发表机构
University of Waterloo（滑铁卢大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出使用Kolmogorov-Arnold网络（KAN）在物理信息框架内求解自由边界偏微分方程，通过残差损失纳入障碍、互补等条件，实验证明KAN在精度和界面解析上优于PINN和残差网络基线。
AI 中文摘要
我们在物理信息框架内研究自由边界问题，使用Kolmogorov-Arnold网络（KAN）近似。所提出的方法通过基于残差的损失函数纳入障碍约束、偏微分方程（PDE）不等式、互补条件和边界条件。我们考虑一个线性椭圆障碍问题、一个非线性$p$-Laplacian障碍问题以及一个时间依赖的单相Stefan问题。所提出的KAN求解器与物理信息神经网络（PINN）和残差网络基线进行比较。数值实验表明，KAN实现了较低的相对$L^2$和$L^\infty$误差，同时准确解析接触区域和移动界面。结果表明，KAN表示为求解自由边界PDE提供了一种有效的替代方案。
英文摘要
We study free-boundary problems within a physics-informed framework using Kolmogorov-Arnold network (KAN) approximations. The proposed approach incorporates obstacle constraints, partial differential equation (PDE) inequalities, complementarity conditions, and boundary conditions through residual-based loss functions. We consider a linear elliptic obstacle problem, a nonlinear $p$-Laplacian obstacle problem, and a time-dependent one-phase Stefan problem. The proposed KAN solver is compared with physics-informed neural network (PINN) and residual-network baselines. Numerical experiments show that KANs achieve low relative $L^2$ and $L^\infty$ errors while accurately resolving contact regions and moving interfaces. The results indicate that KAN representations provide an effective alternative for solving free-boundary PDEs.
