Application of Euler Neural Networks with Soft Computing Paradigm to Solve Nonlinear Problems Arising in Heat Transfer

In this study, a novel application of neurocomputing technique is presented for solving nonlinear heat transfer and natural convection porous fin problems arising in almost all areas of engineering and technology, especially in mechanical engineering. The mathematical models of the problems are exploited by the intelligent strength of Euler polynomials based Euler neural networks (ENN’s), optimized with a generalized normal distribution optimization (GNDO) algorithm and Interior point algorithm (IPA). In this scheme, ENN’s based differential equation models are constructed in an unsupervised manner, in which the neurons are trained by GNDO as an effective global search technique and IPA, which enhances the local search convergence. Moreover, a temperature distribution of heat transfer and natural convection porous fin are investigated by using an ENN-GNDO-IPA algorithm under the influence of variations in specific heat, thermal conductivity, internal heat generation, and heat transfer rate, respectively. A large number of executions are performed on the proposed technique for different cases to determine the reliability and effectiveness through various performance indicators including Nash–Sutcliffe efficiency (NSE), error in Nash–Sutcliffe efficiency (ENSE), mean absolute error (MAE), and Thiel’s inequality coefficient (TIC). Extensive graphical and statistical analysis shows the dominance of the proposed algorithm with state-of-the-art algorithms and numerical solver RK-4.


Introduction
Most of the problems in engineering sciences, especially heat transfer problems, are inherently nonlinear. Except for a limited number of these problems, most of them cannot be solved analytically by using traditional techniques. Linear and nonlinear differential equations were generally solved by integral transformation methods such as the Fourier or Laplace transform. These techniques are used to convert differential equations into a corresponding algebraic system of equations. Nonetheless, applying integral transformation methods was challenging at times [1]. In the 19th century, researchers such as Bellman [2], Cole [3], and O'Malley used the perturbation approach for solving nonlinear problems. In perturbation methods, the choice of small parameters and their exertion in differential equations was one of the challenging tasks for the research community. Nayfeh [4] and Van Dyke [5] improved the method by working on the loss of small parameters during physical verification. Later on, various powerful techniques have been developed to eliminate the small parameters such as the tanh method introduced by Wazwaz [6], artificial parameter method (APM) [7,8], homotopy analysis method (HAM) [9,10], homotopy perturbation method (HPM) [11], modified homotopy perturbation method (MHPM) [12], and iteration perturbation method (IPM) [13]. The application of these methods in the field of fluid dynamics, mechanical engineering, and heat radiation was studied by Rajabi [14], Abbasbandy, and Nadim [15,16]. Nonlinear problems arising in the heat transfer problem has been solved by Yaghoobi and Torabi [1] using the differential transformation method (DTM). The variational iteration method (VIM) was implemented by [17] to study the analytical solution for nonlinear problems. Recently, Kumbinarasaiaha [18,19] uses the Hermite wavelet method (HWM) to investigate the convecting-radiating and cooling of a lumped system with variable specific heat. Several Lagrangian based and ISPH methods are also used to study the heat transfer of various problems arising in fluid dynamics such as double-diffusive natural convection [20], a nanofluid-filled cavity including rotating solid structures [21], trapezoidal cooling microchannel [22], solid particles in an inner cross shape [23], circular enclosure partially saturated with a porous medium [24], nanofluid in a cavity with a partially layered porous medium [25,26], sloshing porous cavity filled with a nanofluid [27,28], and magneto-convective flow of a ferrofluid in a closed space [29,30]. Heat transfer in fins, also known as extended surfaces, has been a subject of interest, which has led to extensive research on the use of porous fins. Kiwan and Al-Nimr [31,32] were the first to investigate the use of porous materials to enhance heat transfer. Kiwan studies the performance of fins in a natural convection environment while Al-Nimr developed a numerical method to analyze the thermal analysis of fins [33]. Analytical models and optimization of porous fins were studied by Kundu and Bhanja [34]. In the last decade, the number of researchers numerically investigate the heat transfer of a rectangular porous fin [35], constructal T-shape porous fin [36], pin fins [37,38], cylindrical porous fins [39], and porous radial fins [40] under different environments. In 2013, Gorla [41,42] studies the influence of variations in thermal conductivity on natural convection and radiation in porous fins. To solve the heat transfer problems in porous fins, Saedodin and Sadeghi [39] applied the Runge-Kutta method for thermal analysis in fins. Kundu [37] investigates optimum design analysis by applying the Adomian decomposition method (ADM). To study heat transfer in longitudinal fins, Darvishi [40] and Moradi [43] adopted the homotopy analysis method (HAM). Several other techniques including the homotopy perturbation method (HPM) [44], spectral collocation method (SCM) [45,46], least square method (LSM) [47,48], variational iterative method (VIM) [49], and differential transformation method (DTM) [50] are used to study the temperature distribution and heat transfer of different fin problems. All of these methods are based on deterministic approaches and have their own merits, applicability, and drawbacks.
In recent times, intelligence-based nature-inspired meta heuristic algorithms have gained the attention of researchers. Some recent applications of soft computing techniques include the saturation process of water and oil through a porous medium during secondary oil recovery [51], heat transfer prediction of supercritical water [52], physics-informed neural networks [53], temperature profiles in longitudinal fin designs [54], wire coating dynamics [55,56], data-driven modeling for boiling heat [57], prediction of turbulent heat transfer [58], the corneal model for eye surgery [59], fuzzy systems [60], infrared, boiling heat transfer investigations [61], neuro-fuzzy modeling is used to predict the summer precipitation in targeted metrological sites [62], and prediction of heat transfer rates for shell-and-tube heat exchangers [63], beam-column designs [64], and nonlinear dusty plasma systems are analyzed with the help of NAR-RBFs neural networks [65]. The plant prorogation algorithm (PPA) and improved PPA are developed to solve a number of constrained and unconstrained engineering optimization problems [66,67]. The above-mentioned algorithms and their applications motivate the authors to build an efficient, reliable, and stable algorithm for analytical solutions of nonlinear heat transfer problems. Salient features of the given study are summarized as • A novel soft computing paradigm is developed to analyze the heat transfer problems arising in mechanical engineering. Euler polynomials based Euler neural networks (ENN's) are constructed to define an unsupervised differential equation model for different problems. Neurons in ENN are optimized by the hybridization of generalized normal distribution (GNDO) algorithm and interior-point algorithm (IPA). The proposed methodology is called the ENN-GNDO-IPA. • The efficiency and correctness of the design scheme are ascertained by comparing its results with state-of-the-art algorithms and a numerical solver based on RK-4 for each case of different scenarios and problems. • To validate the consistency of accuracy and convergence, the proposed algorithm is simulated to obtain a statistical results in terms of Nash-Sutcliffe efficiency (NSE), error in Nash-Sutcliffe efficiency (ENSE), mean absolute error (MAE), and Thiel's inequality coefficient (TIC). • Statistical inferences through minimum, mean, and standard deviations for fitness function and performance indicators further validate the worth of intelligent stochastic strategy in terms of complexity and accuracy.

