An Efficient Stochastic Numerical Computing Framework for the Nonlinear Higher Order Singular Models

Abstract

Abstract The focus of the present study is to present a stochastic numerical computing framework based on Gudermannian neural networks (GNNs) together with the global and local search genetic algorithm (GA) and active-set approach (ASA), i.e., GNNs-GA-ASA. The designed computing framework GNNs-GA-ASA is tested for the higher order nonlinear singular differential model (HO-NSDM). Three different nonlinear singular variants based on the (HO-NSDM) have been solved by using the GNNs-GA-ASA and numerical solutions have been compared with the exact solutions to check the exactness of the designed scheme. The absolute errors have been performed to check the precision of the designed GNNs-GA-ASA scheme. Moreover, the aptitude of GNNs-GA-ASA is verified on precision, stability and convergence analysis, which are enhanced through efficiency, implication and dependability procedures with statistical data to solve the HO-NSDM. Keywords: gudermannian neural network; higher order; singular; genetic algorithms; statistical analysis; active-set approach

1. Introduction

The research community has always taken keen interest in solving the singular nonlinear differential models. Various schemes have been presented to solve such models, which involve singularity. Few traditional schemes fail to solve the singular model, such as the Adams method, Milne predictor-corrector method, Runge–Kutta and Euler’s method, etc. However, few schemes solve the singular models, such as homotopy perturbation method, Adomian decomposition method, Bernoulli collocation method, spectral collocation method, quasi linearization scheme, homotopy analysis method and variation iteration method [1,2,3,4,5,6,7,8,9,10], but all these techniques solve the singular models by taking the approximate values close to zero. However, stochastic numerical heuristic/swarming techniques solve the singular models at exactly zero without approximating. There are not only singular nonlinear models that have been solved by using the stochastic numerical techniques, but also the delayed, prediction, fractional, functional and pantograph differential models have also been treated with the stochastic computing methods [11,12,13]. To mention the importance of the singular models, no one can deny their significance due to the variety of applications in fluid mechanics, relativity theory, dynamics of population evolution, pattern construction and chemical reactors [14,15,16,17].

An important and historical singular system is the Emden–Fowler model, discovered centuries ago by working on a spherical cloud of gas along with the classical thermodynamic law. Additional applications of the singular systems are catalytic diffusion reactions [18], isothermal gas spheres [19], density state of gaseous stars [20], stellar formulation [21], electromagnetic theory [22], quantum mechanics [23], oscillating magnetic fields [15] and mathematical physics [24]. The general form of this model is provided as: 𝑦 ″ ( ℜ ) + 𝜓 ℜ 𝑦 ′ ( ℜ ) + ℎ ( ℜ ) 𝑔 ( 𝑦 ) = 𝑓 ( ℜ ) , 𝑦 ( 0 ) = 𝑎 , 𝑦 ′ ( 0 ) = 0 . (1) where ℎ ( ℜ ) and 𝑔 ( 𝑦 ) are the identified functions with input ℜ and solution y, respectively. 𝜓 ≥ 1 represents the shape factor, ℎ ( ℜ ) is the forcing function and a is used as a constant. The models (1) give different values for different values of 𝑔 ( 𝑦 ) , e.g., temperature deviation, interior polytrophic stars structure, radiative cooling, gas clouds and galaxy clusters modelling [14,15,16,17]. The model presented in system (1) becomes a Lane–Emden equation for ℎ ( ℜ ) = 1 and is given as: 𝑦 ″ ( ℜ ) + 𝜓 ℜ 𝑦 ′ ( ℜ ) + 𝑔 ( 𝑦 ) = 𝑓 ( ℜ ) , 𝑦 ( 0 ) = 𝑎 , 𝑦 ′ ( 0 ) = 0 . (2)

