Two-dimensional Klein-Gordon and Sine-Gordon numerical solutions based on deep neural network

IAES International Journal of Artificial Intelligence (IJ-AI)
Vol. 14, No. 2, April 2025, pp. 1548∼1560
ISSN: 2252-8938, DOI: 10.11591/ijai.v14.i2.pp1548-1560 ❒ 1548
Two-dimensional Klein-Gordon and Sine-Gordon
numerical solutions based on deep neural network
Soumaya Nouna1
, Assia Nouna1
, Mohamed Mansouri1
, Ilyas Tammouch2
, Boujamaa Achchab1
1Laboratory LAMSAD, ENSA Berrechid, Hassan First University, Settat, Morocco
2Laboratory of Telecommunications Systems and Decision Engineering, Faculty of Science, Ibn Tofail University, Kenitra, Morocco
Article Info
Article history:
Received Feb 25, 2024
Revised Oct 30, 2024
Accepted Nov 14, 2024
Keywords:
Deep learning
Feedforward neural network
Nonlinear Klein-Gordon
equations
Partial differential equations
Sine-Gordon equations
ABSTRACT
Due to the well-known dimensionality curse, developing effective numerical
techniques to resolve partial differential equations proved a complex problem.
We propose a deep learning technique for solving these problems. Feedforward
neural networks (FNNs) use to approximate a partial differential equation with
more robust and weaker boundaries and initial conditions. The framework called
PyDEns could handle calculation fields that are not regular. Numerical exper-
iments on two-dimensional Sine-Gordon and Klein-Gordon systems show the
provided frameworks to be sufficiently accurate.
This is an open access article under the CC BY-SA license.
Corresponding Author:
Soumaya Nouna
Laboratory LAMSAD, ENSA Berrechid, Hassan First University
Km. 10 Idiroko Road, Canaan Land, Ota, Ogun Settat, Morocco
Email: s.nouna@uhp.ac.ma
1. INTRODUCTION
Partial differential equations (PDEs) can be used to simulate nonlinear processes seen in many
scientific domains, including plasma physics, the physics of solids, chemical kinetics, the dynamics of flu-
ids, hydrodynamics, and the biology of mathematics. Many numerical and analytical solutions have been em-
ployed to solve these issues. Various studies of phenomena in physics across multiple domains of engineering
and science have recently been conducted [1], [2]. This research focuses on the Klein-Gordon and Sine-Gordon
systems, which are significant families of PDEs used to explain many diverse physical phenomena.
The Sine-Gordon system [3] was developed in the 19th century during research into various
differential geometry issues. It is an intermediate hyperbolic differential equation widely applied in describing
and modeling physical phenomena in various engineering and scientific fields, including fluxions propagation,
nonlinear waves, and metal dislocation. This equation has garnered significant interest due to its numerous
soliton solutions [4]. Similarly, the Klein-Gordon equation [5] has also received considerable attention for
studying solitons, soliton interactions in collisionless plasmas, and nonlinear wave systems [6]. The Klein-
Gordon equation represents a motion equation for a scalar or pseudoscalar field, significant for several scientific
applications, including nonlinear optics theory and solid-state physics.
Various numerical approaches have been proposed to solve Sine-Gordon and Klein-Gordon systems,
such as the exp-function model [7], the adomian decomposition method [8], [9], the reduced differential trans-
form approach [10], [11], the variational iteration approach [12], [13], the homotopy perturbation approach
[14], and the homotopy analysis approach [15], [16]. These traditional numerical approaches use a discrete
domain over several finite elements or points, requiring finer meshes for better solutions. This strategy’s com-
Journal homepage: http://ijai.iaescore.com