Convecting-Radiating Cooling of a Lumped System with Variable Specific Heat
Consider the problem of combined convective-radiative cooling of a lumped system [18]. Let V and A be the volume and surface area of the system with specific heat density ρ and T i be the initial temperature. At t = 0, the system is subjected to the convective environment with temperature (T a ) and the coefficient of convective heat transfer is h. In addition, the system loses heat through radiation, and T s denotes the effective sink temperature. Specific heat is denoted by C and is defined as C a is a specific heat at temperature T a andζ is invariable (constant). Using the heat conduction equation, the cooling equation and its initial condition are derived as To solve Equation (2), the following parameters are defined: After modification of parameters, the heat transfer equation will result in the following: at For simplicity, we assume the case when y a = y s = 0; then, we have [1 + ε 1 y] dy dx + y + ε 2 y 4 = 0, with initial condition y(0) = 1.

Cooling of a Lumped System with Variable Specific Heat
Consider the cooling of a lumped system [18]. From Equation (1), specific heat is given as C = C a 1 +ζ(T − T a ) .
C a is a specific heat at temperature T a andζ is invariable (constant). Using the heat conduction equation, the cooling equation and its initial condition can be written as with to solve Equation (9), the following parameters are defined as Subsequently, the governing differential equation reduced to dy dx + εy dy dx + y = 0 (12) with initial condition y(0) = 1.

