Using deep neural network in computational analysis of coupled systems of fractional integro-differential equations

In recent times, the artificial intelligence (AI) based deep neural networks (DNNs) have attracted much attention from researchers. The aforesaid tools provide best solutions in many real world applications. On the other hand, combining the aforementioned tools with the traditional and fractional calculus tools, significant results can be created for comprehensive analysis of many problems of real world phenomenon and engineering disciplines. Systems of differential equations play important roles in modeling various real world problems involving ordinary or fractional order derivatives. Keeping the mentioned importance in minds, this paper investigates a coupled system of fractional integro-differential equations for qualitative and computational analysis. Utilizing the well known conformable fractional derivative, sufficient results are developed to analyze the solution of the system for existence and uniqueness. The fixed point concepts are utilized to advances to the desired results. Stability results are deduced by using the concept of Ulam–Hyers. At the last section the numerical aspect are investigated of the problem by using RK-4 (Runge–Kutta of order four) method under the mentioned fractional derivative. Also, the mentioned DNNs techniques have used to analysis the considered problems from AI perspectives. To demonstrate our adopted results, some important real world examples are presented at the end. The Levenberg–Marquardt training algorithm is used for classifications of various results including root mean squared error (RMSE), mean square error (MSE), regression coefficient by taking a specified numbers of neuron and epoches. Various graphical illustrations for training, testing, validation and all data are presented.