Int J Artif Intell ISSN: 2252-8938 ❒ 1549
puting cost can be greatly increased, hindering its use for highly-dimensional systems. To address these chal-
lenges, many deep learning methods have been proposed for solving high-dimensional problems.
In machine learning, deep learning methods use multiple hidden layers in neural networks. The benefit
of adding hidden layers within the neural network process is that, for certain implementations, computing costs
are reduced exponentially, as is the amount of data required for training [17]. Consequently, these networks
can, in principle, find any solutions to differential equations within low error boundaries. Extensive research
has been conducted on applying neural networks to solve differential equations. For example, the research in
[18], [19] used artificial neural networks and unsupervised neural networks for resolving differential systems.
Other works have also applied deep learning methods or deep neural networks (DNNs) to solve PDEs [20]–[22],
demonstrating that deep learning-based approaches effectively solve mathematical models, including high-
dimensional PDEs.
This paper uses a deep learning technique to resolve the 2-dimensional Sine-Gordon and Klein-
Gordon equations. This research is partly motivated by the concepts presented in [23], where feedforward
neural networks (FNNs) are employed to solve partial differential systems under both boundary and initial con-
ditions. The rest of this paper is structured as follows: section 2 discusses the theory behind FNNs, section 3
introduces our approach to solving PDEs using these networks, and section 4 presents numerical simulations.
Finally, section 5 concludes the article.
2. DEEP LEARNING: FEEDFORWARD NEURAL NETWORK
FNNs [24] are the most basic form of artificial neural network and have many applications in machine
learning. FNNs are also called multilayer neuron networks (MLNs). These model networks are known as
feedforward the information in the neural network propagates only forward, first via the input nodes, then
through the single or many hidden layers, and ultimately through the output nodes. FNNs appear to be a
potential method for solving PDEs. Figure 1 shows an example of an FNN with two hidden layers.
X1
X2
.
.
.
Hidden
Layer
.
.
.
Hidden
Layer
Ỹ
Input
layer
Output
layer
Figure 1. FNN with two hidden layers
Assume an FNN consisting of (M + 1) layers, wherein 0 represents an entry layer, and M represents
an exit layer. Hidden layers consist of the intermediary layers between an input and output layer. Rm neurons
are found in the m-th layer, m = 1, 2, ..., M. Specifically, RM = 1, indicating that the exit layer contains just
a single neuron. Also, every node of the current layer m takes as input each output of layer m − 1 and passes
its proper output onto every node of the following layer. A weighted entry at every node i, i = 1, 2, ..., Rm, in
Two-dimensional Klein-Gordon and Sine-Gordon numerical solutions based on ... (Soumaya Nouna)

