Intelligent Backpropagation Networks with Bayesian Regularization for Mathematical Models of Environmental Economic Systems

The research community of environmental economics has had a growing interest for the exploration of artificial intelligence (AI)-based systems to provide enriched efficiencies and strengthened human knacks in daily live maneuvers, business stratagems, and society evolution. In this investigation, AI-based intelligent backpropagation networks of Bayesian regularization (IBNs-BR) were exploited for the numerical treatment of mathematical models representing environmental economic systems (EESs). The governing relations of EESs were presented in the form of differential models representing their fundamental compartments or indicators for economic and environmental parameters. The reference datasets of EESs were assembled using the Adams numerical solver for different EES scenarios and were used as targets of IBNs-BR to find the approximate solutions. Comparative studies based on convergence curves on the mean square error (MSE) and absolute deviation from the reference results were used to verify the correctness of IBNs-BR for solving EESs, i.e., MSE of around 10−9 to 10−10 and absolute error close to 10−5 to 10−7. The endorsement of results was further validated through performance evaluation by means of error histogram analysis, the regression index, and the mean squared deviation-based figure of merit for each EES scenario.


Introduction
Differential equations-based modeling is extensively use for measuring the behavior of complex systems representing different applications of applied sciences, engineering, and technologies. The role of mathematical modeling in environments such as economic and management studies exploiting supply chain dynamics has been growing in interest in the research community in recent years due to the wide utilization in social, industrial, and commercial organizations; starting from the processing from the raw material to the final products, as well as their effective distribution or delivery to clients [1][2][3][4][5]. Conceptual or theoretical investigations are a mandatory phase in order to enhance the operational and financial benefits for the involved companies, optimizing the total cost with reasonable balance of the inventories [6,7]. Thus, different mathematical representations are constructed for a variety of scenarios to predict the feasible situation more accurately [8][9][10].
For example, to forecast the progress of interrelated economies, one has to consider that the economies of countries with relatively low levels of development have economical losses as the world's leading economies suffer due to the economic crisis; therefore, "predatorvictim"-type [11] mathematical models accurately represent their descriptions with the help of a liability differential system as follows [12]:      dy 1 dt = K 1 y 1 (K 2 − y 1 ) − K 3 y 1 y 2 + K 4 y 1 y 3 , y 1 (0) = c 1 , dy 2 dt = K 4 y 2 (K 6 − y 2 ) − K 7 y 1 y 2 + K 8 y 2 y 3 , y 2 (0) = c 2 , dy 3 dt = K 9 y 1 (K 10 − y 1 ) − K 11 y 1 y 2 + K 12 y 1 y 2 , y 3 (0) = c 3 , where K i is the ith function representation of inputs, y 1 and y 2 stand for economical strong countries or regions, while y 3 is the low economic country representation with suitable conditions. Another environmental economic model based on three class nonlinear differential equations is given as [12]: where y 1 is the population of the region or country, y 2 is the population level that represents the nonharmful characteristic of the environment due to economic activities of the masses, and y 3 is the flora level of the region. Similarly, consider another representation of environmental economic differential systems with different settings of indicators as follows [12]:      dy 1 dt = K 1 y 1 (a − y 1 ) − K 2 y 2 + K 3 y 3 , y 1 (0) = c 1 , dy 2 dt = K 4 y 1 (a − y 1 ) − K 5 (b − y 2 ) + K 6 y 3 , y 2 (0) = c 2 , dy 3 dt = K 7 y 1 + K 8 y 2 , where y 1 is the technical diagnostics implementation cost, y 2 is the elimination cost emergencies consequences, y 3 is the industrial system efficiency, and K i is the ith known function or constants. The existing studies available for analyzing the dynamics of EESs are based on numerical and analytical procedures of deterministic nature, while the stochastic or probabilistic procedure based on the exploitation of artificial intelligence (AI) methodologies is relatively less implemented for the analysis of the EESs, as represented in Equations (1)-(3). Probabilistic methodologies via AI algorithms have been implemented exhaustively for a variety of linear/nonlinear systems arising in a spectrum of applications in social, economic, environment, physical, and engineering disciplines [13][14][15][16][17]. A few illustrations of paramount interest include ecological studies [18], acoustics [19], physics [20][21][22][23][24], bioinformatics [25][26][27][28][29][30][31], fluid dynamics [32][33][34][35], financial mathematics [36,37], and energy [38]. These motivational recent relevant and valuable reported articles inspired authors to investigate the intelligence computing paradigm for numerical treatment and analysis for EESs.
The innovative insights and contributions of the proposed stochastic computing via intelligent backpropagation networks of Bayesian regularization (IBNs-BR) [39,40] are briefly listed as follows: • A novel application of AI-based intelligent backpropagation networks via Bayesian regularization for a nonlinear environmental economic system is presented effectively.

