Geometric Kolmogorov--Arnold Network (GeoKAN)

Authors: Abhijit Sen, Bikram Keshari Parida, Giridas Maiti et al.

arXiv · PDF

Summary

The authors propose GeoKAN, a Kolmogorov-Arnold Network variant that learns a coordinate transformation (a diagonal Riemannian metric) to warp the input space before applying basis functions. Instead of using fixed basis functions on the raw input coordinates, GeoKAN first stretches regions with rapid variation and compresses smoother regions, reallocating representational capacity where it's most needed. They test it on physics-informed learning and differential-equation problems where functions have sharp, localized features.

Main takeaways:

• Learns a geometry-adapted coordinate system (diagonal metric) before basis expansion, rather than using fixed Euclidean coordinates
• Stretches high-variation regions and compresses smooth regions, placing capacity where needed
• Offers three main variants (GeoKAN-NNMetric, GeoKAN-γ, LM-KAN) and basis-specific versions (RBF, Wavelet, Fourier)
• Well-suited to sharp, stiff, or strongly non-uniform regimes in scientific ML and differential equations

Relevance

Not directly related to my persona or midtraining work—this is a specialized architecture for scientific computing and physics problems. It's included for general ML interest but doesn't touch behavior installation, steering, or prompt–fine-tuning equivalence.

Abstract

arXiv:2605.06740v1 Announce Type: new 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.