A new efficient algorithm based on feedforward neural network for solving differential equations of fractional order

Artificial neural network (ANN) have shown great success in various scientific fields over several decades. Recently, one of its variants known as deep feedforward neural network (FNN) led to dramatic improvement in many tasks, including getting more accurate approximation solution for integer-order differential equations. However, its capability on solving FDEs is still remain questionable. Thus, this paper aims to design new scheme based on deep feedforward neural network (FNN) with vectorized algorithm (FNNVA) using selected first-order optimization techniques which are gradient descent (GD), momentum method (MM) and adaptive moment estimation method (Adam) to solve Caputo FDEs. At the first stage, a detailed method formulations on solving Caputo FDEs using FNN are presented. Then, a vectorized algorithm is developed for the scheme to be computationally efficient. The effectiveness and applicability of the scheme were validate on linear and nonlinear FDEs through comparison based on different number of hidden layers with varying learning rates and number of neurons. The results show that FNNVA with Adam technique with one and two hidden layers outperformed among others by appropriate selection value of learning rates and number of neurons respectively. This scheme also provide high accuracy and low computational cost compared to several existing numerical methods. •To design new scheme based on deep feedforward neural network (FNN) with vectorized algorithm (FNNVA).•To propose a detailed method formulations on solving Caputo FDEs using FNN are presented.•A vectorized algorithm is developed for the scheme to be computationally efficient.•The effectiveness and applicability of the scheme were validated on linear and nonlinear FDEs.