•
The reference datasets for IBNs-BR for variants of a nonlinear environmental economic system are accurately assembled by implementation of the Adams numerical solver for different scenarios and are used as targets to find the approximate solutions.

•
The governing mathematical relations of the nonlinear environmental economic system in the form of different differential models representing the fundamental com- The comparative studies based on convergence curves on the mean square error, error histogram analysis, and regression index are used to verify further the correctness of the IBNs-BR for each EES scenario.
The remaining sections of the paper are arranged as follows: Section 2 describes the mathematical representations of EESs for numerical experimentations. Section 3 describes the methodology adopted for solving EESs. Section 4 presents outcomes of the simulation studies of IBNs-BR. Section 5 provides a brief conclusion along with future research openings.

Solution Methodology
The stochastic soft computing platform has been extensively used by the research community for addressing different applications of paramount interest arising in a broad domain of applied science and engineering. A few examples exploiting the strength of AI methods as reliable and effective solution approaches include online learning, scheduling via multi-objective optimization [41], nonlinear Falkner-Skan systems [42], the berth scheduling problem at marine container terminals [43], the entropy generation model [44], density estimation [45], the vehicle routing problem [46], nonlinear Lane-Emden multipantograph delay differential systems [47], the identification of differences between bacterial and viral meningitis [48], the Bouc-Wen hysteresis model for piezostage actuators [49], data classification [50], and the parameter estimation of power signals [51]. All these illustrations prove the worth of heuristics and meta-heuristics methodologies.
The solution methodology for the proposed IBNs-BR to solve EESs consists of two parts: first, dataset generation for IBNs-BR by an appropriate numerical procedure; and secondly, execution of IBNs-BR with the generated data to approximate the solutions of EESs. The generic workflow diagram of the proposed IBNs-BR for EESs is provided in Figure 1. trained/tuned/learned by exploitation of the backpropagation strength of Bayesian regularization procedures [52,53].

The Governing Differential System 3 Reference Numerical Solutions
Create the reference dataset for environment economic system for different scenarios with the help of Adams numerical procedure of predictor corrector by using "NDSolver" routine with method 'Adams'

Environmental Economic Model
Design of intelligent backpropagation networks of Bayesian-Regularization for numerical treatment of nonlinear differential systems representing variants of environmental economic Scenarios

Dataset formulation 4 Adams Numerical Method
Numerical solutions of EESs

IBNs-BR formulation IBNs-BR Execution Steps
if the requirement of step 7 is not fulfilled then continue from step 2 until termination conditions met

Predictor
Step Step n n n n n n n n n n g t y g t y h y y g t y g t y

Analysis on
Error Histogram studies and regression plots for training and testing data of EESs for different Scenarios Figure 1. Process workflow of IBNs-BR for solving EESs.

Adams Method
The numerical procedure of the Adams predictor-corrector scheme was implemented to assemble the reference dataset for the proposed IBNs-BR to find the approximate results for EESs represented in Equations (1)-(3) and the particular scenarios provided in Equations (4)- (7). The detailed mathematical procedure for the reproduction of results with the Adams method is provided in Appendix A, while, in the said study, the Adams procedure is implemented with the auto adjustment of steps from 2 to 12 with the help of the 'NDSolve' routine for the numerical solution of differential equations [52,53], as provided in Mathematica software for solving variants of EESs.

Intelligent Backpropagation Networks of Bayesian-Regularization
The dataset assembled with the Adams numerical procedure was used as standard target outputs for the IBNs-BR. It was constructed with a neuron model formulation with log-sigmoid activation or a windowing kernel, as depicted in Figure 2, in the form of input, hidden, and output layers, while the all-encompassing process block flow is shown in Figure 1. The designed IBNs-BR models were implemented using the neural networks toolbox in Matlab for AI and machine learning paradigms. The weights of networks

Results with Discussion
The numerical experimentation with detailed interpretations of the results for solving EESs as given in Equations (4)-(7) using the IBNs-BR approach is presented here.