Natural Convection Porous Fin with Temperature-Dependent Thermal Conductivity and Internal Heat Generation
Consider a straight porous fin with a length L and a thickness t that is exposed to a convective environment on both sides with temperature T ∞ , as shown in Figure 1. x represents the height of the fin having a base at the origin. To briefly analyze the problem, the following assumptions are considered: (a) Porous medium of a straight fin is isotropic, homogenous, and saturated with fluid in a single phase. (b) Physical parameters except the density of fluid and solid are considered to be constant. (c) Radiative transfers, surface convection, and non-Darcian effects are small (negligible), and only natural convection is considered. In addition, the fin base is responsible for the transfer of heat through pores i.e., no convective heat is transmitted to the surrounding environment. (d) Porous medium and fluid are in thermodynamic equilibrium. (e) Variation of the temperature inside the fin is one-dimensional. Temperature changes with the length and remains static with time. (f) Tip of the fin is adiabatic, and the base of the fin is isolated.
Using the above assumptions, thermal energy based on Darcy's model can be expressed as The velocity v(x) of buoyancy flow in the fin at any point x is obtained by using Darcy's law for dx → 0, Equation (16) can be written as from Fourier's law of heat conduction Using Equation (18) in Equation (17), we get Governing equation of fin is obtained by simplification of Equation (19), we have subjected to boundary conditions given as Internal heat generation and temperature-dependent thermal conductivity are defined as Using Equations (22) and (23) in Equation (14), we get Introduce the following dimensionless parameters: The following dimensionless governing differential equation of the model is obtained: From simplification of Equation (25), we get the differential equation of the form with boundary conditions given as

Adiabatic Tip
Porous Medium Figure 1. Schematic of the straight porous fin geometry with the internal heat generation.

Metallic Annular Fin with Temperature Dependent Thermal Conductivity
In this problem, we consider an optimal design of metallic annular fin as shown in Figure 2, having uniform thickness (t). The fin is attached to the heat transfer surface (base) of the heat exchanger. This exchanger is exposed to the surrounding with ambient temperature T ∞ and temperature-dependent thermal conductivity. The phenomenon causes a mixed conductive-convective heat exchange with the air. To derive the governing equation for annular fin, we assume that the coefficient of heat transfer (h) is constant, and the exchanger is quite symmetric. Temperature distribution within fins can be mathematically presented as [68] with Furthermore, temperature dependent thermal conductivity of the fin is defined as where k ∞ denotes the thermal conductivity at ambient temperature and κ is constant. In order to obtain the corresponding boundary value problem, we define the following dimensionless variables: Now, Equation (28) along with boundary conditions will reduce to subjected to To solve this problem, we focus our attention on the interval [0, 1]; therefore, we define a variable ξ = (λ − 1)x. Now, Equation (30) can be written as

Methodology
The design methodology is comprised of two steps. Initially, feed forward artificial neural networks are used to construct a mean square error (MSE) based fitness function for a mathematical model of heat transfer and natural convection porous fin with temperaturedependent thermal conductivity and internal heat generation. In the second part, the learning procedure is provided for finding solutions of the model using memetic computing in which GNDO is used as a global search technique while IPA is used for the refinement of the local procedure. The designed algorithm ENN-GNDO-IPA is utilized as an optimization mechanism for unknown neurons in the ENN model. The flow chart of the procedure is shown in Figure 3.

Euler Polynomials and ENN Modeling
The classical Euler polynomials are denoted by E n and generally defined by the means of an exponential generating function given as and, explicitly, Euler polynomials are defined as where B k is Bernoulli number for each k. First, seven Euler polynomials are given as Some interesting properties and relations about Euler polynomials can be found in [69,70].   A trial or approximate solutionŷ(x) along with firstŷ (x) and secondŷ (x) derivatives are considered based on feed-forward artificial neural networks aŝ andŷ where α, ξ and β are unknown vectors of W as In order to find approximate solution for the mathematical models, Euler polynomials are used as an activation function i.e., Thus, the updated manifestation for the networks becomeŝ The generic architecture for each problems of heat transfer can be formulated by using the appropriate network from Equations (40)-(42).

Objective Function
In order to find the neurons in ENN structure, objective function ζ is considered as a mean square errors which is formulated as where ζ 1 is an error function for differential equations, while ζ 2 represents the corresponding error function for boundary conditions. The elaborative form of Equations (6), (12), (26), and (32) is given as

Learning Procedure
In order to study the temperature distribution of heat transfer problems under the influence of specific heat, temperature-dependent thermal conductivity, and internal heat generation, corresponding optimization problems in terms of objective functions are formulated. Unknown neurons in ENN architecture are optimized by using a derivative-free technique called a generalized normal distribution optimization (GNDO) algorithm for global search and an interior-point algorithm (IPA) for local convergence of solutions. Figure 3 represents the working of GNDO and IPA. Details of both of the algorithms are given below.