In the above model, the factor 𝑔 ( 𝑦 ) = 𝑦 𝑚 indicates the linearity or nonlinearity of the singular model. For m = 0 and 1, the above model is linear; for the rest of the values, the model has a nonlinear form. The aim of the study is to introduce a stochastic numerical computing framework based on Gudermannian neural networks (GNNs) together with the global and local search genetic algorithm (GA) and active-set approach (ASA), i.e., GNNs-GA-ASA. The designed computing framework GNNs-GA-ASA is tested for the higher order nonlinear singular differential model (HO-NSDM). The general form of the HO-NSDM is given as [25,26]: 𝑦 ( 𝑖 𝑣 ) ( ℜ ) + 𝜓 ℜ 𝑦 ‴ ( ℜ ) + ℎ ( ℜ ) 𝑔 ( 𝑦 ) = 𝑓 ( ℜ ) , 𝑦 ( 0 ) = 𝑎 , 𝑦 ′ ( 0 ) = 𝑏 , 𝑦 ″ ( 0 ) = 𝑐 , 𝑦 ‴ ( 0 ) = 0 . (3)

The implementation of the GNNs-GA-ASA on the above higher order, singular and nonlinear model provides a useful platform to researchers. The stochastic solvers have been applied in diverse applications. Some prominent applications are functional differential systems [27,28], doubly singular nonlinear systems [29], prey-predator models [30], the Thomas–Fermi model [31], HIV infection models [32,33,34], periodic differential models [35] and fractional differential models [36]. Based on this evidence, we are interested in designing a platform based on the GNNs that has never been implanted before to solve the HO-NSDM. A few main and novel geographies of the designed GNNs-GA-ASA based on the comprehensive highlights are provided below.

The design of the GNNs-GA-ASA aims to solve models that involve singularity at the origin, nonlinearity and higher order.

The design of GNNs-GA-ASA is debated in Section 2. Section 3 indicates the statistical procedures. Section 4 shows the result and discussions. The concluding remarks are provided in Section 5.

2. Methodology

In this section, the designed outlines are presented to solve the HO-NSDM, which is separated into two portions:

The descriptions of an objective fitness function that is constructed on the basis of a differential model and its initial conditions; and The hybridization of GA-ASA is described to enhance the formulation of the fitness for HO-NSDM given in Equation (3).

The form of the layer structure, i.e., hidden, input and output ANNs layers with an activation Gudermannian function, are applied as an estimated continuous mapping. The designed systems are presented in the singular, higher order and nonlinear equations to formulate a merit function by summing the MSE for the differential model and the boundary/initial conditions. A merit function is optimized with the integrated strength of global/local techniques by manipulating the GA at the start and then fine tuning with the local and rapid search scheme ASA, i.e., GNN-GA-ASA.

