A physics-informed neural network for solving multi-term space-time fractional differential equations

초록

In this study, we propose an attention-based physics-informed neural networks (PINNs) framework for solving multi-term Riesz-Caputo fractional advection-diffusion equations. The considered model generalizes the classical advection diffusion equation by replacing the first-order temporal derivative with the Caputo fractional derivative of order alpha is an element of(0,1], and the first and second-order spatial derivatives with the Riesz fractional derivatives of orders gamma 1 is an element of(0,1) and gamma 2 is an element of(1,2], respectively. Owing to the presence of multi-term fractional derivatives, conventional numerical methods often encounter challenges related to stability, computational complexity, and approximation accuracy. To address these difficulties, we propose a PINNs framework incorporating attention mechanisms together with hard- and soft-constrained formulations for enforcing initial and boundary conditions. The fractional derivatives are discretized using the L1 scheme for temporal terms and the Grunwald-Letnikov and shifted Grunwald-Letnikov formulations for spatial terms. A systematic comparison is performed among different variants of PINNs like vanilla, self-attention, ResNet, and the proposed location-attention under both hard and soft constraint settings. The numerical results demonstrate that the proposed attention-based framework significantly improves parameter efficiency and solution accuracy while maintaining stable convergence behavior. In particular, the location-based attention mechanism effectively captures the spatially varying and nonlocal dynamics inherent in fractional differential equations. The proposed framework achieves competitive performance compared with existing architectures and conventional numerical approaches, demonstrating its effectiveness for solving complex fractional advection-diffusion problems.

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