Purpose: This paper presents the numerical methods for solving the fractional order (FO) model of the accelerating universe in the adapted gravity (AUAG) based on the five different classes by using the deep neural network stochastic structure. The fractional order derivative provides more accurate solutions of the model in comparison with the integer order derivative. Method: A stochastic deep neural network procedure (DNNP) is presented under the optimization of Bayesian regularization for solving the FO-AUAG. The implementation of DNNP is presented by using two different hidden layers (HLs), sigmoid activation function along with 18 and 35 numbers of neurons in both HLs and the optimization is performed through the BR. An Adam method is programmatic to obtain the dataset by reducing the mean square error (MSE) with the statics of 0.76 as training, and 0.12 for both testing and validation. Three different cases based on the fractional order derivatives are considered in the range 0 and 1. Results: The precision of DNNP is authenticated by using the statistical performances based different operators along with the minor absolute error around 10 -06 to 10 -08 , comparison of the outputs and best training values calculated as 10 -11 to 10 -12 . Novelty: The numerical solutions of the nonlinear FO-AUAG have been presented by using a single layer process along with the Levenberg-Marquardt backpropagated neural networks solver. But no one has applied the DNNP procedure based two hidden layers together with the optimization of Bayesian regularization before to solve the FO-AUAG. The proposed DNNP provides better results as compared to single layer neural network process.
Sabir et al. (Tue,) studied this question.