On Nonlinear Financial Fractional-Order Model Using Artificial Deep Neural Networks

In this manuscript, we investigate a fractional-order conformable three-dimensional chaotic financial model with interest rate, investment demand, and price index compartments. On the application of fixed-point theorems and nonlinear analysis, we establish theoretical results regarding the existence and uniqueness of a solution and also study Ulam–Hyers criteria for the stability of the solution of the considered system. Further, we use the fractional-order Runge–Kutta (RK-4) method to approximate the solution of our problem. Also, deep neural network (DNN) techniques are applied to investigate the model from artificial intelligence (AI) perspectives. Numerical simulation shows that it reproduces accurately the qualitative dynamics and confirms the theoretical stability results of the mentioned system. Subsequently, for the DNN analysis, we follow the Levenberg–Marquardt algorithm using Matlab 2023. Different quantities like the root-mean-square error (RMSE), mean squared error (MSE), and regression coefficient and a comparison with numerical data are presented graphically. Also, absolute errors between numerical values and those predicted by DNNs corresponding to different fractional orders are presented.