Numerical Methods for Fractional Differential Equations Using Physics Informed Neural Networks

Abstract

This paper investigates the numerical solution of fractional differential equations involving the Erdelyi-Kober fractional operator, which play a pivotal role in modeling memory-dependent and anomalous dynamic processes in applied mathematics, physics, and engineering, including anomalous diffusion, viscoelasticity, and heterogeneous heat conduction. We introduce a novel neural network-based framework that integrates Physics-Informed Neural Networks with the Theory of Functional Connections and employs an Extreme Learning Machine for efficient training. A new loss function is constructed using the L1 finite difference scheme in combination with Volterra integral equations, enabling accurate approximation of Erdelyi-Kober fractional derivatives while ensuring real-valued computations and minimal error. The proposed approach presents three main contributions over conventional method: (i) seamless integration of Theory of Functional Connections with Physics-Informed Neural Networks specifically for Erdelyi-Kober fractional derivatives, (ii) a novel loss function formulation tailored to the Erdelyi-Kober operator, and (iii) enhanced accuracy and  computational efficiency. A neural network approach based on L1 is compared with the conventional L1 numerical scheme in order to solve fractional differential equations. The L1 discretization approach is used in a deep learning model in the proposed neural architecture to increase efficiency and accuracy. Using numerical analyses, we demonstrate that the L1 neural network technique achieves reliable convergence and exceeds the accuracy of the traditional L1 method. Root Mean Squared Error (RMSE), Mean Absolute Error (MAE), and coefficient of determination (R²) are performance metrics that further support the model's effectiveness and demonstrate its resilience while handling fractional-order problems. The results show that the combination of neural networks and the L1 scheme provides a strong and accurate solution for solving complex fractional differential equations. Several illustrative examples are provided to demonstrate the efficacy, reliability, and practical relevance of the methodology. The results highlight the capability of the proposed framework to solve complex fractional differential equations effectively, offering a powerful tool for researchers and practitioners in applied mathematics, engineering, and related scientific disciplines.