Multilayer Neural Networks Enhanced with Hybrid Methods for Solving Fractional Partial Differential Equations

Abstract

This paper introduces a novel multilayer neural network technique to solve partial differential equations with non-integer derivatives (FPDEs). The proposed model is a deep feed-forward multiple layer neural network (DFMLNN) that is trained using advanced optimization approaches, namely adaptive moment estimation (Adam) and limited-memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS), which integrate neural networks. First, the Adam method is employed for training, and then the model is further improved using L-BFGS. The Laplace transform is used, concentrating on the Caputo fractional derivative, to approximate the FPDE. The efficacy of this strategy is confirmed through rigorous testing, which involves making predictions and comparing the outcomes with exact solutions. The results illustrate that this combined approach greatly improves both precision and effectiveness. This proposed multilayer neural network offers a robust and reliable framework for solving FPDEs. © 2025 John Wiley & Sons Ltd.