1550 ❒ ISSN: 2252-8938
layer m, for m = 1, 2, ..., M, is provided by the recursive formula:
km
i =
Rm−1
X
s=1
wm
is φm−1(km−1
s ) + bm
i (1)
Where φm represents the activation function of layer m, bm
i denotes a bias of node i in layer m and a weight
between node s in layer m − 1 and node i in layer m is wm
is . For clarity, we replace R0 with r. If the vector
input is X = (X1, X2, ..., Xr) ∈ Rr
, so:
φ0(k0
s) = Xs, s = 1, 2, ..., r (2)
Therefore, a neural network’s output is provided by:
Ỹ = N(X, q) = kM
(X, q) (3)
Where q ∈ Rrq
is a vector of all tunable parameters wm
is and bm
i , where rq represents the number of total
parameters, and kM
comes from the (1). We assumed in (1) a unique activating functional utilized inside m-th
layer with m = 1, 2, ...M − 1.
Giving an entry or a collection of entries, a node’s activation function defines the node’s output. A
neural network would be a linear transformation without the activation function. Also, any neural network
having one hidden layer with a sufficient number of neurons can approximate every function continuously
according to Theorem 2..1. There are many types of activation functions [25], such as the Relu function, which
is defined as follows:
φ(X) = max (0, X) (4)
The Relu activation can also be employed for second or higher-order PDEs. The purpose of FNN
training is to minimize a loss function. Lesser the loss, the better the match between the analytical and predicted
solution. We use optimizer algorithms to determine the best weights and collection biases to minimize the loss
function. There are various optimization approaches, including Broyden-Fletcher-Goldfarb-Shanno (BFGS)
[26], Levenberg-Marquardt [27], limited-memory Broyden-Fletcher-Goldfarb-Shanno (LBFGS) [28], adaptive
moment estimation (ADAM) [29], and stochastic gradient descent (SGD) [30].
Theorem 2..1 (Theorem of universal approximation) The neural networks having the Relu activation func-
tion are the universal approximators. Note that universal approximators are defined in the Definition 2..1.
Definition 2..1 Suppose Ω is a topology space with φ : R → R. We state by saying that any neural network
having activating functions φ would be a universal approximator on Ω only if
P
n(φ) would be dense on C(Ω).
Where C(Ω) is all the continuous functions from Ω through to R and the collection
P
n is all tasks that may be
computed using a neural network with one hidden layer with activating functions φ.
3. DESCRIPTION OF THE APPROACH
This study uses the FNN method to solve PDEs. The approach includes reformulating the problem into
a loss function incorporating the initial and boundary conditions. This loss function is then minimized using
advanced optimization techniques. FNNs are particularly well suited to this task because of their universal
approximation capability [31], as demonstrated in the Theorem 3..1. This property allows neural networks
to accurately approximate any continuous function, which is particularly advantageous for solving complex
PDEs without resorting to traditional discretization methods. This section describes in detail the experimental
procedure, the neural network configurations, the validation criteria, and the optimization techniques employed.
Assume that the first non-linear system is described as follows:
∂2
v(x, y, t)
∂t2
+ λ
∂v(x, y, t)
∂t
= β
∂2
v(x, y, t)
∂x2
+
∂2
v(x, y, t)
∂y2
!
− f(x, y) sin(v(x, y, t))
+ g(x, y, t), (x, y) ∈ Ω1, t ≥ 0
(5)
Int J Artif Intell, Vol. 14, No. 2, April 2025: 1548–1560

According to the next initial conditions:
F(x, y)
(
v(x, y, 0) = ϕ1(x, y), (x, y) ∈ Ω1
∂v
∂t
(x, y, 0) = ϕ2(x, y), (x, y) ∈ Ω1
(6)
And the second non-linear system is described as:
∂2
v(x, y, t)
∂t2
+ α
∂v(x, y, t)
∂t
= γ
∂2
v(x, y, t)
∂x2
+
∂2
v(x, y, t)
∂y2
!
−

µ1v(x, y, t) + µ2vn
(x, y, t)

+ h(x, y, t), (x, y) ∈ Ω2, t 0
(7)
Depending on the initial conditions:
G(x, y)
(
v(x, y, 0) = ψ1(x, y), (x, y) ∈ Ω2
∂v
∂t
(x, y, 0) = ψ2(x, y), (x, y) ∈ Ω2
(8)
Where (5) is known as a nonlinear Sine-Gordon system in two dimensions, with Ω1 ∈ R2
, f(x, y) may be
thought of as just a Josephson density, with ϕ1(x, y), and ϕ2(x, y) representing wave patterns as well as curves
and acceleration, correspondingly. In (7) is named the two-dimensional Klein-Gordon equation which has a
fourfold non-linearity when n = 2 and a cubical non-linearity when n = 3, Ω2 ∈ R2
, and h(x, y, t) seems to be
a known function, ψ1(x, y) and ψ2(x, y) are starting functions. Parameters λ, β, α, µ1, µ2 and γ are constants.
We applied the FNN technique to find a functional vF NN which approaches the real solution v. This
method minimizes the loss functions associated with the two systems, defined as follows:
Ĵ(q) =
∂2
vF NN (x, y, t; q)
∂t2
+ λ
∂vF NN (x, y, t; q)
∂t
−β
∂2
vF NN (x, y, t; q)
∂x2
+
∂2
vF NN (x, y, t; q)
∂y2
!
+f(x, y) sin(vF NN (x, y, t; q)) − g(x, y, t)
2
(x,y)∈Ω1,t≥0,ξ1
+ vF NN (x, y, 0; q) − F(x, y)
2
(x,y)∈Ω1,ξ2
(9)
L̂(q) =
∂2
vF NN (x, y, t; q)
∂t2
+ α
∂vF NN (x, y, t; q)
∂t
−γ
∂2
vF NN (x, y, t; q)
∂x2
+
∂2
vF NN (x, y, t; q)
∂y2
!
+

