AI 中文总结
针对PINNs理论分析局限于线性方程的问题，提出用于求解p-拉普拉斯方程的鲁棒训练框架，结合两类损失建立严格误差估计，扩展至参数场景并通过数值实验验证结果。
AI 中文摘要
随着科学机器学习的兴起，物理信息神经网络（PINNs）已被广泛应用于各类问题。然而，多数PINNs的理论分析仍局限于线性方程，非线性问题的PINNs与经典数值分析之间存在显著差距。为解决该问题，本文提出一种用于求解非线性偏微分方程的PINNs鲁棒训练框架，并开展严格误差分析。具体而言，针对p-拉普拉斯方程，本文引入一种新型损失公式，将在$W^{-1,p'}$中测量的对偶残差损失与在分数阶索伯列夫范数$W^{1-\frac{1}{p},p}$中测量的边界损失相结合，该公式旨在适配弱解的有限正则性，使本文得以建立严格的先验和后验误差估计。此外，本文将所提框架及其分析扩展至参数设置场景，其中指数、源项和边界条件均可随参数变化。最后，本文给出数值实验，验证了上述理论结果。
英文摘要
With the rise of scientific machine learning, physics-informed neural networks (PINNs) have been extensively applied to a wide range of problems. Nevertheless, most theoretical analyses of PINNs remain confined to linear equations, and a substantial gap persists between PINNs and classical numerical analysis for nonlinear problems. To address this issue, we propose a robust training framework for PINNs solving nonlinear partial differential equations, together with a rigorous error analysis. Specifically, for the $p$-Laplace equation, we introduce a novel loss formulation that combines a dual residual loss measured in $W^{-1,p'}$ with a boundary loss measured in a fractional Sobolev norm $W^{1-\frac{1}{p},p}$. This formulation is designed to accommodate the limited regularity of weak solutions and enables us to establish rigorous \textit{a priori} and \textit{a posteriori} error estimates. Moreover, the proposed framework and its analysis are extended to a parametric setting in which the exponent, the source term, and the boundary condition may all vary with the parameters. Finally, we present numerical experiments that substantiate our theoretical findings.
