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K-DAREK: Distance Aware Error for Kurkova Kolmogorov Networks

Masoud Ataei, Vikas Dhiman, Mohammad Javad Khojasteh

TL;DR

Problem: neural networks can be unreliable in safety-critical settings; Approach: K-DAREK, a distance-aware, two-block KKAN that couples spectrally normalized MLPs with spline blocks and derives worst-case error bounds. Contributions: (i) a principled MLP-bound and spline-bound error framework with distance-awareness, (ii) computational advantages over ensembles and Gaussian processes, and (iii) zero-violation guarantees on Real Estate Valuation data. Results: empirical studies show faster inference and scalable performance, with improved safety and reliable uncertainty in real-world tasks and multi-agent control. Impact: enables robust, interpretable, and risk-aware deployment of hybrid neural architectures in control and decision-support systems.

Abstract

Neural networks are powerful parametric function approximators, while Gaussian processes (GPs) are nonparametric probabilistic models that place distributions over functions via kernel-defined correlations but become computationally expensive for large-scale problems. Kolmogorov-Arnold networks (KANs), semi-parametric neural architectures, model complex functions efficiently using spline layers. Kurkova Kolmogorov-Arnold networks (KKANs) extend KANs by replacing the early spline layers with multi-layer perceptrons that map inputs into higher-dimensional spaces before applying spline-based transformations, which yield more stable training and provide robust architectures for system modeling. By enhancing the KKAN architecture, we develop a novel learning algorithm, distance-aware error for Kurkova-Kolmogorov networks (K-DAREK), for efficient and interpretable function approximation with uncertainty quantification. Our approach establishes robust error bounds that are distance-aware; this means they reflect the proximity of a test point to its nearest training points. In safe control case studies, we demonstrate that K-DAREK is about four times faster and ten times more computationally efficient than Ensemble of KANs, 8.6 times more scalable than GP as data size increases, and 7.2% safer than our previous work distance-aware error for Kolmogorov networks (DAREK). Moreover, on real data (e.g., Real Estate Valuation), K-DAREK's error bound achieves zero coverage violations.

K-DAREK: Distance Aware Error for Kurkova Kolmogorov Networks

TL;DR

Problem: neural networks can be unreliable in safety-critical settings; Approach: K-DAREK, a distance-aware, two-block KKAN that couples spectrally normalized MLPs with spline blocks and derives worst-case error bounds. Contributions: (i) a principled MLP-bound and spline-bound error framework with distance-awareness, (ii) computational advantages over ensembles and Gaussian processes, and (iii) zero-violation guarantees on Real Estate Valuation data. Results: empirical studies show faster inference and scalable performance, with improved safety and reliable uncertainty in real-world tasks and multi-agent control. Impact: enables robust, interpretable, and risk-aware deployment of hybrid neural architectures in control and decision-support systems.

Abstract

Neural networks are powerful parametric function approximators, while Gaussian processes (GPs) are nonparametric probabilistic models that place distributions over functions via kernel-defined correlations but become computationally expensive for large-scale problems. Kolmogorov-Arnold networks (KANs), semi-parametric neural architectures, model complex functions efficiently using spline layers. Kurkova Kolmogorov-Arnold networks (KKANs) extend KANs by replacing the early spline layers with multi-layer perceptrons that map inputs into higher-dimensional spaces before applying spline-based transformations, which yield more stable training and provide robust architectures for system modeling. By enhancing the KKAN architecture, we develop a novel learning algorithm, distance-aware error for Kurkova-Kolmogorov networks (K-DAREK), for efficient and interpretable function approximation with uncertainty quantification. Our approach establishes robust error bounds that are distance-aware; this means they reflect the proximity of a test point to its nearest training points. In safe control case studies, we demonstrate that K-DAREK is about four times faster and ten times more computationally efficient than Ensemble of KANs, 8.6 times more scalable than GP as data size increases, and 7.2% safer than our previous work distance-aware error for Kolmogorov networks (DAREK). Moreover, on real data (e.g., Real Estate Valuation), K-DAREK's error bound achieves zero coverage violations.
Paper Structure (12 sections, 2 theorems, 17 equations, 3 figures, 8 tables, 1 algorithm)

This paper contains 12 sections, 2 theorems, 17 equations, 3 figures, 8 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Background
  3. Architecture and Method
  4. Main model architecture
  5. Error Analysis framework for K-DAREK
  6. Experiments and Results
  7. Conclusion
  8. Table of acronyms
  9. Related works
  10. Algorithmic Complexity
  11. Extended Knots
  12. Additional Experiments

Key Result

Theorem 1

For a function $f$ with $k+1$ continuous derivatives (that is $f \in C^{(k+1)}$) and its $k+1$-th-order Lipschitz constant is ${\cal L}^{k+1}_f$, the error at test point $x \in [\boldsymbol{\tau}[j], \boldsymbol{\tau}[j+1])$ is bounded as follows:

Figures (3)

  • Figure 1: (a) K-DAREK architecture, consists of two blocks: MLP and Spline. The MLP Block contains one SNR-MLP for each input dimension. A combination layer unifies the output of SNR-MLPs into a feature vector. The Spline Block is a Spline Layer followed by a combination layer that constructs the model's final output. (b) The error analysis of K-DAREK, has two main components. The SNR-MLP Layer receives inputs $\mathbf{x}$ and computes the feature map of SNR-MLPs, $\zeta$s, along with the linear error \ref{['eq:MLP-error']}. The Spline Layer employs features $\boldsymbol{\psi}$ to calculate the spline values $s_{r,q}$ and spline error using interpolation error and error at the knot as described in \ref{['eq:two-layer-error']}.
  • Figure 2: Cosine function approximation experiment. (a) Error estimation comparison of DAREK and K-DAREK, showing that K-DAREK achieves smoother function fitting without requiring an explicit regularizer. In both models, the error bounds increase as test points move further from the nearest knot, reflecting distance-awareness in the error behaviour. (b)$3\sigma$ uncertainty bounds comparison between an Ensemble of KANs and a GP. The Ensemble's uncertainty bounds lack distance-awareness and do not follow a consistent spatial pattern, in contrast to the DAREK, K-DAREK, and GP. (c) Process time comparison of K-DAREK (green), DAREK (orange), GP (red), and Ensemble (blue) models. Training time (solid line) and inference time (dashed). The error bars indicate the variability observed across repeated runs of the experiment.
  • Figure 3: Multiagent safe control setup. This is one of the successful trajectories using K-DAREK bounds, where the controlled agent safely navigates from the start position to the goal location. The controlled agent and its path are shown in blue, and the surrounding red vehicles represent other agents.

Theorems & Definitions (3)

  • Definition 1: Distance-awareness ataei2025darek
  • Theorem 1: Interpolation error ataei2025darek
  • Theorem 2: Spline error ataei2025darek