Physics Informed Neural Network Method for Solving Delay Hilfer Fractional Differential Equations

In this research, a machine learning method based on physics informed neural network and fractional-order Genocchi wavelets (FGWs) as activation function is explored to solve delay Hilfer fractional differential equations (DHFDEs). In this machine learning algorithm, the FGWs and $\sinh$ functions are used as kernel functions to approximate the solution of DHFDEs. In fact, the solution of DHFDEs is approximated as a combination of the mentioned kernel functions and a set of weights that are learned during the fitting process. We apply the roots of the Legendre functions as training data to develop the algorithm. Then, the training is proposed using the optimizer algorithm. In addition, the error bound of the presented strategy is discussed. Finally, to illustrate the validity and feasibility of our results, three numerical simulation along with several tables and figures are utilized.