Geometry-Aware R-Structured Kolmogorov-Arnold Networks

2026-07-01

AI Summary

This work addresses the challenge of modeling geometric constraints, discontinuities, and implicit boundaries in regression tasks by introducing, for the first time, differentiable R-functions into Kolmogorov–Arnold Networks (KANs). By leveraging R-logic operations to explicitly encode geometric priors, the proposed method precisely characterizes feasible and discontinuous regions. It integrates R-conjunction and R-disjunction with KAN’s additive, multiplicative, and agnostic weighted branching structures, enabling geometry-aware modeling and automatic validation of prior feasibility. This integration substantially enhances model interpretability and adaptability. Evaluated on regression problems involving circular and rectangular supports, the approach reduces RMSE by up to 67% and demonstrates markedly superior boundary localization accuracy and predictive performance compared to baseline methods.

Abstract

We propose a novel hybrid neural architecture, the Geometry-aware R-Structured Kolmogorov-Arnold Network (GRS-KAN), which integrates V.L.Rvachev's R-functions into the Kolmogorov-Arnold Network (KAN) framework. The proposed approach combines two complementary modeling mechanisms: smooth nonlinear structure is learned by KAN branches, while known geometric or logical constraints are encoded analytically using differentiable R-functions. This enables explicit representation of discontinuities, feasible regions, and implicit geometric boundaries within a trainable neural architecture. The framework implements differentiable logical operations through R-conjunctions and R-disjunctions, allowing complex geometric supports to be represented analytically and incorporated directly into regression models. Several GRS-KAN variants are introduced, including additive, multiplicative, and agnostic branch-weighted architectures. The method is demonstrated on regression problems involving discontinuities with circular and rectangular supports. Numerical experiments show that explicit geometric encoding substantially improves predictive accuracy and boundary localization compared with standard KANs. In the considered benchmarks, geometry-aware GRS-KAN models reduce test RMSE by up to 67% while simultaneously improving interpretability through explicit analytical representation of the learned geometric structure. The agnostic variant further demonstrates the ability to automatically determine whether geometric priors are beneficial for a given learning task.

Problem

• geometric constraints
• discontinuities
• Kolmogorov-Arnold Networks
• regression
• boundary representation

Innovation

• R-functions
• Kolmogorov-Arnold Networks
• geometry-aware modeling
• differentiable logical operations
• analytical geometric constraints