µ1vF NN (x, y, t; q) − µ2vn
F NN (x, y, t; q)

−h(x, y, t)
2
(x,y)∈Ω2,t0,ξ1
+ vF NN (x, y, 0; q) − G(x, y)
2
(x,y)∈Ω2,ξ2
(10)
In (9) represents the loss function of (5) and (6), whereas in (10) represents the loss function of (7) and (8).
ξ1, ξ2 design the measurements. q represents a neural network’s vector of parameters weights and biases. The
approximate solution vF NN , used in the equations and loss functions, is generated by the output of the FNN
described in the previous section (see in (3)) and is expressed as (11):
vF NN = kM
(x, y, t; q). (11)

1552 ❒ ISSN: 2252-8938
Our approach uses FNNs to approximate the solutions of Sine-Gordon and Klein-Gordon PDEs. First,
we transform these problems into loss functions incorporating initial and boundary conditions, which allows
optimization techniques to be used to train the neural networks. The backpropagation method, detailed in [32],
is used to determine the derivatives of the function vF NN concerning the network inputs. TensorFlow [33]
offers an automatic differentiation package in inverse mode, capable of calculating each differential of the loss
function and producing the necessary gradients.
Next, we apply SGD to minimize the loss functions Ĵ and L̂, thereby optimizing the neural network
parameters. The Algorithms 1 and 2 describe in detail the specific steps of this procedure for the Sine-Gordon
and Klein-Gordon equations respectively, taking into account the particularities of each equation.
Algorithm 1 The deep neural network algorithm for nonlinear Sine-Gordon equation
1. Determine the FNN model (layers, neurons, and the activation function)
2. Initializing FNN parametric q0
3. Process FNN (ξ1, ξ2)
4. Repeating
5. Generating c1 and c2 points from Ω1 × [0, T] and Ω1 × {0}. Get distributed ξ1, ξ2.
6. Utilizing sampling lots, approximate the loss (9):
Ĵ(q) =
X
(xj ,yj ,tj )∈c1
∂2
vF NN (xj, yj, tj; q)
∂t2
+ λ
∂vF NN (xj, yj, tj; q)
∂t
− β
∂2
∂x2
+
∂2
∂y2
!
+ f(xj, yj) sin(vF NN (xj, yj, tj; q)) − g(xj, yj, tj)
!2
+
X
(xj ,yj ,0)∈c2
vF NN (xj, yj, 0; q) − F(xj, yj)
!2
7. Consider taking a step of descending gradient:
qm+1 = qm − δm∇qĴ(qm)
8. Until its convergence.
9. Final Process
Theorem 3..1 Consider fN : [0, 1]r
→ [0, 1] as the function computed using a single-layer Relu neural
network with r inputs, one output, and s Hidden Nodes. Thus, there also existed an alternative Relu neural
network, having m Hidden Layers with a width of r + 2, that computes the identical function fN .
The theorem’s proof utilizes an approach that uses r nodes of every layer for storing the first input,
one for computing each node of an originally constructed network, and the final node for storing the result of
summing functions.
Proof. As fN can be produced using a single-layer Relu network having r hidden nodes, it takes the following
expression:
fN (X) = φ(
m
X
j=1
wjφ(lj(X)) + b)
Wherein every lj has a format of lj = Yj · X + bj where Yj ∈ Rr
, bj ∈ R and φ is the Relu activation
function. As [0, 1]r
is a compact function, and Relu network-generated function is continuous. The term
Pj
i=1 wiφ(li(X)) is at a minimum for every j ⩽ m, so there is a number S 0 like that:

S +
j
X
i=1
wiφ(li(X)) 0
For all X ∈ [0, 1]r
as well as all j ⩽ m.
We now have a novel network where every hidden layer utilizes r nodes for copying the originally
entered. Furthermore, every j-th hidden layer also has an additional node which calculates function φ(lj(X))
based on the input nodes that were replicated in the preceding layer, plus a final node which calculates:
φ(S +
j−1
X
i=1
wiφ(li(X))) = S +
j−1
X
i=1
wiφ(li(X))
By making the linear composition of both supplementary nodes from the preceding layer. Briefly, the j-th layer
until j ⩽ m shall calculate the following function:
Hj
(X) = (X1, ..., Xr, φ(lj(X)), φ(S +
j−1
X
i=1
wiφ(li(X)))).
Finally, the last layer calculates:
Hm+1
(X) = φ

wmφ(lm(X)) + φ(S +
Pm−1
i=1 wiφ(li(X)) − S + b)

= φ(b +
Pm
j=1 wjφ(lj(X)))
= fN (X)
That finishes this proof with the use of m hidden layers, every one of width r + 2.
Algorithm 2 The deep neural network algorithm for nonlinear Klein-Gordon equation
1. Determine the FNN model (layers, neurons, and the activation function)
2. Initializing FNN parametric q0
3. Process FNN (ξ1, ξ2)
4. Repeating
5. Generating d1 and d2 points from Ω2 × [0, T] and Ω2 × {0}. Get distributed ξ1, ξ2.
6. Utilizing sampling lots, approximate the loss (10):
L̂(q) =
X
(xj ,yj ,tj )∈d1
∂2
∂t2
+ α
∂vF NN (xj, yj, tj; q)
∂t
− γ
∂2
∂x2
+
∂2
∂y2
!
+

µ1vF NN (xj, yj, tj; q) − µ2vn
F NN (xj, yj, tj; q)

− h(xj, yj, tj)
!2
+
X
(xj ,yj ,0)∈d2
vF NN (xj, yj, 0; q) − G(xj, yj)
!2
7. Consider taking a step of descending gradient:
qm+1 = qm − δm∇qĴ(qm)
8. Until its convergence.
9. Final process

1554 ❒ ISSN: 2252-8938
4. NUMERICAL RESULTS
In this section, we apply the method described above to test examples associated with nonlinear Sine-
Gordon and Klein-Gordon systems in two-dimensional space. The approximate solutions obtained by FNNs,
implemented with the TensorFlow open-source framework, are compared with the exact solutions. Each ex-
periment was performed with a learning rate of α = 0.005 and using the hyperbolic tangent activation function
(Tanh). The results show the effectiveness of our approach in finding accurate solutions to the PDEs under
consideration.
4.1. Problem 1
Let us examine the inhomogeneous bidimensional Sine-Gordon problem as follows:
∂2
v(x, y, t)
∂t2
=
∂2
v(x, y, t)
∂x2
+
∂2
v(x, y, t)
∂y2
− sin(v(x, y, t))
+ sin(sin(x + y + t)) + sin(x + y + t), (x, y) ∈ Ω1, t ∈ [0, 1]
(12)
Under the following initial conditions:
(
v(x, y, 0) = sin(x + y), (x, y) ∈ Ω1
∂v
∂t
(x, y, 0) = cos(x + y), (x, y) ∈ Ω1
(13)
With Ω1 = [0, 4] × [0, 4], for which the analytical solution to this problem can be defined as follows:
v(x, y, t) = sin(x + y + t) (14)
Firstly, the FNN, named v̂(x, y, t) = vF NN (x, y, t), was constructed for approximating a solution of
(12) and (13). In terms of the network structure, a neural network is composed of five hidden layers, the first
one containing 40 neurons, the second one containing 50 neurons, the third one containing 60 neurons, the
fourth one containing 70 neurons, and the fifth one containing 80 Neurons. Then the following step consists of
training the neural network parameters. We continuously optimize these parameters through minimization of
the loss (9) generated based on the information from initial conditions. In addition, we choose the space-time
dimension data for training a neural network. Spatio-temporal domain training data are randomly selected, and
the quantity is 100 training data for each space and time domain.
Figure 2(a) presents the FNN model prediction derived by entrainment, and it is observed the resulting
prediction is almost the same as the exact solution. Several moments are chosen for comparison between the ap-
proximated and the correct solution to evaluate the prediction accuracy. Figure 2(b) compares the approximate
solution with the exact solution for different times t = 0.1 and t = 0.9. As a result, Figure 2(b) demonstrates
that the FNN approaches and the exact solutions are significantly consistent, suggesting that the neural network
models constructed have a high capacity for solving PDEs. Furthermore, this example’s relative error has been
computed as equal to 10−4
, which confirms the performance of this approach. These results show that our
method offers comparable or even better accuracy than the approaches mentioned above, demonstrating the
significant value of our work.
4.2. Problem 2
Here we have the Sine-Gordon model, which can be used as a bi-dimensional equation:
∂2
v(x, y, t)
∂t2
=
∂2
v(x, y, t)
∂x2
+
∂2
v(x, y, t)
∂y2
− sin(v(x, y, t)), (x, y) ∈ Ω1, t ∈ [0, 1] (15)
Depending on initial conditions:



v(x, y, 0) = 4 tan−1
(exp(x + y)), (x, y) ∈ Ω1
∂v
∂t
(x, y, 0) = −
4 exp(x + y)
1 + exp(2x + 2y)
, (x, y) ∈ Ω1
(16)
Thus, Ω1 = [−2, 2] × [−2, 2], and the correct solution to the above problem is given as:
v(x, y, t) = 4 tan−1
(exp(x + y − t)) (17)

x
0
1
2
3
4
y
0
1
2
3
4
v
(
x
,
y
,
t
)
−0.5
0.0
0.5
Analytical Solution
x
0
1
2
3
4
y
0
1
2
3
4
̂
v
(
x
,
y
,
t
)
−0.5
0.0
0.5
FNN Solution
(a)
(b)
Figure 2. Comparison of analytic solution and FNN solution for the two-dimensional Sine-Gordon equation:
(a) analytical and approximated solution and (b) comparing the analytical and approximated solutions
We have built a FNN for approximating the solution of (15) and (16), denoted as v̂(x, y, t). For the
analysis of a neural network, we have used the same way as problem 1, in which we have to build an FNN
with five hidden layers with the same nodes used in the previous problem. Then, we trained the neural network
parameters. We continuously optimize these parameters while minimizing a loss (9), which is generated based
on the information from the initial conditions. Furthermore, as in the previous experiment, we select the spatial-
temporal dimension data for training an FNN. Also, spatiotemporal domain training data are randomly selected,
and the quantity is 100 training data for each spatial as well as the temporal domain.
In Figure 3(a), we can observe a prediction by the FNN model, where we can see that the approximated
solution is similar to the analytical solution. Different time points from the prediction outputs are selected for
comparison against the analytical solution to verify the accuracy of this approximation. Figure 3(b) shows com-
parisons between the accurate solution and the approximate solution for the time points t = 0.2 and t = 0.5.
This figure shows that the neural network approximations and the analytical solutions are very consistent, in-
dicating that our model can successfully solve the two-dimensional Sine-Gordon equation. Finally, the relative
error of our approach was calculated to be 10−4
, validating its performance. These results show that our method
offers comparable or better accuracy than existing approaches, demonstrating the significant value of our work.
4.3. Problem 3
The following equation is a bi-dimensional nonlinear Klein-Gordon equation (7) in which n = 2,
α = 0, γ = 1, µ1 = 0, and µ2 = 1:
∂2
v(x, y, t)
∂t2
=
∂2
v(x, y, t)
∂x2
+
∂2
v(x, y, t)
∂y2
−v2
(x, y, t) − xy cos(t) + x2
y2
cos2
(t), (x, y) ∈ Ω2, t 0
(18)

