IAES Inter national J our nal of Articial Intelligence (IJ-AI) V ol. 14, No. 6, December 2025, pp. 5172 ∼ 5182 ISSN: 2252-8938, DOI: 10.11591/ijai.v14.i6.pp5172-5182 ❒ 5172 Deep neural netw ork solutions to Newell-Whitehead-Segel equations Soumaya Nouna 1 , Ilyas T ammouch 2 , Assia Nouna 1 , Mohamed Mansouri 1 1 Laboratory LAMSAD, Department of Mathematics and Informatics, Hassan First Uni v ersity of Settat, Berrechid, Morocco 2 Laboratory of T elecommunications Systems and Decision Engineering, F aculty of Science, Ibn T of ail Uni v ersity , K enitra, Morocco Article Inf o Article history: Recei v ed Dec 3, 2024 Re vised Oct 22, 2025 Accepted No v 8, 2025 K eyw ords: Articial neural netw orks Deep learning Deep neural netw ork NeuroDif fEq Ne well-Whitehead-Se gel equations Abstract In this w ork, we use the deep neural netw ork (DNN) approach called NeuroDif fEq, and the unied nite dif ference e xponential approach for obtaining the approximated and e xact solutions of Ne well-Whitehead-Se gel systems that are essential for the biology of mathematics. A unied approach w as used to generate se v eral solutions for solitary w a v es of those systems. The approximated solutions for selected studies are e xplored using the NeuroDif fEq approach, which is the articial neural netw orks (ANN) approach and is based upon trial approximate solution (T AS). The comparison between the obtained approximated solutions and the analytical solutions indicates that the applied method has pro v ed an ef cient as well as a highly successf ul approach to solving v arious types of the Ne well-Whitehead-Se gel equations. This is an open access article under the CC BY -SA license . Corresponding A uthor: Soumaya Nouna Laboratory LAMSAD, Department of Mathematics and Informatics, Hassan First Uni v ersity of Settat Berrechid, Morocco Email: s.nouna@uhp.ac.ma 1. INTR ODUCTION A more general form of the Ne well-Whitehead-Se gel system is as (1): ∂ v ∂ t = β ∂ 2 v ∂ x 2 + cv + k v p , x ∈ [ a, b ] , t ∈ [0 , T ] (1) W ith an initial condition as (2): v ( x, 0) = f ( x ) (2) And the boundary conditions as (3): v ( a, t ) = g 1 ( t ) , v ( b, t ) = g 2 ( t ) , t ⩾ 0 . (3) Where c, k , and β represent the real numbers, and p represents the inte gers of positi v e numbers. Ne well-Whitehead-Se gel equations [1], [2] described dynamic properties around the point of bifurcation of the Re yleigh-Benard con v ecti v e o w for the binary uid mixtures. These equations are found throughout v arious applications in science, including mathematical biological sciences , quantic mechanical sciences, and the ph ysics of plasma. Dif ferent approaches are being proposed for solving Ne well Whitehead Se gel equations: laplace adomian decomposition approach, cubic B-spline collocation approach, nite dif ference scheme, He’ s v ariational iteration approach, and homotop y perturbation approach [3]-[6]. The problem with these traditional J ournal homepage: http://ijai.iaescor e .com Evaluation Warning : The document was created with Spire.PDF for Python.
