基于PINN的Kolmogorov-Arnold网络与RAR-D自适应采样解决椭圆界面问题
PINN-Based Kolmogorov-Arnold Networks with RAR-D Adaptive Sampling for Solving Elliptic Interface Problems
AI总结：
本文提出基于Kolmogorov-Arnold网络的PINN架构，结合RAR-D自适应采样策略，解决椭圆界面问题，实现更小网络规模和更快收敛的高精度解法。
摘要：
物理信息神经网络（PINNs）已成为求解偏微分方程（PDEs）的流行且强大的框架，利用神经网络近似解的同时，将PDE约束、边界条件和界面跃变条件直接嵌入损失函数中。然而，大多数现有PINN方法基于多层感知机（MLPs），这可能需要较大的网络规模和大量的训练才能获得高精度，尤其是对于复杂的界面问题。在本文中，我们提出了一种基于Kolmogorov-Arnold网络（KANs）的新型PINN架构，该架构在选择激活函数方面具有更大的灵活性，并能用更少的参数表示函数。具体而言，我们引入了双KANs结构，该结构在子域之间耦合两个KANs，并显式强制界面条件。为进一步提高训练效率和收敛性，我们整合了RAR-D自适应采样策略以动态细化训练点。在椭圆界面问题的数值实验中，得到的计算域内误差分布更加均匀，这表明我们的基于PINN的KANs在更小的网络规模和更快的收敛速度下实现了更高的精度，相较于标准PINNs。
英文摘要：
Physics-Informed Neural Networks (PINNs) have become a popular and powerful framework for solving partial differential equations (PDEs), leveraging neural networks to approximate solutions while embedding PDE constraints, boundary conditions, and interface jump conditions directly into the loss function. However, most existing PINN approaches are based on multilayer perceptrons (MLPs), which may require large network sizes and extensive training to achieve high accuracy, especially for complex interface problems. In this work, we propose a novel PINN architecture based on Kolmogorov-Arnold Networks (KANs), which offer greater flexibility in choosing activation functions and can represent functions with fewer parameters. Specifically, we introduce a dual KANs structure that couples two KANs across subdomains and explicitly enforces interface conditions. To further boost training efficiency and convergence, we integrate the RAR-D adaptive sampling strategy to dynamically refine training points. Numerical experiments on the elliptic interface problems yield more uniform error distributions across the computational domain, which demonstrates that our PINN-based KANs achieve superior accuracy with significantly smaller network sizes and faster convergence compared to standard PINNs.
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