Generalized Normal Distribution Optimization Algorithm
The generalized normal distribution optimization (GNDO) algorithm is a novel metaheuristic algorithm proposed by Yiying Zhang [71] inspired by the normal distribution model. In GNDO, the position of the individual is updated with the help of a normal distribution curve. The authors' motivation for using GNDO algorithm is that it does not require any controlling parameters and any prior information about the problem. GNDO needs essential population size and specific terminal conditions before execution. The working procedure of the GNDO algorithm is divided into exploitation and exploration. Exploitation: During the process of exploitation, the algorithm searches to find better solutions around a search space that contains the current positions of all individuals. A model is developed for the relationship between normal distribution and individuals in populations which is given as where a t i represents the trial vector of the ith individual, η, d i and m i denote penalty factor, variance, and median, respectively, which are defined as where a, b, λ 1 and λ 2 are random numbers between 0 and 1. M denotes generalized mean, which is given as The above discussed parameters are used to find that x t best represents the current best position of the individual which is further modified by the process of exploration. Exploitation: Exploitation is a search of finding the global best solution in the entire population space. This process is based on three random variables which are expressed as Local information sharing Global information sharing (53) where β is adjustment parameter, λ 3 , λ 4 are random parameters subjecting to normal distribution are a 1 , a 2 are trail vectors, which are expressed as where p1, p2, and p3 are three random integers selected from 1 to N and p1 = p2 = p3 = i. Using the above procedure, GNDO gives the global best solution for the problem.

Interior Point Algorithm
Interior point algorithm (IPA) is a local search technique used to fine-tune the unknown weights in ENN structure. IPA is a derivative-based technique that is derived from Lagrange multipliers [72]. It involves scaling function, maximum perturbations, and type of derivative. IPA is incorporated with the global best simulations of GNDO for optimization of heat transfer and convection porous fin problems in the hybridization process. Some recent applications of IP algorithm are numerical solutions for correction of array failure [73], simulation of viscoplastic fluid flows [74], and decentralized optimal power flow of multi-area interconnected power systems [75].
The procedural steps of the hybridized ENN-GNDO-IPA algorithm are presented graphically through Figure 3. The performance of the algorithm is dependent on parameter settings which are shown in Table 1. During the analysis of the proposed algorithm, it is observed that a slight change in these parameters results in premature convergence. Therefore, a lot of experience, care, and experimentations are needed for the selection of optimal parameters of the metaheuristic ENN-GNDO-IPA algorithm. A detailed explanation of the working procedure of the proposed technique is given in Algorithm 1.

end if end for
The iteration is updated as t = t + 1.

Performance Indices
In this section, the performance of the design scheme for solving different problems of heat transfer is studied by incorporating performance indicators in terms of mean absolute error (MAE), Theil's inequality coefficient (TIC), root mean square error (RMSE), Nash-Sutcliffe efficiency (NSE), and error in Nash-Sutcliffe efficiency (ENSE). Mathematical formulation of these indicators are given as [51].
where y m is analytical solution andŷ m represents the approximate solution by the proposed algorithm; N also denotes the grid points.

Numerical Experimentation and Discussion
To evaluate the proposed method, problems with different scenarios are considered as shown in Figure 5.
Problems 1 to 4 are given as follows: Problem 1: Convecting-radiating cooling of a lumped system with variable specific heat. In this problem, nonlinear Equation (6) along with initial condition Equation (7) are considered to study the temperature distribution under the effect of variations in specific heat. An unsupervised fitness function for governing equation of the model is described below: To briefly study the model, two scenarios are considered. In scenario-I, we assume that ε 1 = 0, and ε 2 varies from 0 to 1 with a step size of 0.1. In addition, in scenario-II, ε 2 is assumed to be 1, and ε 1 varies from 0 to 1 with step size 0.1.

2 3
Convecting-radiating cooling of a lumped system with variable specific heat Cooling of a lumped system with variable specific heat Natural convection porous fin with temperature-dependent thermal conductivity and internal heat generation = .