Int J Artif Intell ISSN: 2252-8938 ❒ 5173 methods for solving equations is the y can f ail before achie ving the best-approximated solutions, and the y are not promising in terms of accurac y . In the absence of discretization processes, articial neural netw orks (ANNs) are alternati v es. ANN is e xtensi v ely implemented for solving se v eral problems in scientic computing [7], [8]. ANN-based techniques ha v e also been de v eloped to address partial dif ferential equations (PDEs). F or instance, Malek and Be idokhti [9] combined articial neural netw orks with the Nelder –Mead simple x method to obtain accurate numerical approximations of nonlinear PDEs. Furthermore, Sirignano and Spiliopoulos [10] introduced the deep Galerkin method (DGM), a fully mesh-free neural netw ork frame w ork capable of solving high-dimensional PDEs through stochastic training. Building on these adv ances, Nouna et al. [11] demonstrated the ef fecti v eness of deep neural netw orks in approximating nonlinear equations such as the Klein–Gordon and Sine–Gordon models, while Nouna et al. [12] e xtended these concepts to a broader class of PDEs, conrming the rob ustness and v ersatility of ANN-based solv ers for scientic computing. In parallel, another neural-netw ork-based strate gy , the ph ysics-informed neural netw ork (PINN), w as introduced by Raissi et al. [13], and further enhanced by Guo et al. [14] through residual-based adapti v e renement mechanisms. ANNs’ ability to solv e problems in PDEs has certain benets, such as dif ferentiable and continuous approximated solutions, strong interpolati v e features, and reduced memories. Further benets of the ANN include the ability to use automatically dif ferentiating utilities, for e xample, PyT orch [15] and T ensorFlo w [16], which allo ws the researchers to de v elop more straightforw ard approaches for solving the problems of PDE [17]. In this research, we present a methodology to resolv e the Ne well-Whitehead-Se gel models using a model of ANN kno wn as NeuroDif fEq, as the ANN approximator ha ving trial approximate solution (T AS) applied [18]. The structure of this paper is as follo ws: in section 2, we pro vide the e xact solutions to the Ne well-Whitehead-Se gel models. Section 3 presents the NeuroDif fEq approach for solving these systems. Numerical solutions for v arious Ne well-Whitehead-Se gel types using the NeuroDif fEq approach are introduced and compared with their e xact solutions in section 4. Finally , in section 5, we conclude our w ork. 2. EXA CT SOLUTIONS In this section, we describe some concepts in a unied approach [19]-[21] to nding some anal ytical solutions to by (1). T o nd the w a v e solutions, the transformation as in (4) is used: v ( x, t ) = v ( κ ) , κ = x − εt (4) In (1), the result in (5) is obtained: cv + k v p + εv ′ + β v ′′ = 0 (5) with v ′ = dv κ . In (5) can be inte grated if p = 2 or p = 3 . Theref o r e, we restrict the search to the solutions of (1) in both cases. 2.1. First case In the case of p = 2 , the (5) w ould ha v e a form as (6): cv + k v 2 + εv ′ + β v ′′ = 0 (6) A unied approach states that the requisite solutions are e xpressed in the follo wing terms as in (7): v ( κ ) = Σ m i =0 q i Φ i ( κ ) (7) Auxiliary feature Φ( κ ) fullls the follo wing auxiliary equation as in (8): Φ ′ ( κ ) = Σ n i =0 k i Φ i ( κ ) (8) with q i and k i being the real constants determined at some later . The homogeneous equilibrium between v ′′ and v 2 in (6) gi v es m = 2( n − 1) , n ⩾ 1 . The e xact solutions of the w a v es for (1) are found here when n = 2 . According to (7) and (8), we get (9) and (10): v ( κ ) = q 0 + q 1 Φ( κ ) + q 2 Φ 2 ( κ ) (9) Deep neur al network solutions to Ne well-Whitehead-Se g el equations (Soumaya Nouna) Evaluation Warning : The document was created with Spire.PDF for Python.
