Transformed Diffusion-Wave fPINNs: Enhancing Computing Efficiency for PINNs Solving Time-Fractional Diffusion-Wave Equations

TL;DR

This paper introduces transformed Diffsuion-Wave fractional PINNs (tDWfPINNs), a novel, efficient mesh-free neural network method for solving time-fractional diffusion-wave equations with reduced computational costs and maintained accuracy.

Contribution

The paper presents a new integrand transformation technique for PINNs that significantly reduces the computational cost of fractional derivative evaluation in time-fractional PDEs.

Findings

tDWfPINNs outperform traditional methods in efficiency.

Gauss-Jacobi quadrature generally yields better results than Monte Carlo.

Adaptive sampling with RAD improves solution accuracy for complex dynamics.

Abstract

We propose transformed Diffsuion-Wave fractional Physics-Informed Neural Networks (tDWfPINNs) for efficiently solving time-fractional diffusion-wave equations with fractional order $\alpha\in(1,2)$. 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 $\alpha\in(1,2)$, 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.