from the end data $u(x,2)$, $N_{b}=20$ points pertaining to the periodic
boundary data, and $N_{f}=1000$ points from the spatio-temporal solution zone.
Figures 8(a1) and (c1) exhibit the 2D and 3D profiles of the latent solution
$\widehat{u}(x,t)$ learned by a 7-hidden-layer deep PINN $f_{mchn}(x,t)$ with
10 neurons per layer combined with a hyperbolic tangent activation function.
The comparison of learning solution (red dashed line), numerical solution
(blue solid line) and exact solution (green plus) is shown at three different
times $t=0.1,\,0.8$, and $1.5$ (see Fig. 8(b1)). The $\mathbb{L}_{2}$-norm
error between learning solution $\widehat{u}(x,t)$ and numerical solution is
6.02e-02.

ii) We here consider $c=-0.7,\,k_{1}=-0.15$ and $k_{2}=-1$ in the initial
condition (38) such that the initial condition is a bright peakon. Similarly,
we choose the spatial region $x\in[-10,\,10]$ with 256 Fourier modes, and the
temporal region is $t\in[0,2]$ with time-step $\Delta t=0.004$. As a result,
we produce the data-set for the PINN pertaining to the mCH-Novikov equation.
The training data-set used in the PINN $f_{mchn}(x,t)$ consists of randomly
chosen $N_{int}=50$ points from the initial data $u(x,0)$, $N_{end}=20$ points
from the end data $u(x,2)$, $N_{b}=20$ points pertaining to the periodic
boundary data, and $N_{f}=1000$ points from the spatio-temporal solution zone.
Figures 8(a2) and (c2) exhibit the 2D and 3D profiles of the latent solution
$\widehat{u}(x,t)$ learned by a 6-hidden-layer deep PINN $f_{mchn}(x,t)$ with
10 neurons per layer combined with a hyperbolic tangent activation function.
The comparison of learning solution (red dashed line), numerical solution
(with blue solid line) and exact solution (green plus) is shown at three
different times $t=0.1,\,0.8$, and $1.3$ (see Fig. 8(b2)). The
$\mathbb{L}_{2}$-norm error between learning solution $\widehat{u}(x,t)$ and
numerical solution is 1.31e-02.

Figure 9: The mCH-Novikov equation. (a) The data-driven periodic peakon
solution resulted from the PINN; (b) The comparisons between the learning,
numerical, and exact periodic peakon solutions at the distinct times $t=0.1$,
$0.8$, and $1.5$. The $\mathbb{L}_{2}$-norm error between the learning and
numerical periodic peakon solutions is 8.23e-02; (c) The 3D profile of the
learning periodic peakon solution.

Case 2. The mCH-Novikov equation (15) can be shown to possess the periodic
peakon solution
$u(x,t)=\sqrt{3c/[k_{1}(2\cosh^{2}\pi+1)+3k_{2}\cosh^{2}\pi]}\cosh(x-ct-2\pi\lfloor(x-ct)/(2\pi)\rfloor-\pi$
with $c>0$. We consider the stationary periodic peakon solution at $t=0$ as
the initial value condition

$\displaystyle
u(x,0)=\sqrt{\frac{3c}{k_{1}(2\cosh^{2}\pi+1)+3k_{2}\cosh^{2}\pi}}\cosh\left(x-2\pi\left\lfloor\frac{x}{2\pi}\right\rfloor-\pi\right),\quad
c>0$ (40)

and the periodic boundary condition $u(L_{1},t)=u(L_{2},t)$, as well as
$c=0.2,\,k_{1}=-0.25,\,k_{2}=1/3$.

Similarly, we use the pseudo-spectral approach to generate the data-set for
the PINN pertaining to the mCH-Novikov equation in the spatio-temporal region
$(x,t)\in[-3\pi,3\pi]\times[0,2]$ with the 512 Fourier modes in the $x$
direction and time-step $\Delta t=0.004$. The training data-set used in the
PINN consists of randomly chosen $N_{int}=60$ points from the initial data
$u(x,0)$, $N_{end}=0$ point from the end data $u(x,2)$, $N_{b}=10$ points
pertaining to the periodic boundary data, and $N_{f}=2000$ points from the
spatio-temporal solution zone. The hidden solution $\widehat{u}(x,t)$ is
learned by a 6-hidden-layer deep PINN $f_{mchn}(x,t)$ with 20 neurons per
layer combined with a hyperbolic tangent activation function.

