fPINN-DeepONet: A Physics-Informed Operator Learning Framework for Multi-term Time-fractional Mixed Diffusion-wave Equations

2026-05-15

AI Summary

This study addresses the lack of efficient and highly accurate numerical methods for solving time-fractional mixed diffusion-wave equations (TFMDWEs) involving multi-order Caputo derivatives under variable fractional orders and noisy data. To this end, the authors propose the fPINN-DeepONet framework, which uniquely integrates an L₂-type discretization scheme with physics-informed Deep Operator Networks (DeepONets) to achieve first-order accuracy in approximating Caputo derivatives of order β ∈ (1,2). The method enables spatiotemporal variable-order fractional modeling while demonstrating robustness against data noise and computational efficiency. Extensive numerical experiments confirm its high accuracy and broad applicability across diverse scenarios.

Abstract

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.

Problem

• time-fractional mixed diffusion-wave equations
• multi-term fractional PDEs
• variable fractional order
• noisy data
• fractional partial differential equations

Innovation

• physics-informed operator learning
• fractional PDEs
• L2 approximation
• DeepONet
• time-fractional diffusion-wave equations