Dataset Formulation
The first phase of the methodology implemented for the solution of EESs with the help of the Adams numerical procedure is illustrated in this section to create a dataset for all four variants presented in Equations (4)-(7). The system (4) for implementation of the Adams procedure is given as follows: Similarly, one can develop the expressions for systems in 5, 6, and 7. The dataset created all four EESs for inputs between 0 to 10 with a step size of 0.1 using the Adams solver with default parameters, i.e., accuracy goal, stoppage criteria, and relative error tolerances. The dataset based on solutions, i.e., y 1 , y 2 , and y 3 , for all four EESs for 101 inputs was formulated and is presented for fewer selected inputs in Table 1, for systems in 4, 5, 6, and 7, respectively, with 3 decimal places of accuracy, while these reference solutions are provided in Appendix B, Tables A1-A4, for up to 15 decimal places of accuracy. The IBNs-RB were trained with reference data with higher decimal places of accuracy to avoid the rounding of error issues.

Implementation of IBNs-BR to EESs
The created dataset was used arbitrarily for training, 80% of 303 input grid points, and testing, i.e., 15% of 303 input grid points, by the IBNs-BR procedure, i.e., 15 hidden neurons, log-sigmoid transfer function, and single input, output, and hidden layers, for solving EESs in Equations (4)- (7). The procedure of the IBNs-BR implemented using the 'nftool' routine in the neural network toolbox of Matlab software for all four problems of EESs and results on the basis of the mean square error (MSE), i.e., average absolute of the difference between estimated targets and original targets of reference numerical outcomes, are plotted in Figures 3-5. Figure 3a-d provide the convergence against epoch number for solving a set of Equations (4)- (7), by the designed IBNs-RB, respectively. The final execution parameters of IBNs-RB in terms of epochs, consume time, performance, gradient, µ, effective number, and sum of squared are illustrated for EESs in (4)-(7) in Figure 4a-d, respectively.
The fitness plots with respect to the target outputs along with the absolute error from reference outputs are illustrated in Figure 5a-d for the four EES differential models. One may observe from Figure 3 that the performance was in close vicinity of 10 −10 , while the time taken by IBNs-RB was around 1000 ± 100 s. The testing, best, and training curves on MSE lay around 10 −10 for all four systems given in Equations (4)-(7), as shown in Figure 4. The value of the gradient lay around 10 −5 to 10 −7 , as predicted from Figure 5. The information presented in Figure 5 shows that the maximum value of absolute error lay around 10 −4 to 10 −6 for solving each EES differential model by IBNs-RB.
A detailed analysis of the performance of the IBNs-RB was conducted with the help of histogram studies and regression values for each EES variant. The results of error histogram plots and regression are provided in Figures 6 and 7, respectively. One may see from subfigures of Figure 7 that the maximum instances-based error bar with values 6.72 × 10 −7 , −3.90 × 10 −7 , −1.20 × 10 −5 , and 2.59 × 10 −6 contained the desired zero error line, for EESs in Equations (4)-(7), respectively, which show the reasonable precision/accuracy of IBNs-RB. Additionally, the values of regression were consistently equal to the desired unity values for training, testing, and all datasets for all four variants of EESs solved with the IBNs-RB, as shown in Figure 7a-d. The large overfitting or consistent accuracy of the proposed ANNs-BR scheme in each sample point in training and testing, i.e., the value of regression index R, was very close to its optimal unity value in the case of perfect modeling. However, if the percentages of sampling data were changed drastically, i.e., 50% training and 50% testing, the consistency accuracy in the training sample would still be achieved, i.e., value of R close to unity, but the overfitting observed in testing was not large, and all samplebased data plots of regression, i.e., the values of R, were not as close to unity. Regression plots of the exhaustive numerical experimentation with R close to its optimal value further endorsed the worth of the proposed IBNs-BR. Additionally, we have also critically reviewed the recent reported articles of supervised neural networks [32,33,35,39,46] and found the same trend of regression plots.    The fitness plots with respect to the target outputs along with the absolute error from reference outputs are illustrated in Figures 5a-d for the four EES differential models. One may observe from Figure 3 that the performance was in close vicinity of 10 −10 , while the time taken by IBNs-RB was around 1000 ± 100 s. The testing, best, and training curves on MSE lay around 10 −10 for all four systems given in Equations (4)-(7), as shown in Figure 4. The value of the gradient lay around 10 −5 to 10 −7 , as predicted from Figure 5. The information presented in Figure 5 shows that the maximum value of absolute error lay around 10 −4 to 10 −6 for solving each EES differential model by IBNs-RB.  Figures 6 and 7, respectively. One may see from sub-figures of Figure 7 that the maximum instances-based error bar with values 6.72 × 10 −7 , −3.90 × 10 −7 , −1.20 × 10 −5 , and 2.59 × 10 −6 contained the desired zero error line, for EESs in Equations (4)-(7), respectively, which show the reasonable precision/accuracy of IBNs-RB. Additionally, the values of regression were consistently equal to the desired unity values for training, testing, and all datasets for all four variants of EESs solved with the IBNs-RB, as shown in Figures 7a-d. The large overfitting or consistent accuracy of the proposed ANNs-BR scheme in each sample point in training and testing, i.e., the value of regression index R, was very close to its optimal unity value in the case of perfect modeling. However, if the percentages of sampling data were changed drastically, i.e., 50% training and 50% testing, the consistency accuracy in the training sample would still be achieved, i.e., value of R close to unity, but the overfitting observed in testing was not large, and all sample-based data plots of regression, i.e., the values of R, were not as close to unity. Regression plots of the exhaustive numerical experimentation with R close to its optimal value further endorsed the worth of the proposed IBNs-BR. Additionally, we have also critically reviewed the recent reported articles of supervised neural networks [32,33,35,39,46] and found the same trend of regression plots.

