Solving ill-posed problems faster using fractional-order Hopfield neural network

Abstract

The Hopfield neural network (HNN) is one of the most used neural network architectures, it has been used to solve ill-posed problems with great success. Some works in recent years have incorporated fractional-order derivative with respect to time into original Hopfield neural network model. The motivation for this proposal is that it will be included nonlocal features in the original model. However, these studies were restricted to answer questions about the existence, uniqueness and stability of the equilibrium point for the neural network equation system used. The question that was addressed here is how the fractional-order Hopfield neural network (FHNN) can be used to solve ill-posed problems in Physical-Chemistry research field and how they compare to conventional model. For this purpose, fractional-order derivative with respect to time was included on HNN equations, obtained from Lyapunov function defined by 2-norm of the residual function. Therefore, the FHNN model used here was different from that used in previous work. Three prototype problems for the examination of the FHNN model were presented and discussed. In all of them, the solution found by FHNN model was achieved in less time when compared to HNN model. In this work, the Mittag-Leffler stability of the modified FHNN model is demonstrated and used to discuss how fractional order affects the solution. This work shows how the improvement in Hopfield neural network model, using fractional derivative order, gained competitive advantages over integer order Hopfield neural networks.