Intelligent computing with the knack of Bayesian neural networks for functional differential systems in Quantum calculus model

The purpose behind this research is to utilize the knack of Bayesian solver to determine numerical solution of functional differential equations arising in the quantum calculus models. Functional differential equations having discrete versions are very difficult to solve due to the presence of delay term, here with the implementation of Bayesian solver with means of neural networks, an efficient technique has been developed to overcome the complication in the model. First, the functional differential systems are converted into recurrence relations, then datasets are generated for converted recurrence relations to construct continuous mapping for neural networks. Second, the approximate solutions are determined through employing training and testing steps on generated datasets to learn the neural networks. Furthermore, comprehensive statistical analysis are presented by applying various statistical operators such as, mean squared error (MSE), regression analysis to confirm both accuracy as well as stability of the proposed technique. Moreover, its rapid convergence and reliability is also endorsed by the histogram, training state and correlation plots. Expected level for accuracy of suggested technique is further endorsed with the comparison of attained results with the reference solution. Additionally, accuracy and reliability is also confirmed by absolute error analysis.

Automated summary

The purpose of this research is to utilize a Bayesian solver with neural networks to determine numerical solutions of functional differential equations arising in quantum calculus models. The difficulty in solving functional differential equations with discrete versions is overcome by converting them into recurrence relations and generating datasets for neural networks. The proposed technique is confirmed to be accurate and stable through comprehensive statistical analysis, including mean squared error and regression analysis. The convergence and reliability of the technique are further supported by histogram, training state, and correlation plots, as well as by comparison with a reference solution and absolute error analysis.