发表机构
American University of Beirut（贝鲁特美国大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对一维多孔介质方程反问题，提出两阶段PINN训练框架，解决标准PINN反问题对初始猜测敏感、仅局部收敛的问题，提升了鲁棒性与未知参数恢复可靠性。
AI 中文摘要
多孔介质方程（PME）为$u_t = \Delta(u^m)$（$m > 1$），是一类退化非线性抛物型偏微分方程，广泛应用于多孔介质流体流动、等离子体传热、种群动力学等物理场景，其特点是非线性扩散与有限传播速度。本文采用物理信息神经网络（PINN）研究一维PME正问题与反问题的数值解，并将结果与经典数值方法、现有解析解及人工构造解对比。尽管PINN为正、反问题求解提供了灵活框架，但标准反问题公式对初始猜测值敏感，仅能实现局部收敛。为解决该问题，本文提出一种用于反问题的新型两阶段PINN训练框架，显著提升了收敛稳定性，即使初始猜测值较差，也能可靠恢复未知参数。总体而言，所提方法表明PINN是一维PME经典方法的灵活、精确替代方案，引入的两阶段训练策略大幅提升了PINN在反问题中的鲁棒性，为扩展至更复杂几何与高维情形奠定了坚实基础。
英文摘要
The Porous Medium Equation (PME), given by $u_t = Δ(u^m)$ for $m > 1$, is a degenerate nonlinear parabolic partial differential equation that arises in various physical applications such as fluid flow in porous media, heat transfer in plasmas, and population dynamics. It is known for its nonlinear diffusion and finite propagation speed. In this paper, we study numerical solutions of the one-dimensional direct and inverse PME using Physics-Informed Neural Networks (PINNs), and compare them with classical numerical methods and available analytical and manufactured solutions. While PINNs provide a flexible framework for solving both forward and inverse problems, we show that the standard inverse formulation suffers from a strong sensitivity to the initial guess, leading to only local convergence. To address this issue, we propose a novel two-stage PINN training framework for the inverse problem, which significantly improves convergence stability and allows reliable recovery of the unknown parameter even for poor initial guesses. Overall, the proposed approach demonstrates that PINNs are a flexible and accurate alternative to classical methods for the 1D PME, and the introduced two-stage training strategy substantially improves their robustness in inverse problems, providing a solid basis for extensions to more complex geometries and higher-dimensional cases.
Comments54 pages
