Geometric Kolmogorov--Arnold Network (GeoKAN)

TL;DR

GeoKAN introduces geometry-aware neural networks that learn a Riemannian metric to adapt input representations, improving approximation in non-uniform, physics-informed, and differential-equation tasks.

Contribution

The paper proposes a family of geometry-aware KAN models with learned metrics, enabling adaptive resolution and improved performance in scientific machine learning applications.

Findings

GeoKAN reallocates representational capacity to regions with rapid variation.

The models are effective for sharp, stiff, and localized regimes in differential equations.

Multiple variants demonstrate flexibility in function approximation and physics-informed learning.

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-$\gamma$, 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.