1556 ❒ ISSN: 2252-8938
Under the next initial conditions:
(
v(x, y, 0) = xy, (x, y) ∈ Ω1
∂v
∂t
(x, y, 0) = 0, (x, y) ∈ Ω1
(19)
With Ω1 = [0, 4] × [0, 4], in which the correct solution for this problem could be given in the following form:
v(x, y, t) = xy cos(t) (20)
The FNN has been designed for approximating a solution of the (19), (20), indicated by the value
v̂(x, y, t). A neural network is composed of six hidden layers. The first one contains 50 nodes, the second one
contains 60 nodes, the third one contains 70 nodes, the fourth one contains 70 nodes, the fifth one contains
80 nodes, and the sixth one containing 90 nodes. We optimize these parameters continuously by minimizing a
loss (10), which is generated based on information coming from initial conditions. The space-time field training
data are randomly chosen, and the quantity of it is 100 training data for each space and time field.
Figure 4(a) represents the approximations of a Klein-Gordon system produced using FNNs, in which
the approximated solution looks precisely like the analytical solution. Figure 4(b) compares the approximation
and the analytical solution for different time points t = 0.5 and t = 0.9 to test the performance of the model.
This figure shows that an approximate solution is similar to an analytical solution, confirming that our neural
network method can successfully simulate the Klein-Gordon equations. In addition, the relative error was
calculated to be 10−4
, confirming the effectiveness of our approach. These results show that our method offers
comparable or better accuracy than existing approaches, demonstrating the significant value of our work.
(a)
(b)
Figure 3. Visualization of numerical and analytical solutions of the Sine-Gordon equation at various times:

(a)
(b)
Figure 4. Comparison of numerical and analytical solutions for the two-dimensional Klein-Gordon equation:
4.4. Problem 4
Let’s considered the bidimensional nonlinear Klein-Gordon equation (7) in which n = 3, α = 0,
γ = 1, µ1 = 0, and µ2 = 1:
∂2
v(x, y, t)
∂t2
=
∂2
v(x, y, t)
∂x2
+
∂2
v(x, y, t)
∂y2
− v3
(x, y, t)
+ cos(x) cos(y) cos(t) + cos3
(x) sin3
(y) sin3
(t), (x, y) ∈ Ω2, t 0
(21)
Under the following initial conditions:
(
v(x, y, 0) = 0, (x, y) ∈ Ω1
∂v
∂t
(x, y, 0) = cos(x) cos(y), (x, y) ∈ Ω1
(22)
With Ω1 = [0, 4]×[0, 4], in which the analytical solution for this problem could be given in the following form:
v(x, y, t) = cos(x) cos(y) sin(t) (23)
We have constructed the FNN for approximating the solution of (21) and (22), nominated by v̂(x, y, t).
To analyze the neural network, we have used the same procedure as in problem 3, i.e. we have to build a six-
hidden layer neural network of the same nodes used for a preceding problem. We continuously optimize these
parameters by minimizing a loss (10), which is generated based on the information from the initial conditions.
In addition, training data the space and time domains are randomly selected, and the amount of 100 of training
data for each space-time domain.
In Figure 5(a), an approach of an FNN is illustrated, where we can observe that the approximated
solution is identical to the exact resolution. Hence, we pick several instances in the prediction outputs for com-
parison to the accurate solution for testing the precision and accuracy of this approximation. Figure 5(b) shows
the difference between an analytical solution and an approximate solution for different times t = 0.7 and t = 1.
This figure shows that the FNN approximations and the precise solutions are significantly consistent, indicating
that our model can solve this equation successfully. For this problem, the relative error was calculated to be