Approximate Solutions
Absolute Errors

RMSE, ENSE
Metallic annular fin with temperature dependent thermal conductivity

Variations in
Variations in 4 Figure 5. Graphical overview of problems along with different cases studied in this paper.
The objective function Equation (61) for scenario I and scenario II has been optimized by executing the proposed algorithm for 100 independent runs. Approximate solutions for the influence of variations in specific heat ε 2 and ε 1 on temperature distributions are disclosed in Tables 2 and 3, respectively. Figure 6 shows comparison of our solutions with numerical solver RK-4 (ode45). It can be seen that temperature distribution varies directly with an increase in ε 2 , while it varies inversely with variations in ε 1 . Furthermore, to validate the results, we have compared the solutions with VIM, HPM, DTM, and exact solutions as shown in Figure 7. The calculated values of absolute errors (AE) in Tables 4 and 5 Tables 6 and 7, respectively. The convergence of objective function for both scenarios is shown in Figure 9.  x  Table 3. Approximate solutions for different cases of scenario-II of problem 1.
x  Table 4. Absolute errors in solutions obtained by proposed technique for variations in ε 2 with ε 1 = 0.
x  Table 5. Absolute errors in solutions obtained by proposed technique for variations in ε 1 with ε 2 = 1.
x  Table 6. Performance analysis on absolute errors in terms of minimum and mean values obtained by the proposed algorithm during 100 independent executions for each case of scenario-I of problem 1.  Table 7. Performance analysis on absolute errors in terms of minimum and mean values obtained by the proposed algorithm during 100 independent executions for each case of scenario-II of problem 1.  The detailed evaluation of fitness function and performance measures is carried out by statistical performance in terms of minimum, mean, and standard deviation as shown in Tables 8 and 9, and best so far design weights for solutions of different scenarios are given in Table 10. It can be seen that values of fitness function and performance measures lie around 10 −5 to 10 −14 , 10 −8 to 10 −12 , 10 −9 to 10 −12 , 10 −8 to 10 −12 and 0 to 10 −16 , respectively.  Table 9. Statistical analysis of performance indicators for scenario-II or problem 1.  Table 10. Unknown parameters in the ENN structure obtained for the optimization of fitness function corresponding to different scenarios of problem 1. Cases Approximate series solution obtained by the proposed technique for Scenario-II (ε 1 = 0.2, 0.4, 0.6, 0.8 and 1.0) of problem 1.
Problem 2: Cooling of a lumped system with variable specific heat. In this problem, Equations (12) and (13) are considered to study the influence of variations in ε on the cooling of a lumped system. In order to investigate the model, an objective function is constructed In this problem, ε is varied from 0 to 1 with a step size of 0.1. The optimization of fitness function Equation (73) is carried out executing a proposed algorithm for 100 independent trails. The convergence plot of the fitness function is shown in Figure 14, which shows that the design scheme converges for all cases of the cooling lumped system. To access the level of accuracy, approximate solutions and absolute errors are calculated as shown in Tables 11 and 12, respectively. Furthermore, to extend the correctness of the design scheme, Figure 15 is plotted that shows the comparison of results with state-of-the-art techniques including VIM, DTM, and exact solution along with numerical solver RK-4 (ode45  Table 13, it can be noticed that the proposed technique approaches the exact solution with mean errors that lie around 10 −9 to 10 −13 . The values of performance indices measuring the objective value, mean absolute error, Theil's inequality coefficient, root mean square error, and error in Nash-Sutcliffe efficiency are tabulated in Table 14, representing the minimum values, mean values, and standard deviations. Convergence of RMSE, MAE, TIC, and ENSE during 100 independent trials are plotted through Figures 16-19. Unknown neurons in ENN structure are given in Table 15 that are used to obtain the approximate solution for a cooling lumped system with variable specific heat.        Table 15. Unknown parameters in ENN structure obtained for the optimization of fitness function corresponding to different values of specific heat in problem 2. Cases Problem 3: Natural convection porous fin with temperature-dependent thermal conductivity and internal heat generation. In this problem, a mathematical model of convection porous fin with temperaturedependent thermal conductivity and internal heat generation is considered as presented by Equations (26) and (27). The ENN-GNDO-IPA algorithm is applied to study the temperature distribution of convective porous fin under the influence of variation in β, S h , γ, and Q. An unsupervised objective function in terms of mean square errors is given as Furthermore, to study the model extensively, we have considered four scenarios as follows.
Optimization of fitness equation Equation (80) for each scenario of problem 3 is conducted with the help of soft computing technique ENN-GNDO-IPA by executing it for 100 independent trials. The convergence of fitness value during the learning procedure is plotted in Figure 19. The approximate solutions for scenarios I, II, III, and IV by the proposed technique are dictated in Table 16. From Figure 20, it can be seen that nonlinear thermal conductivity parameters i.e., β, γ, and Q increases, the dimensionless temperature distribution in the fin decreases, while, with the increase in S h , temperature distribution increases. Absolute errors in our solution are presented in Table 17 and graphically illustrated through Figure 21. It can be seen that absolute errors for each scenario lie around 1.59 × 10 −9 to 3.76 × 10 −15 , 9.67 × 10 −11 to 5.59 × 10 −18 , 1.15 × 10 −11 to 6.90 × 10 −14 , and 1.19 × 10 −11 to 1.57 × 10 −14 .
The accuracy of the proposed algorithm is measured by the results of fitness function and performance indicators. Tables 18 and 19 show that the design scheme is convergent, and results are approaching zero. Bar graphs are plotted in Figure 22, which shows that mean values of fitness function, MAE, TIC, RMSE, and ENSE for different scenarios lie around 10 −5 to 10 −8 , 10 −5 to 10 −6 , 10 −5 to 10 −7 , 10 −5 to 10 −6 and 10 −8 to 10 −10 , respectively. Furthermore, normal probability curves given in Figure 23 show the robustness of technique. Unknown neurons used in the process of optimization for the best solution of each scenario are presented in Tables 20-23.               Problem 4: Metallic annular fin with temperature dependent thermal conductivity. In this problem, heat transfer and temperature distribution in metallic annular fin with temperature dependent thermal conductivity have been investigated by using the proposed algorithm. An unsupervised objective function for the problem is given as Furthermore, to briefly study the model, different scenarios are considered depending on variations in several parameters such as β and Bi. Approximate solutions obtained by the ENN-GNDO-SQP algorithm for different values of β are dictated in Table 24 and graphically illustrated through    Furthermore, to validate the accuracy of the design scheme, 100 independent executions have been carried out. The results for the convergence of fitness function in each run are shown in Figure 26a. The results of fitness function for different values of β lie around 10 −4 to 10 −7 . Boxplot analysis shows that mean values for MAD, TIC, RMSE, and ENSE lie around 10 −3 to 10 −5 , 10 −4 to 10 −6 , 10 −3 to 10 −5 , and 10 −6 to 10 −9 , respectively. Plots of optimized neurons for obtaining best solutions are shown in Figure 27. Results dictate that the design algorithm achieved accurate and overlapping results with minimum absolute errors in comparison with the techniques available in the latest literature.  I  II  III  I  II  III  I  II  III  I  II  III   10