5174 ❒ ISSN: 2252-8938 Φ ′ ( κ ) = k 0 + k 1 Φ( κ ) + k 2 Φ 2 ( κ ) . (10) By substituting (9) with (10) in (6) and setting Φ( κ ) ’ s coef cients equal to null, the solutions of the parameters q i and k i , i = 0 , 1 , 2 ha v e the follo wing form as in (11): q 0 = − 3( k 1 + λ ) 2 β 2 k , q 1 = − 6 k 2 ( k 1 + λ ) β k , q 2 = − 6 k 2 2 β k , ε = 5 λβ , c = 6 λ 2 β , λ = q k 2 1 − 4 k 0 k 2 . (11) By using (11) in (9) and solving auxiliary equation pro vided from (10), the solution to (1) is obtained as (12): v 1 ( x, t ) = − 3 λ 2 β 2 k   1 − 2 tanh λ 2 κ + tanh 2 λ 2 κ ! (12) with κ = x − εt and v 1 ( x, t ) is the e xact solution. 2.2. Second case In the case of p = 3 , the (5) w ould ha v e a form as (13): cv + k v 3 + εv ′ + β v ′′ = 0 (13) Considering a homogeneous equilibrium between v ′′ and v 3 in (13), we obtain m = n − 1 , n ⩾ 1 . Lik e wise, the e xact solutions of the w a v es to (1) are found when n = 2 . According to (7) and (8), we get (14) and (15): v ( κ ) = q 0 + q 1 Φ( κ ) (14) Φ ′ ( κ ) = k 0 + k 1 Φ( κ ) + k 2 Φ 2 ( κ ) . (15) By substituting (14) with (15) in (13) and setting Φ( κ ) ’ s coef cients equal to null, the solutions of the parameters q i and k i ha v e the (16) form: q 0 = ( k 1 + λ ) √ − β √ 2 k , q 1 = k 2 √ − β √ k , ε = 2 λβ , c = 2 λ 2 β , λ = q k 2 1 − 4 k 0 k 2 , β > 0 . (16) By using (16) in (14) and solving auxiliary equation pro vided from (15), the solution to (1) is obtained as (17): v 2 ( x, t ) = − λ √ − β √ 2 k   1 + tanh λ 2 κ ! (17) with κ = x − εt and v 2 ( x, t ) is the e xact solution. 3. AR TIFICIAL NEURAL NETW ORK SOLUTIONS This section introduces the ANN architecture. F ollo wed by the NeuroDif fEq method which uses the ANN architecture. T o approximate solutions for Ne well Whitehead Se gel equations. 3.1. The articial neural netw ork ANN comprises R + 1 layers, where 0 layer represents an input layer while R layer represents an output layer . 0 < r < R layers represent hidden layers. Ac ti v ation functions inside hidden layers are an y acti v ation function, lik e rectied linear units, h yperbolic tangents, and sigmoids. By def ault, we will use sigmoid functions (see denition 3.1.) for hidden layers, which possess uni v ersal approximation capability (see theorem 3.1.) [22]. Ev ery node of an ANN is biased, which includes output and not input nodes, while connections between nodes in the follo wing layers are presented as matrices of weights (see Figure 1). Int J Artif Intell, V ol. 14, No. 6, December 2025: 5172–5182 Evaluation Warning : The document was created with Spire.PDF for Python.
Int J Artif Intell ISSN: 2252-8938 ❒ 5175 . . . . . . . . . . . . x t b r − 1 1 b r − 1 l b r − 1 m b r 1 b r k b r m ˆ v = y R (0) -layer (0) ( r − 1) -hidden layer ( r ) -hidden layer ( R ) -layer Figure 1. ANN architecture Let b r k represent a bias for neuron k at layer r . w r k l represent the weights from neuron l at layer r − 1 to neuron k at layer r . The r -layer acti v ation function is denoted as φ r . F or the output of neuron k at layer r , we denote it as y r k . The important quantity that is widely utilized is kno wn as the weighted input, dened as (18): h r k = Σ l w r k l φ r − 1 ( h r − 1 l ) + b r k (18) Where the summation is computed on e v ery input of the neuron k inside the r -layer . Namely number of neurons in ( r − 1) -layer . W eighted entry (18) can ob viously be described as an output of the preceding layer in the (19) w ay: h r k = Σ l w r k l y r − 1 l + b r k (19) Where output y r − 1 l = φ r − 1 ( h r − 1 l ) represents the weighted input acti v ation. Since we are w orking on ANNs, we prefer the (18) since this naturally describes the recursion of pre vious weighted inputs into the ANN. As a denition, we ha v e the (20): φ 0 ( h 0 k ) = y 0 k = x k (20) End all recursion. If we remo v e subscripts, it is possible to write (18) as a con v enient v ector e xpression as (21): h r = W r φ r − 1 ( h r − 1 ) + b r = W r y r − 1 + b r (21) in which e v ery component of the v ectors h r and y r is gi v en respecti v ely by h r k and y r k , with an acti v ation function applied per component. The matrix elements W r can be g i v en using W r k l = w r k l . Using an y of the Deep neur al network solutions to Ne well-Whitehead-Se g el equations (Soumaya Nouna) Evaluation Warning : The document was created with Spire.PDF for Python.