1558 ❒ ISSN: 2252-8938
10−4
, confirming the effectiveness of our approach. These results show that our method offers comparable or
better accuracy than existing approaches, even for complex equations, demonstrating the power and usefulness
of our approach and justifying further research.
(a)
(b)
Figure 5. Numerical and exact solutions for the Klein-Gordon equation with cubic non-linearity: (a) analytical
and approximated solution and (b) comparing the analytical and approximated solutions
4.5. Discussion and comparison
Our study investigated the use of FNNs for approximating solutions of PDEs, filling a gap in previous
research that has not explicitly addressed their influence on accuracy and efficiency in complex spatiotemporal
contexts. By comparing our results with those of existing traditional methods, we have shown that increasing
the number of nodes in hidden layers improves accuracy without compromising computational efficiency. Al-
though this study validated the approach for specific PDEs, future research could extend its application to even
more complex and high-dimensional equations. In conclusion, our results highlight the significant potential
of neural networks for solving complex scientific and engineering problems and justify further research in this
area.
5. CONCLUSION
In this study, we proposed an approach involving solutions of PDEs by applying a methodology of
deep learning designs, especially the feed-forward neural network. We have conducted experiment analyses for
four major PDEs, including two-dimensional Sine-Gordon and Klein-Gordon, to determine the effectiveness
of the approach by comparing its outputs to accurate solutions and estimating its mistakes. The approach
obtained successful experimental results through the power of the neural network function approximation. The
FNN have been deployed via the TensorFlow open source Framework. This approach has shown promise to
solve this initial set of issues. Nevertheless, more numeric experimentation has to be made to reinforce our
results. Even with the successful implementation of those novel deep learning techniques for solving PDEs,
several significant issues concerning theory and practice need to be explored and discussed in more detail.

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BIOGRAPHIES OF AUTHORS
Soumaya Nouna is a researcher at the Systems Analysis and Modelling and Decision
Support Research Laboratory at Hassan First University in Settat. She is an expert in mathematics,
machine learning and deep learning. A doctoral researcher in mathematics and computer science, she
brings a wealth of experience to her field. Her skills include the analysis of differential equations, and
machine learning algorithms. She is also the author of numerous research articles and is constantly
seeking to advance in her areas of expertise. She can be contacted at email: s.nouna@uhp.ac.ma.
Assia Nouna is a researcher at the Systems Analysis and Modelling and Decision Support
Research Laboratory at Hassan First University in Settat. A doctoral researcher in mathematics and
computer science. She is currently working on deep learning and satellite imagery for agricultural
applications. Her research aims to enhance agricultural practices through precise soil analysis, im-
proving crop management and yield predictions. Additionally, she has contributed to various projects
and publications in the field, demonstrating her expertise in applying advanced computational tech-
niques to solve real-world problems. She can be contacted at email: a.nouna@uhp.ac.ma.
Mohamed Mansouri received the Ph.D. degree in mechanical engineering and engineer-
ing sciences from the Faculty of Science and Technology, Hassan First University, Settat, Morocco,
and from L’INSA, Rouen, France, in 2013. He is currently a Professor and researcher at the National
School of Applied Sciences in Berrechid, Department of Electrical Engineering and Renewable En-
ergies. His research interests include mechano-reliability study, industrial engineering, optimization
of shape and reliability optimization of coupled fluid-structure systems, and energy storage systems.
He can be contacted at email: m.mansouri@uhp.ac.ma.
Dr. Ilyas Tammouch is a distinguished researcher and academic affiliated with Ibn
Tofail University in Kenitra, Morocco. His expertise spans multiple advanced fields within artifi-
cial intelligence, including machine learning, deep learning, data analysis, and evaluation systems.
He has contributed extensively to the scientific community, focusing on innovative approaches to
harness data-driven insights for practical applications in education, healthcare, and various indus-
trial sectors. His research not only emphasizes technical advancements but also explores the ethical
and societal impacts of technology. He is actively involved in academic collaborations and knowl-
edge exchange initiatives, both within Morocco and internationally. He can be contacted at email:
ilyas.tammouch@uit.ac.ma.
Boujamaa Achchab is a professor and director at ENSA Berrechid, Hassan 1st University,
specializing in applied mathematics and computer science. He completed his Ph.D. at Université
Claude Bernard Lyon 1 in 1995. His research focuses on numerical analysis, mathematical modeling,
and computational finance. Notable works include simulations of the black-scholes equation and
studies on stochastic processes. He is proficient in various mathematical and simulation software,
with strong analytical skills and experience in collaborative research projects. He can be contacted at
email: achchab@yahoo.fr.

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