A Deep Learning Framework for Solving Fractional Partial Differential Equations

Abstract

This research focuses on the study and solution of fractional partial differential equations (FPDEs), a critical area in mathematical analysis. FPDEs pose significant challenges due to their complexity, often requiring extensive computational resources to solve. Given the scarcity of exact solutions, numerical methods have been a primary approach for tackling FPDEs. However, these methods often yield substantial but limited results. The ongoing quest for more effective solutions has led researchers to explore new methodologies. Recent advancements in deep learning (DL), particularly in deep neural networks (DNNs), offer promising tools for solving FPDEs due to their exceptional function-approximation capabilities, demonstrated in diverse applications such as image classification and natural language processing. This research addresses the challenges of solving FPDEs by proposing a novel deep feedforward neural network (DFNN) framework. The method integrates the Laplace transform for memory-efficient Caputo derivative approximations and demonstrates superior accuracy across various examples. The results highlight the framework's versatility and computational efficiency, establishing it as a powerful tool for solving FPDEs.