Figures 9(a) and (c) exhibit the 2D and 3D profiles of the latent solution
$\widehat{u}(x,t)$ learned by a 6-hidden-layer deep PINN $f_{mchn}(x,t)$ with
20 neurons per layer combined with a hyperbolic tangent activation function.
The comparison of learning solution (red dashed line), numerical solution
(blue solid line) and exact solution (green plus) is shown at three different
times $t=0.1,\,0.8$ and $1.5$ (see Fig. 9(b)). The $\mathbb{L}_{2}$-norm error
between learning solution $\widehat{u}(x,t)$ and numerical solution is
8.23e-02.

Figure 10: The generalized mCH equation (11) with $k=2$: (a1,a2) The data-
driven peakon solution resulted from the PINN; (b1, b2) The comparisons
between the learning, numerical, and exact peakon solutions at the distinct
times (b1) $t=0.5,\,0.8,\,1.4$ or (b2) $t=0.1,\,0.4,\,0.7$. The
$\mathbb{L}_{2}$-norm errors $\widehat{u}(x,t)$ between learning and numerical
peakon solutions are (b1) 5.29e-02 and (b2) 3.84e-02; (c1,c2) The 3D profiles
of the learning bright peakon and periodic peakon solutions. $c=0.15$ for (a1,
b1, c1) and $c=0.1$ for (a2, b2, c2).

###  3.7 The generalized mCH equation with quintic nonlinearity

In the subsection, we use the PINN to consider the data-driven peakon and
periodic peakon solutions of the initial-boundary value problem of the
generalized mCH equation (11) with $k=2$.

Case 1. The generalized mCH equation (11) with $k=2$ admits the peakon
solution [41] $u(x,t)=\sqrt[4]{15c/8}\,e^{-|x-ct|}$ with $c>0$. We here
consider the the stationary peakon solution at $t=0$ as the initial condition

$\displaystyle u(x,0)=\sqrt[4]{\frac{15c}{8}}\,e^{-|x|},\quad c>0,$ (41)

and the periodic boundary $u(-L,t)=u(L,t)$. The considered PINN
$f_{gmch}(x,t)$ is written as

$\displaystyle
f_{gmch}(x,t):=u_{t}-u_{xxt}+[(u^{2}-u_{x}^{2})^{2}(u-u_{xx})]_{x}.$ (42)

We here consider $c=0.15$ in the initial condition (43), and use the Fourier
pseudo-spectral method (i.e., one can take Fourier transform in space, and
choose the explicit fourth-order Runge-Kutta method in time) to simulate the
generalized mCH equation (11) with $k=2$ and the initial value condition (43)
and periodic boundary condition. The spatial region is $x\in[-10,\,10]$ with
640 Fourier modes, and the temporal region is $t\in[0,1.5]$ with time-step
$\Delta t=0.003$. As a result, we produce the data-set for the PINN pertaining
to the generalized mCH equation. The training data-set used in the PINN
$f_{gmch}(x,t)$ consists of randomly chosen $N_{int}=30$ points from the
initial data $u(x,0)$, $N_{end}=10$ points from the end data $u(x,1.5)$,
$N_{b}=10$ points pertaining to the periodic boundary data, and $N_{f}=2000$
points from the spatio-temporal solution zone. Figures 10(a1) and (c1) exhibit
the 2D and 3D profiles of the latent solution $\widehat{u}(x,t)$ learned by a
5-hidden-layer deep PINN $f_{gmch}(x,t)$ with 10 neurons per layer combined
with a hyperbolic tangent activation function. The comparison of learning
solution (red dashed line), numerical solution (blue solid line) and exact
solution (green plus) is shown at three different times $t=0.5,0.8$, and $1.4$
(see Fig. 10(b1)). The $\mathbb{L}_{2}$-norm error between learning solution
$\widehat{u}(x,t)$ and numerical solution is 5.29e-02.