5176 ❒ ISSN: 2252-8938 abo v e denitions, a feed-forw ard algorithm to calculate output y R , based on input x , can be gi v en as (22): h 1 = W 1 x + b 1 h 2 = W 2 φ 1 ( h 1 ) + b 2 h 3 = W 3 φ 2 ( h 2 ) + b 3 . . . h R − 1 = W R − 1 φ R − 2 ( h R − 2 ) + b R − 1 h R = W R φ R − 1 ( h R − 1 ) + b R ˆ v = y R = φ R ( h R ) . (22) The ef cienc y of output ˆ v can be determined by the calculation of the loss function. Consequently , the last step is to minimize the loss function with the stochastic gradient descent (SGD) optimizer . In addition, the output ˆ v ( x, t, θ ) of the ANN is the approximate of v ( x, t ) in the (1)-(3) where x and t represent inputs, and θ represents the weights and biases of the adjustable parameters. Then we can easily e xpress n th deri v ati v es for ˆ v ( x, t, θ ) via automatic dif ferentiation (AD) (1) as (23) and (24): ∂ n ˆ v ∂ t n ( x, t, θ ) = ∂ n φ R ( h R ) ∂ t n , n = 1 , (23) ∂ n ˆ v ∂ x n ( x, t, θ ) = ∂ n φ R ( h R ) ∂ x n , n = 1 , 2 . (24) Denition 3.1.: a Sigmoid function [23] can be dened as a function of mathematics that associates its input with a v alue in the range of 0 to 1, which produces a curv e in the shape of an S. Ho we v er , a sigmoid function is dened mathematically as (25): φ ( x ) = 1 1 + e − x = e x e x + 1 (25) Where φ ( x ) represents a sigmoid function on input x. Theorem 3.1.: as sume that φ is bounded, strictly monotonic, and increasing. Consider Ω as an m -dimensional compact sub-set by R M . The space of continuous real functi ons which are dened in Ω is denoted as C (Ω) . So, for an y gi v en function h , h : Ω − → R , some real ε > 0 , and x ∈ Ω , then there e xists a set of real v alues w k , b k , b l , and w k l , so that we can dene (26): H ( x ) = Σ k w k φ (Σ l w k l x l + b l ) + b k (26) As the approximate implementation for a function h , as depicted in (27): | H ( x ) − h ( x ) | ⩽ ε. (27) 3.2. The Neur oDiffEq appr oach The NeuroDif fEq approach uses ANN to solv e the Ne well-Whitehead-Se gel equations (1)-(3). This approach dif fers in se v eral aspects, including the generation of data points, the denit ion of boundary conditions, and the choice of loss functions. Furthermore, to solv e these equations with NeuroDif f Eq , the primary concept in v olv es constructing the T AS, denoted as ˜ v ( x, t ; θ ) [18]. The NeuroDif fEq algorithm denes in the follo wing w ays: i) An ( M × N ) -grid point set E = ( x, t ) , with x = ( x 1 , ..., x M ) , t = ( t 1 , ..., t N ) within the [ a, b ] × [0 , T ] domain, supplied by the ANNs. An y distrib ution you choose, such as uniform distrib ution and equal spacing, can be used to generate points. ii) De v elopment of the T AS, a well-kno wn function, is composed of input E and output ˆ v ( x, t ; θ ) . Moreo v er , T AS is dened in the (28) form [24]: ˜ v ( x, t ; θ ) = B ( x ) + G ( x, t ; ˆ v ) (28) Int J Artif Intell, V ol. 14, No. 6, December 2025: 5172–5182 Evaluation Warning : The document was created with Spire.PDF for Python.
