Bernstein Neural Networks Method for Solving Variable Order Fractional Mixed Volterra-Fredholm Integro-Differential Equations

Abstract

This paper deals with the drawback of the semi-group properties in variable order fractional mixed Volterra-Fredholm integro-differential equations (VO-FVFIDEs) under Caputo derivative operator. The variable order of the equation is converted to piecewise constant functions by partition it into sub-intervals. The existence and uniqueness of solutions are investigated. A novel technique using Bernstein Neural Network (BernsteinNN) method is proposed to obtain the approximate solution for the FVFIDEs, which used the basis of Bernstein polynomials instead of the activation function in the artificial neural network method. The loss function is developed by adding the L2 regularization for parameter terms and the hyper-parameter λ to ensure the stability of training process and to control the regularization strength, respectively. Adam optimization approach is  applied to training the neural networks and the model performance is computed using the mean square error.  The validity of the presented method is demonstrated through the presented example.