用物理约束的深度算子学习法求解分数阶扩散波方程
fPINN-DeepONet: A Physics-Informed Operator Learning Framework for Multi-term Time-fractional Mixed Diffusion-wave Equations
- 结合L2逼近与算子学习，构建新型物理信息框架
- 对变分数阶、带噪声数据均保持高精度与鲁棒性
- 适合求解复杂分数阶偏微分方程，尤其适用于动态参数场景
本文提出一种用于求解多术语时间分数阶混合扩散-波动方程（TFMDWEs）的物理信息深度算子学习框架。首先推导出针对阶数β∈(1,2)的Caputo分数阶导数的$L_2$逼近方法，实现一阶精度。在此基础上，提出fPINN-DeepONet框架，将算子学习与$L_2$逼近相结合，高效求解分数阶偏微分方程（FPDEs）。该框架成功应用于固定与可变分数阶方程，展现良好泛化能力。通过一系列数值实验验证模型性能，涵盖空间时间动态变化的分数阶、含噪声数据等场景，结果表明该框架在精度、鲁棒性和效率方面表现优异。
原文摘要 · Abstract (English)
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.
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