2.1. ANNs Modeling by Using Gudermannian Function

The models based on ANNs with the activation Gudermannian function are introduced by the scientists to solve different systems that arise in many areas. The HO-NSDM has been solved by using the stochastic procedures by implementing the Log-sigmoid function as a neural network. The Gudermannian function has never been implemented nor explored to solve the HO-NSDM. Thus, we are interested in exploring Gudermannian neural networks for the research community to solve HO-NSDM. The hidden, input and output layer structure based on the feed-forward GNNs are applied to provide the continuous mapping solution form as: 𝑦 ^ ( ℜ ) = ∑ 𝑖 = 1 𝑛 𝑎 𝑖 𝑄 ( 𝑝 𝑖 ℜ + 𝑞 𝑖 ) , (4) 𝑦 ^ ( 𝑖 𝑣 ) ( ℜ ) = ∑ 𝑖 = 1 𝑛 𝑎 𝑖 𝑄 ( 𝑖 𝑣 ) ( 𝑝 𝑖 ℜ + 𝑞 𝑖 ) , (5) where 𝑎 𝑖 , 𝑝 𝑖 and 𝑞 𝑖 are ith components of 𝑎 , 𝑝 and 𝑞 vectors, and m and n indicate the derivative order and neurons. The Gudermannian function 𝑄 ( ℜ ) = 2 tan − 1 [ exp ( ℜ ) ] − 1 2 𝜋 along with its mth derivatives work as an objective function presented in the Equations (6) and (7), respectively. The use of Gudermannian function in Equations (4) and (5) give the new mathematical form as: 𝑦 ^ ( ℜ ) = ∑ 𝑖 = 1 𝑛 𝑎 𝑖 ( 2 tan − 1 𝑒 ( 𝑝 𝑖 ℜ + 𝑞 𝑖 ) − 𝜋 2 ) , (6) 𝑦 ^ ′ ( ℜ ) = ∑ 𝑖 = 1 𝑛 𝑎 𝑖 2 𝑝 𝑖 𝑒 ( 𝑝 𝑖 ℜ + 𝑞 𝑖 ) 1 + 𝑒 2 ( 𝑝 𝑖 ℜ + 𝑞 𝑖 ) , 𝑦 ^ ″ ( ℜ ) = ∑ 𝑖 = 1 𝑛 2 𝑎 𝑖 ( − 2 𝑝 𝑖 2 𝑒 3 ( 𝑝 𝑖 ℜ + 𝑞 𝑖 ) ( 1 + 𝑒 2 ( 𝑝 𝑖 ℜ + 𝑞 𝑖 ) ) 2 + 𝑝 𝑖 2 𝑒 ( 𝑝 𝑖 ℜ + 𝑞 𝑖 ) 1 + 𝑒 2 ( 𝑝 𝑖 ℜ + 𝑞 𝑖 ) ) , 𝑦 ^ ‴ ( ℜ ) = ∑ 𝑖 = 1 𝑛 2 𝑎 𝑖 ( 8 𝑝 𝑖 3 𝑒 5 ( 𝑝 𝑖 ℜ + 𝑞 𝑖 ) ( 1 + 𝑒 2 ( 𝑝 𝑖 ℜ + 𝑞 𝑖 ) ) 3 − 8 𝑝 𝑖 3 𝑒 3 ( 𝑝 𝑖 ℜ + 𝑞 𝑖 ) ( 1 + 𝑒 2 ( 𝑝 𝑖 ℜ + 𝑞 𝑖 ) ) 2 + 𝑝 𝑖 3 𝑒 ( 𝑝 𝑖 ℜ + 𝑞 𝑖 ) 1 + 𝑒 2 ( 𝑝 𝑖 ℜ + 𝑞 𝑖 ) ) , 𝑦 ^ ( 𝑖 𝑣 ) ( ℜ ) = ∑ 𝑖 = 1 𝑛 2 𝑎 𝑖 ( − 48 𝑝 𝑖 4 𝑒 7 ( 𝑝 𝑖 ℜ + 𝑞 𝑖 ) ( 1 + 𝑒 2 ( 𝑝 𝑖 ℜ + 𝑞 𝑖 ) ) 4 + 72 𝑝 𝑖 4 𝑒 5 ( 𝑝 𝑖 ℜ + 𝑞 𝑖 ) ( 1 + 𝑒 2 ( 𝑝 𝑖 ℜ + 𝑞 𝑖 ) ) 3 − 26 𝑝 𝑖 4 𝑒 3 ( 𝑝 𝑖 ℜ + 𝑞 𝑖 ) ( 1 + 𝑒 2 ( 𝑝 𝑖 ℜ + 𝑞 𝑖 ) ) 2 + 𝑝 𝑖 4 𝑒 ( 𝑝 𝑖 ℜ + 𝑞 𝑖 ) 1 + 𝑒 2 ( 𝑝 𝑖 ℜ + 𝑞 𝑖 ) ) . (7)