Case 2. The generalized mCH equation (11) with $k=2$ possesses the periodic
peakon solution $u(x,t)=\sqrt[4]{15c\,{\rm
csch}\pi/(8\cosh^{4}\pi+4\cosh^{2}\pi+3)}\cosh(x-ct-2\pi\lfloor(x-ct)/(2\pi)\rfloor-\pi)$
with $c>0$. We consider the stationary periodic peakon solution at $t=0$ as
the initial value condition

$\displaystyle u(x,0)=\sqrt[4]{\frac{15c\,{\rm
csch}\pi}{8\cosh^{4}\pi+4\cosh^{2}\pi+3}}\cosh\left(x-2\pi\left\lfloor\frac{x}{2\pi}\right\rfloor-\pi\right),\quad
c>0$ (43)

and the periodic boundary condition $u(-L,t)=u(L,t)$, as well as $c=0.1$.

Similarly, we use the pseudo-spectral approach to generate the data-set for
the PINN pertaining to the generalized mCH equation (11) with $k=2$ in the
spatio-temporal region $(x,t)\in[-3\pi,3\pi]\times[0,1]$ with the 512 Fourier
modes in the $x$ direction and time-step $\Delta t=0.002$. The training data-
set used in the PINN consists of randomly chosen $N_{int}=60$ points from the
initial data $u(x,0)$, $N_{end}=0$ point from the end data $u(x,2)$,
$N_{b}=10$ points pertaining to the periodic boundary data, and $N_{f}=2000$
points from the spatio-temporal solution zone. The hidden solution
$\widehat{u}(x,t)$ is learned by a 6-hidden-layer deep PINN $f_{gmch}(x,t)$
with 20 neurons per layer combined with a hyperbolic tangent activation
function.

Figures 10(a2) and (c2) exhibit the 2D and 3D profiles of the latent solution
$\widehat{u}(x,t)$ learned by a 6-hidden-layer deep PINN $f_{gmch}(x,t)$ with
20 neurons per layer combined with a hyperbolic tangent activation function.
The comparison of learning solution (red dashed line), numerical solution
(blue solid line) and exact solution (green plus) is shown at three different
times $t=0.1,\,0.4$ and $0.7$ (see Fig. 10(b2)). The $\mathbb{L}_{2}$-norm
error between learning solution $\widehat{u}(x,t)$ and numerical solution is
3.84e-02.

##  4 Conclusions and discussions

In summary, we have applied the PINN deep learning to successfully explore the
data-driven peakon and periodic peakon solutions of some nonlinear dispersive
equations with some initial-boundary value conditions such as the Camassa-Holm
(CH) equation, Degasperis-Procesi equation, modified CH equation, $b$-family
equation, Novikov equation, mCH-Novikov equation, and etc. Moreover, the PINN
deep learning approach can also be extended to other nonlinear peakon
equations given by Eqs. (16) and (17), and higher-dimensional coupled
nonlinear dispersive equations with peakon solutions. These questions will be
studied in another literature.

Acknowledgements

This work is partially supported by the NSFC under Grant Nos. 11731014 and
11925108.

Appendix A. The DP equation

In the appendix, we use the PINN deep learning approach to consider the data-
driven peakon and periodic peakon solutions of the initial-boundary value
problem of the DP equation (7). Though the CH equation and DP equation both
admit the same peakon and periodic peakon solutions, but their Lax pairs are
different, where the CH equation has the second-order Lax pair [2], and the DP
equation admits the third-order Lax pair [5].

Case 1. Since the DP equation (7) also admit the peakon solution [5]
$u(x,t)=ce^{-|x-ct|}$, thus we consider the initial value condition (23) and
periodic boundary condition $u(-L,t)=u(L,t)$. The considered PINN
$f_{dp}(x,t)$ is chosen as

$\displaystyle f_{dp}(x,t):=u_{t}-u_{xxt}+4uu_{x}-3u_{x}u_{xx}-uu_{xxx}.$ (44)

i) We take $c=0.6$ in the initial condition (23) and use the pseudo-spectral
method to generate the data-set for the PINN pertaining to the DP equation (7)
in the spatio-temporal region $(x,t)\in[-5,5]\times[0,3]$ with the 256 Fourier
modes in the $x$ direction and time-step $\Delta t=0.006$. The training data-
set used in the deep PINN $f_{dp}(x,t)$ consists of randomly chosen
$N_{int}=10$ points from the initial data $u(x,0)$, $N_{end}=10$ points from
the end data $u(x,3)$, $N_{b}=10$ points pertaining to the periodic boundary
data, and $N_{f}=1000$ points from the spatial-temporal solution zone. The
hidden solution $\widehat{u}(x,t)$ can be trained by the the 5-hidden-layer
deep PINN $f_{dp}(x,t)$ with 10 neurons per layer and a hyperbolic tangent
activation function.

