IGA-KAN: Isogeometric Analysis with Physics-Informed Closed-Form Kolmogorov-Arnold Networks for Forward and Inverse PDEs

2026-10-05

AI Summary

This study addresses the limited accuracy of neural solvers and the insufficient error control in traditional isogeometric analysis (IGA) by proposing a gradient-free closed-form fitting mechanism. Specifically, it enhances IGA solutions via local Kolmogorov–Arnold Networks (KANs), employing hybrid basis functions to integrate strong-form equations with boundary data, thereby improving accuracy without requiring optimizers. Furthermore, a maximum-principle-based a posteriori safeguarding strategy is incorporated to reconcile computational efficiency with precision. Experimental results demonstrate that the proposed approach reduces L2 and H1 errors by 4.2 to 220 times compared to KANs trained from scratch, while achieving a 167-fold improvement in source term recovery accuracy for inverse problems.

Abstract

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.

Problem

• Isogeometric Analysis
• Partial Differential Equations
• Kolmogorov-Arnold Networks
• Inverse Problems
• Numerical Accuracy

Innovation

• Isogeometric Analysis
• Kolmogorov-Arnold Networks
• Closed-Form Fitting
• Physics-Informed
• Superconvergence