For the solution of Equation (3), the GNNs models for 𝑦 ^ , 𝑦 ^ ′ , 𝑦 ^ ″ , 𝑦 ^ ‴ and 𝑦 ^ ( 𝑖 𝑣 ) are applied, as provided in Equation (7). The objective function in the sense of mean squared error (MSE) is given as: 𝜉 𝐹 = 𝜉 𝐹 − 1 + 𝜉 𝐹 − 2 (8) 𝜉 𝐹 − 1 = 1 𝑁 ∑ 𝑖 = 1 𝑁 ( 𝑦 ^ ( 𝑖 𝑣 ) 𝑖 + 𝑘 ℜ 𝑖 − 1 𝑦 ^ ‴ 𝑖 + ℎ 𝑖 𝑔 ( 𝑦 ^ 𝑖 ) − 𝑓 𝑖 ) 2 , (9) 𝜉 𝐹 − 2 = 1 4 ( ( 𝑦 ^ 0 − 𝑎 ) 2 + ( 𝑦 ^ ′ 0 − 𝑏 ) 2 + ( 𝑦 ^ ″ 0 − 𝑐 ) 2 + ( 𝑦 ^ ‴ 0 ) 2 ) , (10) where 𝜉 𝐹 − 1 and 𝜉 𝐹 − 2 are the MSEs associated with the general form of Equation (3) and the related ICs, respectively. The terms 𝑁 ℎ = 1 , 𝑦 ^ 𝑖 = 𝑦 ^ ( ℜ 𝑖 ) , ℜ 𝑖 = 𝑖 ℎ , 𝑔 𝑖 = 𝑔 ( 𝑦 ) and 𝑓 𝑖 = 𝑓 ( 𝑦 ) . A suitable optimization technique is assumed for the learning of W = [ 𝑎 , 𝑝 , 𝑞 ], i.e., a weight vector, and the objective Function (8) tends to be zero.

2.2. Optimization Procedures: GA-SQP

The weights based on the GNNs are proficient by functioning the combined strength in terms of GAs along with ASA, i.e., GA-ASA. The graphical representations of the designed GNNs-GA-ASA to present the solution of HO-NSDM are illustrated in Figure 1.

Global search efficiency of the GA, introduced at the end of the 19th century by Holand, is explored to obtain the weight vector (W) using the GNNs in the existing investigation. The population’s formulation with participant outcomes, i.e., chromosomes or individuals in GA, is accomplished by applying the real numbers with some bounds in a determinate interval. GA has been pragmatic in numerous applications, including heterogeneous bin packing optimization [37], emergency logistics humanitarian preparation [38], second order singular models [39,40], wellhead back pressure control system [41], the electricity consumption modeling [42], image steganography [43], collaborative filtering recommender system [44] and manufacturing systems [45].

The optimization of the decision variables is performed initially through GNNs by using GA, and after adequate trials, the GA performance is considerably improved by using fine tuning with the suitable rapid local process by taking the global best GA values as a preliminary weight. Subsequently, an efficient ASA scheme is implemented for updating the parameters of the network. In recent decades, ASA has been applied in numerous areas, e.g., unconstrained minimax problems [46], linear MPC [47], water distribution model to control the flow [48], optimal control problem governed by partial differential equation [49], elastodynamic frictional contact problems [50] and constrained node-based shape optimization [51]. The procedural structure of the flow diagram using the proposed GNNs-GA-ASA is shown in Figure 1, and essential details are given in the pseudocode form through the optimization of GA-ASA in Table 1.

