The motive of this work is to provide the numerical performances of the monkeypox transmission mathematical model by using a novel deep neural network process with eleven and twenty-two neurons in the hidden layers. The purpose to provide the deep neural network stochastic process is to obtain more accurate solutions of the monkeypox transmission mathematical system. This process is enhanced by using an activation radial basis function in both layers for solving the monkeypox transmission mathematical model along with the implementation of the Bayesian regularization optimization scheme. The presentation of the mathematical dynamical model has two categories, human and rodent. The human dynamics is classified into, susceptible, exposed, infectious, clinically ill human and recovered individuals. The rodent is divided into three forms, susceptible, exposed, and infected. A dataset is presented with the Adam approach that is processed using the training, testing, and certification procedure by taking the data as 0.13, 0.12 and 0.15. The correctness is observed through the matching of the results and the statistical plots are plotted using the regression, state transition, error histograms and correlation.

The current work presents the numerical solutions of the robotic system in the process of coronavirus. The stochastic performances using the modeling of Gudermannian neural networks (GDMNNs) are provided along with the global search genetic algorithm (GA) and rapid interior-point scheme (IPS), i.e., GDMNNs-GAIPS. An error function using the differential form of the model is created and then optimized by applying the hybridization of GA-IPS. The correctness and accuracy of the stochastic procedure GDMNNs-GAIPS is examined by using the comparison of the proposed and reference results. The reliability and substantiation of the proposed GDMNNs-GAIPS is authenticated by using the statistical operators based on the mean square error, Theil inequality coefficient and variance account for. Forty numbers of independent trials along with ten numbers of hidden neurons have been used to solve the mathematical model of robotic system to detect the positive cases of COVID-19.

The current investigations present the numerical solutions of the novel singular nonlinear fifth-order (SNFO) system of multi-pantograph differential model (SMPDM), i.e., SNFO–SMPDM. The novel SNFO–SMPDM is obtained using the sense of the second kind of typical Emden–Fowler and prediction differential models. The features of shape factor, pantograph along with singular points are provided for all four obtained classes of the SNFO–SMPDM. The extensive use of the singular models is observed in the engineering and mathematical systems, e.g., inverse systems and viscoelasticity or creep systems. For the correctness of the proposed novel SNFO–SMPDM, one case of each class is numerically handled by applying supervised neural networks (SNNs) along with the optimization of Levenberg–Marquardt backpropagation scheme (LMBS), i.e., SNNs–LMBS. A dataset using the traditional variational iteration scheme is designed to compare the proposed results of each case of SNFO–SMPDM. The obtained approximate solutions of each class using the novel SNFO-SMPDM are presented based on the training (80%), authentication (10%) and testing (10%) measures to evaluate the mean square error. Fifteen numbers of neurons, and sigmoid activation function are used in this SNN process. To authenticate the competence, and precision of SNFO–SMPDM, the numerical simulations are accessible by applying the relative measures of regression, error histogram plots, and correlation.

This study designs the Morlet wavelet neural network (MWNN) for the numerical performance of the second-order delay differential perturbed singular model (DD-PSM). These stiff singular models are always challenging for the research community to numerically present their results. The DD-PSM is used as an objective function, and its boundary conditions are assembled and then optimised using the computing hybrid proficiency of the global genetic algorithm (GA) and local active-set approach (ASA). Details of the singularity, shape factor, perturbed and delay terms based on the DD-PSM are also provided. Three problems of the DD-PSM are presented and numerically solved using the MWNN–GA–ASA. The precision of the MWNN–GA–ASA is studied by comparing the proposed solution-based DD-PSM and exact solutions. Moreover, a comparison of the MWNN with the Meyer wavelet neural network is presented. The reliability, convergence, correctness and constancy of the numerical scheme are observed by different statistical performances.

Abstract The goal of this research is to design the Gudermannian neural networks (GNNs) to solve a type of two-point nonlinear singular boundary value problems (TPN-SBVPs) that arise within thermal-explosion theory. The results of these investigation are provided for different neurons (4, 12 and 20), as well as absolute error along with the time complexity. For solving the TPN-SBVPs, a genetic algorithm (GA) and sequential quadratic programming (SQP) are used to optimize the error function. The accuracy of designed GNNs is provided by using a hybrid GA–SQP combination, which is based on a comparison of obtained and actual solutions. Furthermore, statistical analysis of the data is proposed in order to establish the competence as well as effectiveness of designed and the efficacy of the designed computing framework for solving the TPN-SBVPs.

In this work, a swarming computational procedure is presented for the numerical treatment of the dynamical model of the susceptible, exposed, infected, and recovered (SEIR) classes that portrayed the spreading of Zika virus. The artificial neural network procedures (ANNPs) have been applied to solve the SEIR mathematical model for spreading of the Zika virus together with the hybridization efficiency of global swarming and local search schemes. The global particle swarm optimization (PSO) and local search active-set algorithm (ASA) have been proposed to solve the model. An error based objective function is presented for the SEIR differential model and then optimized by the hybrid computing efficiency of PSO-ASA. Five neurons, fifteen variables of each class and ten numbers of trials have been used to solve the SEIR mathematical model for spreading of the Zika virus. The correctness of the proposed computing ANNPs-PSO-ASA is observed by using the comparison of the obtained and reference solutions along with the performances of the absolute error, ranges around 10−06 to 10−08. The reliability of the designed computing ANNPs-PSO-ASA technique is observed by using the statistical operator performances on single/multiple trials for the SEIR system for spreading of the Zika virus dynamics.

Abstract The aim of the current study is to present the numerical solutions of a nonlinear second-order coupled Emden–Fowler equation by developing a neuro-swarming-based computing intelligent solver. The feedforward artificial neural networks (ANNs) are used for modelling, and optimization is carried out by the local/global search competences of particle swarm optimization (PSO) aided with capability of interior-point method (IPM), i.e., ANNs-PSO-IPM. In ANNs-PSO-IPM, a mean square error-based objective function is designed for nonlinear second-order coupled Emden–Fowler (EF) equations and then optimized using the combination of PSO-IPM. The inspiration to present the ANNs-PSO-IPM comes with a motive to depict a viable, detailed and consistent framework to tackle with such stiff/nonlinear second-order coupled EF system. The ANNs-PSO-IP scheme is verified for different examples of the second-order nonlinear-coupled EF equations. The achieved numerical outcomes for single as well as multiple trials of ANNs-PSO-IPM are incorporated to validate the reliability, viability and accuracy.