Dynamical Analysis of EESs
The collective behavior or dynamics of all four variants of EESs was also studied, and the results of the IBNs-RB are plotted in Figure 8 only with the reference numerical outcomes. The dynamics of EES differential models Equations (4)- (7)

Dynamical Analysis of EESs
The collective behavior or dynamics of all four variants of EESs was also studied, and the results of the IBNs-RB are plotted in Figure 8 only with the reference numerical outcomes. The dynamics of EES differential models Equations (4)-(7) are portrayed in Figure 8a-d, respectively. Consistent matching/overlapping of the results of the proposed IBNs-RB and numerical solutions are seen for each case. In order to access the accuracy level, the values of absolute error are plotted in Figure 9 for each EES scenario. The values of absolute error lay around 10 −8 to 10 −4 , 10 −8 to 10 −5 , 10 −8 to 10 −4 , and 10 −8 to 10 −4 for EESs in Equations (4)-(7), respectively, as portrayed in Figure 9a-d.

Conclusions
The concluding remarks are summarized as follows:  The purpose of this study was the exploration of the artificial intelligence-based computing paradigm for the numerical treatment of mathematical models representing the environmental economic systems using the competency of intelligent backpropagation networks of Bayesian regularization.  The governing relations of the system model were presented in the form of differen- Figure 9. Comparison of AE for IBNs-BR from reference numerical outcomes for all four variants of environmental economic system.

Conclusions
The concluding remarks are summarized as follows: • The purpose of this study was the exploration of the artificial intelligence-based computing paradigm for the numerical treatment of mathematical models representing the environmental economic systems using the competency of intelligent backpropagation networks of Bayesian regularization.

•
The governing relations of the system model were presented in the form of differential models to portray the dynamics of fundamental compartments or indicators for economic and environmental parameters.

•
The reference datasets of EESs were assembled by the Adams numerical solver for different scenarios and were used effectively as inputs and targets of IBNs-BR to predict the approximate solutions for each scenario.

•
The comparative studies based on convergence curves on the mean square error and absolute deviation from the reference results consistently verified the correctness of IBNs-BR for solving EESs with an accuracy of the order 10 −5 to 10 −8 . • The endorsement of results was further validated through performance evaluation by means of error histogram analyses and the regression index for each EES scenario. • Besides the advantage of consistent precision, stability, and robustness, the limitation of IBNs-BR depended mainly on the availability of the quality dataset for nonlinear systems, which is normally restricted for particular tasks and originations.

Conflicts of Interest:
The authors declare no conflict of interest.