Int J Artif Intell ISSN: 2252-8938 ❒ 5177 A ( x ) has been chosen to respect boundary conditions, and G ( x, t ; ˆ v ) has been selected as zero for all ( x, t ) at the boundary . This gi v es the T AS automatically satisfying boundary conditions, whate v er the output of the ANN. Ho we v er , this method is also similar in concept to the trial function [25], although the trial function tak es a dif ferent form. It is also possible to modify t he T AS for the initial conditions (2). Generally , solutions that are transformed ha v e the (29) form [18]: ˜ v ( x, 0; θ ) = v ( x, 0) + xt (1 − x )(1 − t ) ˆ v ( x, t ; θ ) . (29) iii) The rial approximate solution has been implemented to minimize the (30) loss function: L ( θ ) = L 1 ( θ ) + µ L 2 ( θ ) . (30) The rst term of L can be represented as (31): L 1 ( θ ) = ∂ ˜ v ∂ t ( x, t ; θ ) − β ∂ 2 ˜ v ∂ x 2 ( x, t ; θ ) − c ˜ v ( x, t ; θ ) − k ˜ v p ( x, t ; θ ) 2 ( x,t ) ∈ E (31) Where (31) repre sents an approximate solution to the equation itself, with E consisting of the nite points in the equation’ s domain, then the second term is dened as (32): L 2 ( θ ) = ˜ v ( x, 0; θ ) − f ( x ) 2 x ∈ [ a,b ] + ˜ v ( a, t ; θ ) − g 1 ( t ) 2 t ∈ [0 ,T ] + ˜ v ( b, t ; θ ) − g 2 ( t ) 2 t ∈ [0 ,T ] (32) L 2 satises both the initial and boundary conditions. A weighting coef cient called µ indicates the importance of both error components. This f actor is arbitrarily determined in practice. A description of the algorithm for this method is gi v en in Algorithm 1. Algorithm 1 NeuroDif fEq-Algorithm to solv e the Ne well-Whitehead-Se gel equation 1 : Consistently generate 150 × 150 points E = { x n , t n } 150 n =1 within the [0 , 1] domain. 2 : While iter < 0 iteration Do 3 : F or e v ery ( x, t ) ∈ E Do 4 : T o Calculate ˆ v = ( x, t, θ ) = φ R ( h R ) , ∂ ˆ v ∂ t ( x, t, θ ) = ∂ φ R ( h R ) ∂ t , ∂ 2 ˆ v ∂ x 2 ( x, t, θ ) = ∂ 2 φ R ( h R ) ∂ x 2 5 : Construct the T AS that is satised by initial condition of 2 in the follo wing w ay ˜ v ( x, 0; θ ) = v ( x, 0) + xt (1 − x )(1 − t ) ˆ v ( x, t ; θ ) 6 : Calculate a loss function as follo ws: L ( θ ) = L 1 ( θ ) + µ L 2 ( θ ) , While L 1 ( θ ) = ∂ ˜ v ∂ t ( x n , t n ; θ ) − β ∂ 2 ˜ v ∂ x 2 ( x n , t n ; θ ) − c ˜ v ( x n , t n ; θ ) − k ˜ v p ( x n , t n ; θ ) 2 , L 2 ( θ ) = ˜ v ( x n , 0; θ ) − f ( x n ) 2 + ˜ v ( a, t n ; θ ) − g 1 ( t n ) 2 + ˜ v ( b, t n ; θ ) − g 2 ( t n ) 2 . 7 : End F or 8 : Minimizing the loss, update weights and biases using the SGD-optimizer . 9 : End While 4. THE NUMERICAL RESUL TS Se v eral types of Ne well-Whitehead-Se gel equat ions are used to demonstrate the performance of the NeuroDif fEq approach. F or all the numerical e xamples belo w , we implemented an ANN with v e hidden Deep neur al network solutions to Ne well-Whitehead-Se g el equations (Soumaya Nouna) Evaluation Warning : The document was created with Spire.PDF for Python.
