الملخص
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.
الكلمات المفتاحية
الموضوع
بيانات النشر
- المجلة
- غير متاح
- وصول مفتوح
- وصول مفتوح أخضر
اقتبس هذه المقالة
APA 7
Naraghi, S., Parand, K., Sadr, A., & Rahmati, D. (2026). IGA-KAN: Isogeometric Analysis with Physics-Informed Closed-Form Kolmogorov-Arnold Networks for Forward and Inverse PDEs. https://omanscience.com/ar/articles/iga-kan-isogeometric-analysis-with-physics-informed-closed-form-kolmogorov-arnold-networks-for-forward-and-inverse-pdes
MLA 9
Naraghi, Sima, et al. "IGA-KAN: Isogeometric Analysis with Physics-Informed Closed-Form Kolmogorov-Arnold Networks for Forward and Inverse PDEs." https://omanscience.com/ar/articles/iga-kan-isogeometric-analysis-with-physics-informed-closed-form-kolmogorov-arnold-networks-for-forward-and-inverse-pdes.
شيكاغو (المؤلف–التاريخ)
Naraghi, Sima, Kourosh Parand, Amirhossein Sadr, and Dara Rahmati. 2026. "IGA-KAN: Isogeometric Analysis with Physics-Informed Closed-Form Kolmogorov-Arnold Networks for Forward and Inverse PDEs." https://omanscience.com/ar/articles/iga-kan-isogeometric-analysis-with-physics-informed-closed-form-kolmogorov-arnold-networks-for-forward-and-inverse-pdes.
هارفارد
Naraghi, S., Parand, K., Sadr, A. and Rahmati, D. (2026) 'IGA-KAN: Isogeometric Analysis with Physics-Informed Closed-Form Kolmogorov-Arnold Networks for Forward and Inverse PDEs', Available at: https://omanscience.com/ar/articles/iga-kan-isogeometric-analysis-with-physics-informed-closed-form-kolmogorov-arnold-networks-for-forward-and-inverse-pdes.
فانكوفر
Naraghi S, Parand K, Sadr A, Rahmati D. IGA-KAN: Isogeometric Analysis with Physics-Informed Closed-Form Kolmogorov-Arnold Networks for Forward and Inverse PDEs. https://omanscience.com/ar/articles/iga-kan-isogeometric-analysis-with-physics-informed-closed-form-kolmogorov-arnold-networks-for-forward-and-inverse-pdes
IEEE
S. Naraghi, K. Parand, A. Sadr, and D. Rahmati, "IGA-KAN: Isogeometric Analysis with Physics-Informed Closed-Form Kolmogorov-Arnold Networks for Forward and Inverse PDEs," https://omanscience.com/ar/articles/iga-kan-isogeometric-analysis-with-physics-informed-closed-form-kolmogorov-arnold-networks-for-forward-and-inverse-pdes.