A High-Performance Neural Network Algorithm Using a Legendre Ensemble-Based Extreme Learning Machine for Solving Fractional Partial Differential Equations

Abstract

The recent advancement in the use of machine learning techniques across various fields has paved the way for innovative approaches to solving fractional partial differential equations (FPDEs), particularly those utilizing neural networks (NNs). These methods enable efficient representation of complete solutions, leveraging the universal approximation capabilities of neural networks. This study presents a neural network-based method that utilizes the ensemble extreme learning machine (EN-ELM) to efficiently solve FPDEs considered in the sense of the Caputo fractional derivative. The proposed approach incorporates Legendre polynomials to expand input features and employs the radial basis function as the activation function for hidden layer neurons. The EN-ELM framework, enhanced with cross-validation, ensures improved accuracy, stability, and reduced computational complexity. Numerical experiments are conducted to validate the approach, demonstrating its superior accuracy, execution time, and error minimization compared to some known methods. The results confirm the robustness and effectiveness of the proposed method for solving FPDEs.