5178 ❒ ISSN: 2252-8938 layers, where the rst layer has 20 neurons, the second has 30 neurons, the third has 40 neurons, the fourt h has 50 neurons, and the last has 60 neurons. The solutions approximated by the Neurodifeq approach were compared with the e xact solutions. The dif ferent results obtained are illustrated using v arious gures and tables. 4.1. Pr oblem 1 When p = 2 , then Ne well-Whitehead-Se gel’ s equation is as (33) ∂ v ∂ t = β ∂ 2 v ∂ x 2 + cv + k v 2 , x ∈ [0 , 1] , t ∈ [0 , 1] (33) with the initial condition as in (34): v ( x, 0) = − 3 λ 2 β 2 k 1 − 2 tanh( λx 2 ) + tanh 2 ( λx 2 ) (34) the boundary conditions as in (35) v (0 , t ) = − 3 λ 2 β 2 k 1 − 2 tanh( λ 2 ( − at )) + tanh 2 ( λ 2 ( − at )) , v (1 , t ) = − 3 λ 2 β 2 k 1 − 2 tanh( λ 2 (1 − at )) + tanh 2 ( λ 2 (1 − at )) (35) and the e xact solution as in (36): v ex ( x, t ) = − 3 λ 2 β 2 k 1 − 2 tanh( λ 2 ( x − at )) + tanh 2 ( λ 2 ( x − at )) (36) with c = 6 λ 2 β , a = 5 λβ , k = 0 . 5 , β = 0 . 2 , and λ = 0 . 8 which represent real-v alued constants, we e v aluated the ef fecti v eness of the NeuroDif fEq approach in solving the second dif ferential by (33). Figure 2 illustrates the comparison between the e xact solution and the approximate solution obtained using an ANN. The gure presents both solutions: the curv e on the right corresponds to the e xact solution, while the curv e on t he left represents the ANN-based approximation. Figure 2. The e xact and the ANN-solution of (33) W e can observ e that the tw o curv es o v erlap almost perfectly , indicating that our deep l earning-based model is capable of accurat ely reproducing the beha vior of the dif ferential equation. This visual comparison highlights the ef fecti v eness of the ANN in solving the second di f ferential equation. The approximate solution aligns precisely with the e xact one, conrming the model’ s ability to capture the fundamental dynamics of the problem. Int J Artif Intell, V ol. 14, No. 6, December 2025: 5172–5182 Evaluation Warning : The document was created with Spire.PDF for Python.
Int J Artif Intell ISSN: 2252-8938 ❒ 5179 W e also assessed the model’ s performance using T able 1, which reports the loss v al ues for di f ferent numbers of netw ork layers. As in the pre vious case, we observ ed that the loss decreases as the number of hidden layers increases, thereby emphasizing the impro v ed accurac y and rob ustness of the model with increased architectural comple xity . T able 1. V alues of the loss of (33) at v arious layers numbers Hidden layers 3 -layers 4 -layers 5 -layers Neurons 20 30 40 20 30 40 50 20 30 40 50 60 Errors 1 . 9 × 10 − 5 9 . 4 × 10 − 6 9 × 10 − 6 4.2. Pr oblem 2 When p = 3 , then Ne well-Whitehead-Se gel’ s equation is as (37): ∂ v ∂ t = β ∂ 2 v ∂ x 2 + cv + k v 3 , x ∈ [0 , 1] , t ∈ [0 , 1] (37) with the initial condition as in (38): v ( x, 0) = − λ r β 2 k − 1 + tanh( λx 2 ) (38) and the boundary conditions as in (39): v (0 , t ) = − λ r β 2 k − 1 + tanh( λ 2 ( − at )) , v (1 , t ) = − λ r β 2 k − 1 + tanh( λ 2 (1 − at )) (39) with c = 2 λ 2 β , k = 0 . 5 , β = − 0 . 2 , and λ = 0 . 8 represent the real numbers and a = 3 λβ represents v elocity . The e xact solution is as depicted in (40): v ex ( x, t ) = − λ r β 2 k − 1 + tanh( λ 2 ( x − at )) . (40) W e ha v e compared the e xact solution with the approximate solution obtained using an ANN to e v aluate the ef fecti v eness of our deep learning-based approach to solving the third dif ferential (37). Figure 3 i llustrates this comparison using 3D vis ualizations. The gure on the left sho ws the e xact solution, while the gure on the right displays the approximate solution obtained using the ANN. Figure 3. The e xact and the ANN-solution of (37) It can be seen that the tw o surf aces o v erlap closely , indicating that our deep learning-based m odel is capable of accurately reproducing the beha vior of the dif ferential equation. This 3D visual comparison Deep neur al network solutions to Ne well-Whitehead-Se g el equations (Soumaya Nouna) Evaluation Warning : The document was created with Spire.PDF for Python.
