Enhanced numerical solutions of fractional differential equations via fibonacci neural networks

The authors of this paper introduce Fibonacci Neural Networks (FNNs) as a novel approach for solving Differential Equations (DE) of any order. The FNN is structured with input, hidden, and output layers, where Fibonacci polynomials of various degrees are utilized as activation functions in the hidden layer. The trial solution of the differential equation is represented as the output of the FNN, with adjustable parameters (weights) that are refined through backpropagation during the network’s training process. To evaluate the effectiveness of this method, seven differential problems with known exact solutions are solved, allowing for a thorough assessment of accuracy. The proposed FNN approach is then compared with existing techniques, revealing higher accuracy and improved performance in solving the given problems. After verifying the accuracy of the proposed method, it was utilized to simulate the motion of an object attached to a spring within a Newtonian fluid. Furthermore, the effects of varying parameters on this motion were examined. The sensitivity of the FNN to different hyperparameters was also analyzed.