Geometric Kolmogorov--Arnold Network (GeoKAN)

2026-05-07

AI Summary

This work addresses the limitations of conventional Kolmogorov–Arnold Networks (KANs) in adaptively handling scientific computing problems characterized by abrupt variations, strong non-uniformity, or local stiffness. To overcome this, the authors propose Geometry-aware KAN (GeoKAN), which introduces a learnable diagonal Riemannian metric into the KAN framework for the first time. This enables dynamic adjustment of local representation resolution through geometrically adaptive coordinate transformations. By integrating basis function expansions—such as radial basis functions, wavelets, and Fourier series—with feature mixing mechanisms, the framework yields several variants, including GeoKAN-NNMetric, GeoKAN-γ, and LM-KAN. Empirical results demonstrate that GeoKAN significantly enhances approximation accuracy and generalization capability for sharp, stiff, and highly non-uniform functions in scientific machine learning and surrogate modeling of differential equations.

Abstract

We introduce Geometric Kolmogorov--Arnold Networks (GeoKANs), a family of geometry-aware KAN-type models in which approximation is carried out in learned, geometry-adapted coordinates rather than in fixed Euclidean input coordinates. GeoKAN achieves this by learning a diagonal Riemannian metric that warps the input before basis expansion and feature mixing. The learned metric provides a geometric inductive bias through local length scaling and volume distortion, and in physics-informed settings it also affects the differential structure seen by the model. Within this framework, we develop three main variants, namely GeoKAN-NNMetric, GeoKAN-$γ$, and LM-KAN. For LM-KAN, we further consider three basis-specific versions, LM-KAN-RBF, LM-KAN-Wav, and LM-KAN-Fourier. These variants allow us to study geometry-aware KAN models both as general function approximators and as surrogates in physics-informed learning. By stretching regions with rapid variation and compressing smoother regions, GeoKAN reallocates representational resolution in a task-dependent manner, allowing the model to place capacity where it is most needed. As a result, GeoKAN is well suited to sharp, stiff, localized, and strongly non-uniform regimes arising in scientific machine learning and differential-equation problems.

Problem

• function approximation
• scientific machine learning
• differential equations
• non-uniform regimes
• geometry-aware modeling

Innovation

• Geometric Kolmogorov–Arnold Network
• Riemannian metric learning
• geometry-aware function approximation
• physics-informed machine learning
• adaptive representation