3. Performance Measures

The performance operators based on the mean absolute deviation (MAD), Nash Sutcliffe efficiency (NSE) and Theil’s inequality coefficient (TIC) are presented to solve the HO-NSDM. The mathematical formulae of these operators are provided as: TIC = 1 𝑛 ∑ 𝑘 = 1 𝑛 ( 𝑦 𝑘 − 𝑦 ^ 𝑘 ) 2 ( 1 𝑛 ∑ 𝑘 = 1 𝑛 𝑦 𝑘 2 + 1 𝑛 ∑ 𝑘 = 1 𝑛 𝑦 ^ 𝑘 2 ) , (11) MAD = 1 𝑛 ∑ 𝑘 = 1 𝑛 | 𝑦 𝑘 − 𝑦 ^ 𝑘 | , (12) NSE = { 1 − ∑ 𝑘 = 1 𝑛 ( 𝑦 𝑘 − 𝑦 ^ 𝑘 ) 2 ∑ 𝑘 = 1 𝑛 ( 𝑦 𝑘 − 𝑦 ¯ 𝑘 ) 2 , 𝑦 ¯ 𝑘 = 1 𝑛 ∑ 𝑘 = 1 𝑛 𝑦 𝑘 (13) ENSE = 1 - NSE . (14)

4. Results and Discussions

In this section, the detailed results of solving the HO-NSDM for three different cases using the GNNs-GA-ASA are presented.

Consider an HO-NSDM using an exponential function given as:

The reference solution of the HO-NDSM is ln ( 1 + ℜ 4 ) and the objective function is given as: 𝜉 𝐹 = 1 𝑁 ∑ 𝑖 = 1 𝑁 ( ℜ 𝑖 𝑦 ^ ( 𝑖 𝑣 ) + 3 𝑦 ^ ‴ − 96 ℜ 𝑖 ( 5 ℜ 𝑖 8 − 10 ℜ 𝑖 4 + 1 ) 𝑒 − 4 𝑦 ^ ) 2 + 1 4 ( ( 𝑦 ^ 0 ) 2 + ( 𝑦 ^ ′ 0 ) 2 + ( 𝑦 ^ ″ 0 ) 2 + ( 𝑦 ^ ‴ 0 ) 2 ) (16)

Consider a highly nonlinear HO-NSDM using an exponential function given as:

The reference solution of the HO-NDSM is ( 1 + ℜ 4 ) − 1 2 and the objective function is given as: 𝜉 𝐹 = 1 𝑁 ∑ 𝑖 = 1 𝑁 ( ℜ 𝑖 3 𝑦 ^ ( 𝑖 𝑣 ) + 12 ℜ 𝑖 2 𝑦 ^ ‴ + 36 ℜ 𝑖 𝑦 ^ ″ + 24 𝑦 ^ ′ + 60 ℜ 𝑖 3 ( 3 ℜ 𝑖 8 − 18 ℜ 𝑖 4 + 7 ) 𝑦 ^ 9 ) 2 + 1 4 ( ( 𝑦 ^ 0 − 1 ) 2 + ( 𝑦 ^ ′ 0 ) 2 + ( 𝑦 ^ ″ 0 ) 2 + ( 𝑦 ^ ‴ 0 ) 2 ) (18)

Consider a highly nonlinear HO-NSDM using an exponential function given as:

The reference solution of the HO-NDSM is ( 1 + ℜ 4 ) 1 4 and the objective function is given as: 𝜉 𝐹 = 1 𝑁 ∑ 𝑖 = 1 𝑁 ( ℜ 𝑖 2 𝑦 ^ ( 𝑖 𝑣 ) + 4 ℜ 𝑖 𝑦 ^ ‴ + 2 𝑦 ^ ″ − 3 ℜ 𝑖 2 ( 12 ℜ 𝑖 8 − 53 ℜ 𝑖 4 + 12 ) 𝑦 ^ − 15 ) 2 + 1 4 ( ( 𝑦 ^ 0 − 1 ) 2 + ( 𝑦 ^ ′ 0 ) 2 + ( 𝑦 ^ ″ 0 ) 2 + ( 𝑦 ^ ‴ 0 ) 2 ) . (20)

