Transformed Diffusion-Wave fPINNs: Enhancing Computing Efficiency for PINNs Solving Time-Fractional Diffusion-Wave Equations
For researchers solving time-fractional PDEs, this work offers a more efficient mesh-free method, though it is an incremental improvement over existing PINN approaches.
The paper introduces tDWfPINNs, which use an integrand transformation to reduce computational costs of fractional derivative evaluation in PINNs for time-fractional diffusion-wave equations, achieving superior efficiency without sacrificing accuracy.
We propose transformed Diffsuion-Wave fractional Physics-Informed Neural Networks (tDWfPINNs) for efficiently solving time-fractional diffusion-wave equations with fractional order $α\in(1,2)$. Conventional numerical methods for these equations often compromise the mesh-free advantage of Physics-Informed Neural Networks (PINNs) or impose high computational costs when computing fractional derivatives. The proposed method avoids first-order derivative calculations at quadrature points by introducing an integrand transformation technique, significantly reducing computational costs associated with fractional derivative evaluation while preserving accuracy. We conduct a comprehensive comparative analysis applying this integrand transformation in conjunction with both Monte Carlo integration and Gauss-Jacobi quadrature schemes across various time-fractional PDEs. Our results demonstrate that tDWfPINNs achieve superior computational efficiency without sacrificing accuracy. Furthermore, we incorporate the proposed approach into adaptive sampling approaches such as the residual-based adaptive distribution (RAD) for the time-fractional Burgers equation with order $α\in(1,2)$, which exhibits complex solution dynamics. The experiments show that the Gauss-Jacobi method typically outperforms the Monte Carlo approach; however, careful consideration is required when selecting the number of quadrature points. Overall, the proposed tDWfPINNs offer a significant advancement in the numerical solution of time-fractional diffusion-wave equations, providing an accurate and scalable mesh-free alternative for challenging fractional models.