发表机构
Shahid Beheshti University; Institute for Cognitive and Brain Sciences（沙希德·贝赫什提大学; 认知与脑科学研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
提出IGA-KAN，用闭式拟合的局部Kolmogorov-Arnold网络增强等几何分析，在正逆PDE求解中显著提升精度，误差降低达两个数量级。
AI 中文摘要
等几何分析（IGA）能在精确的NURBS几何上精确求解偏微分方程，而神经求解器虽然无网格，但精度通常低几个数量级，且通常通过非凸优化训练，缺乏误差控制。我们提出IGA-KAN，利用局部Kolmogorov-Arnold网络（以闭式拟合）来改进IGA解而非替代它。IGA Galerkin求解产生u_h；在每个节点顶点补片上，将Kolmogorov-Arnold岭模型拟合到方程的强形式、精确边界数据和u_h，并通过IGA帽函数混合这些模型。在内函数固定时，拟合是一个批量线性最小二乘问题，无需优化器、学习率或初始化。由最大原理界启发的后验保护措施决定在何处使用局部模型，其余区域保留IGA解。在八个具有精确解的基准测试中（其中五个来自文献，一个还定义在基于MRI脑切片拟合的域上），该方法在Galerkin未知量不变的情况下，将IGA的误差在参考网格上降低了L^2范数4.2至90倍、H^1范数4.1至220倍，其L^2误差比在相同方程和固定预算下从头训练的最佳Kolmogorov-Arnold网络小6至6×10^4倍。在逆问题中，它从一次无噪声观测中恢复未知常数源，精度比IGA高167倍。该增益归因于Galerkin解局部平均的超收敛性。
英文摘要
Isogeometric analysis (IGA) solves partial differential equations accurately on exact NURBS geometry, whereas neural solvers are mesh-free but often orders of magnitude less accurate and typically trained by non-convex optimization without error control. We propose IGA-KAN, which uses local Kolmogorov-Arnold networks, fitted in closed form, to improve the IGA solution instead of replacing it. An IGA Galerkin solve produces u_h; on every knot-vertex patch a Kolmogorov-Arnold ridge model is fitted to the strong form of the equation, the exact boundary data and u_h, and the models are blended by IGA hat functions. With fixed inner functions the fit is one batched linear least-squares problem, without optimizer, learning rate or initialization. An a posteriori safeguard, motivated by a maximum-principle bound, decides where local models are used, keeping the IGA solution elsewhere. On eight benchmarks with exact solutions, five from the literature and one also posed on a domain fitted to a brain slice from MRI, the method reduces the error of IGA, at an unchanged number of Galerkin unknowns, by factors of 4.2 to 90 in L^2 and 4.1 to 220 in H^1 on the reference meshes, and its L^2 error is 6 to 6x10^4 times smaller than that of the best Kolmogorov-Arnold network trained from scratch on the same equations with a fixed budget. In an inverse problem it recovers an unknown constant source from one noise-free observation 167 times more accurately than IGA. The gain is attributed to the superconvergence of local averages of the Galerkin solution.
Comments29 pages, 14 figures, 9 tables. Code and notebooks: https://github.com/Sima-Naraghi/iga-kan