The graphical representations of the design GNNs-GA-ASA for each problem of the HO-NSDM are provided in Figure 2, Figure 3, Figure 4 and Figure 5. The optimization performances of the designed method are provided for 30 independent executions using the hybrid combination of GA-ASA. Figure 2 indicates the weights set along with the result comparisons using the GA-ASA. It is observed that the obtained results overlapped with the exact solutions for each problem of the HO-NDSM. To find the resulting similarities, the AE performances based on the obtained and exact solutions are plotted in Figure 3. Figure 3a shows that the best values of the AE are found around 10−4 to 10−6 for Problem 1, 10−3 to 10−5 for Problem 2 and 10−4 to 10−5 for Problem 3. Figure 3b indicates the best values of the AE found around 10−8 to 10−10 for Problem 1, and 10−5 to 10−7 for Problems 2 and 3. Figure 3c authenticates the best values of the performance indices for each problem of the HO-NDSM. It is observed that the fitness values for Problem 1 are found at around 10−6 to 10−8, while the fitness values for Problems 2 and 3 are close to 10−6. The MAD and TIC values for each problem of the HO-NDSM are around 10−4 to 10−6 and 10−8 to 10−10. The ENSE performance for each problem are found around 10−7 to 10−8 for Problem 1, 10−6 to 10−7 for Problem 2 and 10−6 to 10−8 for Problem 3. The best values of FIT, MAD, TIC and ENSE found in appropriate ranges to solve each problem of the HO-NDSM.

The graphical illustrations of the statistical operators based on the fitness values, MAD, TIC and ENSE are drawn in Figure 4, Figure 5, Figure 6 and Figure 7 for each problem of the HO-NDSM. The convergence performance of 𝜉 𝐹 , MAD, ENSE and TIC is obtained for 30 independent executions to solve each problem of the HO-NDSM. It is noticed that the FIT values, MAD performances, TIC measures and ENSE operators achieve satisfactory levels of accuracy and around 75% of the executions achieved an accurate level of precision based on the FIT, MAD, TIC and ENSE.

To find the reliability of GNNs-GA-ASA, the statistical performances for 30 implementations based on minimum (Min), Median (Med), Mean and semi-interquartile range (S.I.R) are presented to solve the HO-NDSM. The mathematical form of the S.I.R is − 0.5 ( 𝑄 1 − 𝑄 3 ) , and the 𝑄 1 and 𝑄 3 values represent the first and third quartiles. The Min, Med, Mean and S.I.R operatives are given in Table 2 for the HO-NDSM. The independent trials of the present GNNs-GA-ASA approach for Min error are called the best runs. One can observe that the suitable Min values are calculated at around 10−5 to 10−6 for each problem of the HO-NDSM. Likewise, the Mean values for each problem of the HO-NDSM are calculated at around 10−1 to 10−2, while the Med and S.I.R values for each problem of the HO-NDSM are found around 10−2 to 10−3. Table 3 shows the computational cost of GNNs-GA-ASA based on completing iterations, count of functions and executed time during the process to present the decision variables of the network.

5. Conclusions

The motivation of the current work was to examine the numerical outcomes of the nonlinear higher order singular system by operating the hybrid computing intelligent strength based on GNNs under the optimization of GAs along with ASA. Several major conclusions of the obtained performances are drawn:

A neuro-evolution was designed based on computing the GNNs-GA-ASA approach efficiently for the nonlinear higher order multi-singular differential systems. The designed GNNs-GA-ASA is a suitable procedure to solve the stiff singular problems effectively. The precision and accuracy of the GNN-GA-ASA was authenticated by comparing the obtained measures with the obtainable exact results to solve each variant of the HO-NSDM. The performance achieved through the AE values for solving the HO-NSDM authenticated good measures at around 10−4 to 10−6 for Problem 1, 10−3 to 10−5 for Problem 2 and 10−4 to 10−5 for Problem 3. The statistical assessments and analysis of 30 independent trials based on the GNNs-GA-ASA were implemented to establish the convergence and accuracy of the designed approach to solve three different variants of the HO-NSDM.

In the future, the presented GNNs-GA-ASA will be extended to solve the non-autonomous evolution equations [52,53,54].