Figures 11(a1) and (c1) exhibit the 2D and 3D profiles of the latent bright-
peakon solution $\widehat{u}(x,t)$. The comparison of learning solution (red
dashed line), numerical solution (blue solid line) and exact solution (green
plus) is shown at three different times $t=0.3,\,1.5$, and $2$ (see Fig.
11(b1)). The $\mathbb{L}_{2}$-norm error between learning solution
$\widehat{u}(x,t)$ and numerical solution is 2.74e-02.

Figure 11: The DP equation. (a1,a2) The data-driven peakon solutions resulted
from the PINN; (b1,b2) The comparisons between the learning, numerical, and
exact peakon solutions at the distinct times (a1) $t=0.3,\,1.5$, and $2$, (a2)
$t=0.4,\,1.3$, and $2.4$; The $\mathbb{L}_{2}$-norm errors $\widehat{u}(x,t)$
between learning and numerical peakon solutions are (b1) 2.74e-02 and (b2)
3.29e-02; (c1,c2) The 3D profiles of the learning bright and dark peakon
solutions. $c=0.6$ for (a1, b1, c1) and $c=-1.05$ for (a2, b2, c2).

Figure 12: The DP equation. (a) The data-driven periodic peakon solution
resulted from the PINN, and three distinct tested times; (b) The comparisons
between the learning, numerical, and exact periodic peakon solutions at the
distinct times $t=0.4$, $0.9$, and $1.9$. The $\mathbb{L}_{2}$-norm error
between the learning and numerical periodic peakon solutions is 1.96e-02; (c)
The 3D profile of the learning periodic peakon solution.

ii) For the case $c=-1.05$ in the initial condition (23), we consider the
spatio-temporal region $(x,t)\in[-7,7]\times[0,3]$ with 256 Fourier modes and
time-step $\Delta t=0.006$. The training data-set used in the 5-hidden-layer
deep PINN $f_{dp}(x,t)$ with 20 neurons per layer consists of randomly chosen
$N_{int}=10$ points from the initial data $u(x,0)$, $N_{end}=10$ points from
the end data $u(x,3)$, $N_{b}=10$ points pertaining to the periodic boundary
data, and $N_{f}=1000$ points from the spatial-temporal solution zone. Figures
11(a2) and (c2) exhibit the 2D and 3D profiles of the latent solution
$\widehat{u}(x,t)$. The comparison of learning solution (red dashed line),
numerical solution (blue solid line) and exact solution (green plus) is shown
at three different times $t=0.4,1.3$ and $2.4$ (see Fig. 11(b2)). The
$\mathbb{L}_{2}$-norm error between learning solution $\widehat{u}(x,t)$ and
numerical solution is 3.29e-02.

Case 2. Since the DP equation (7) admit the same periodic peakon solution as
one of the CH equation, thus we consider the initial value condition (25) with
$c=-0.5$ and periodic boundary conditions. Similarly, The considered spatio-
temporal region in the pseudo-spectral method is
$(x,t)\in[-16\pi/5,14\pi/5]\times[0,2]$ with 512 Fourier modes and time-step
$\Delta t=0.004$. The training data-set used in the 6-hidden-layer deep PINN
$f_{dp}(x,t)$ with 20 neurons per layer consists of randomly chosen
$N_{int}=90$ points from the initial data $u(x,0)$, $N_{end}=10$ points from
the end data $u(x,2)$, $N_{b}=10$ points pertaining to the periodic boundary
data, and $N_{f}=1000$ points from the spatial-temporal solution zone by
minimizing the MSE loss (19).

Figures 12(a) and (c) exhibit the 2D and 3D profiles of the latent periodic
peakon solution $\widehat{u}(x,t)$. The comparison of learning solution (red
dashed line), numerical solution (with blue solid line) and exact solution
(green plus) is shown at three different times $t=0.4,\,0.9$, and $1.9$ (see
Fig. 12(b)). The $\mathbb{L}_{2}$-norm error between learning solution
$\widehat{u}(x,t)$ and numerical solution is 1.96e-02.

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