5180 ❒ ISSN: 2252-8938 between the tw o solutions conrms the success of our method in solving the third dif ferential equation. The approximate solution aligns v ery well with the e xact solution, demonstrating the model’ s ability to capture the essential features of the problem. W e also e v aluated the ef fecti v eness of our model using T able 2, which presents the loss v alues for dif ferent numbers of netw ork layers. As mentioned abo v e, we observ e that increasing the number of layers leads to a decrease in loss. This result underlines the impro v ed performance of the model with higher architectural comple xity . T able 2. V alues of the loss of (37) at v arious layers numbers Hidden layers 3 -layers 4 -layers 5 -layers Neurons 20 30 40 20 30 40 50 20 30 40 50 60 Errors 1 . 8 × 10 − 5 5 × 10 − 6 3 × 10 − 6 5. CONCLUSION In this article, we e xamined the general Ne well-Whitehead-Se gel equations both numerically and analytically by applying the NeuroDif fEq and Unied approaches, respecti v ely . The approximate solutions obtained were compared with the e xact analytical solutions, sho wing that the NeuroDif fEq approach pro vides reliable and ef cient approximations with minimal manual interv ention. The results demonstrate a strong agreement with the e xact solutions, conrming that deep learning particularly ANNs is a promising tool for solving dif ferential equations, including both linear and nonlinear cases in applied sciences. Ho we v er , this study also presents some l imitations. The accurac y and con v er gence of the neural netw ork models can be sensiti v e to h yperparameter choices and the selection of training data points. Further more, the computational cost may increase signicantly for comple x or hi gh-dimensional problems. Future research should e xplore adapti v e training techniques, automated h yperparameter optimization, and the application of these models to more comple x boundary conditions or irre gular geometries . Inte grating ph ysics-informed architectures with domain kno wledge could also further impro v e both ef cienc y and generalization. FUNDING INFORMA TION This research recei v ed no specic grant from an y funding agenc y in the public, commercial, or not-for -prot sectors. Authors state no funding in v olv ed. A UTHOR CONTRIB UTIONS ST A TEMENT This journal uses the Contrib utor Roles T axonomy (CRediT) to recognize indi vidual author contrib utions, reduce authorship disputes, and f acilitate collaboration. Name of A uthor C M So V a F o I R D O E V i Su P Fu Soumaya Nouna ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ Ilyas T ammouch ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ Assia Nouna ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ Mohamed Mansouri ✓ ✓ ✓ ✓ ✓ C : C onceptualization I : I n v estig ation V i : V i sualization M : M ethodology R : R esources Su : Su pervision So : So ftw are D : D ata Curation P : P roject Administration V a : V a lidation O : Writing - O riginal Draft Fu : Fu nding Acquisition F o : F o rmal Analysis E : Writing - Re vie w & E diting CONFLICT OF INTEREST ST A TEMENT The authors declare that the y ha v e no kno wn competing nancial interests or personal relat ionships that could ha v e appeared to inuence the w ork reported in this paper . Authors state no conict of interest. Int J Artif Intell, V ol. 14, No. 6, December 2025: 5172–5182 Evaluation Warning : The document was created with Spire.PDF for Python.
Int J Artif Intell ISSN: 2252-8938 ❒ 5181 D A T A A V AILABILITY The authors conrm that the data supporting the ndings of this study are a v ailable within the artic le. No additional datasets were generated or analyzed during the current study . REFERENCES [1] N. S. Elg azery , “ A periodic solution of the Ne well-Whitehead-Se gel (NWS) w a v e equation via fractional calculus, ” J ournal of Applied and Computational Mec hanics , v ol. 6, pp. 1293–1300, 2020, doi: 10.22055/J A CM.2020.33778.2285. [2] N. H. T uan, R. M. Ganji, and H. Jaf ari, “ A numeri cal study of fractional rheological models and fractional Ne well-Whitehead-Se gel equation with non-local and non-singular k ernel, ” Chinese J ournal of Physics , v ol. 68, pp. 308–320, 2020, doi: 10.1016/j.cjph.2020.08.019. [3] P . P .-On, “Laplace adomian decomposition method for solving ne well-whitehead-se gel equation, ” Applied Mathematical Sciences , v ol. 7, no. 129–132, pp. 6593–6600, 2013, doi: 10.12988/ams.2013.310603. 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