Conclusions
In this paper, we have investigated different heat transfer problems arising in various engineering fields. The mathematical models of these problems are presented by highly nonlinear differential equations with initial and boundary conditions. We summarize our finding as follows: • A novel computing framework is designed for solving nonlinear problems using neural networks model based on Euler polynomials. The model is further optimized by using the global optimization mechanism of the generalized normal distribution optimization (GNDO) algorithm and the brilliance of local search mechanism of the interior point algorithm. The proposed technique is called the ENN-GNDO-IPA algorithm. • The accuracy of the proposed technique is demonstrated by the comparison of results with numerical solver RK-4 (ode45), and state-of-the-art algorithms including VIM, HPM, DTM, ADM, and exact solutions. Statistics illustrate that the proposed algorithm reaches accuracy with absolute errors having precision of up to 13-14 decimal places. • The results dictate that the temperature distribution of problems 1, 2, 3, and 4 increases with increases in ε 2 , ε, β, γ, Q and γ while possessing an inverse relation with ε 1 , S h and Bi.

•
The magnitude of fitness value calculated during 100 autonomous executions for each scenario of different problems shows the smoothness and convergence of the proposed technique, which further validates its worth. • Extensive graphical and numerical analysis of performance indicators including MAE, TIC, RMSE, and ENSE and their global values further validate the correctness, stability, and reliability of the design scheme. In the future, the idea of Euler polynomials based artificial neural networks can be extended to solve partial and fractional differential equations representing various real world problems. Funding: This research has been funded by the Dirección General de Investigaciones of Universidad Santiago de Cali.

Data Availability Statement:
The data that support the findings of this study are available from the corresponding author upon reasonable request.

Conflicts of Interest:
The authors of this manuscript declare that they have no conflict of interest regarding this manuscript.