fPINN-DeepONet: A Physics-Informed Operator Learning Framework for Multi-term Time-fractional Mixed Diffusion-wave Equations
This work provides a novel method for solving fractional PDEs, which are important in modeling anomalous diffusion and wave propagation, but the improvements are incremental over existing PINN and DeepONet approaches.
The authors developed fPINN-DeepONet, a physics-informed operator learning framework for solving multi-term time-fractional mixed diffusion-wave equations, achieving first-order accuracy for the Caputo fractional derivative. Numerical experiments demonstrated accuracy, robustness, and efficiency for fixed and variable fractional-order PDEs, including noisy data scenarios.
In this paper, we develop a physics-informed deep operator learning framework for solving multi-term time-fractional mixed diffusion-wave equations (TFMDWEs). We begin by deriving an $L_2$ approximation, which achieves first-order accuracy for the Caputo fractional derivative of order $β\in (1,2)$. Building upon this foundation, we propose the fPINN-DeepONet framework, a novel approach that integrates operator learning with the $L_2$ approximation to efficiently solve fractional partial differential equations (FPDEs). Our framework is successfully applied to both fixed and variable fractional-order PDEs, demonstrating the framework's versatility and broad applicability. To evaluate the performance of the proposed model, we conduct a series of numerical experiments that involve dynamically varying fractional orders in both space and time, as well as scenarios with noisy data. These results highlight the accuracy, robustness, and efficiency of the fPINN-DeepONet framework.