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

TL;DR

This paper introduces fPINN-DeepONet, a physics-informed operator learning framework that efficiently solves multi-term time-fractional diffusion-wave equations with high accuracy and robustness, even with noisy data.

Contribution

The paper presents a novel integration of operator learning with an $L_2$ approximation for fractional derivatives, enabling effective solutions to complex fractional PDEs.

Findings

Achieves first-order accuracy for Caputo fractional derivatives of order (1,2).

Demonstrates versatility with fixed and variable fractional orders.

Shows robustness and efficiency in noisy data 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 $\beta \in (1,2)$. 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.