Construction of the Kolmogorov-Arnold networks using the Newton-Kaczmarz method

TY - JOUR

T1 - Construction of the Kolmogorov-Arnold networks using the Newton-Kaczmarz method

AU - Poluektov, Michael

AU - Polar, Andrew

N1 - Publisher Copyright: © The Author(s) 2025.

PY - 2025/8

Y1 - 2025/8

N2 - It is known that any continuous multivariate function can be represented exactly by a composition functions of a single variable—the so-called Kolmogorov-Arnold representation. It can be a convenient tool for tasks where it is required to obtain a predictive model that maps some vector input of a black box system into a scalar output. In this case, the representation may not be exact, and it is more correct to refer to such structure as the Kolmogorov-Arnold model (or, as more recently popularised, ‘network’). Construction of such model based on the recorded input–output data is a challenging task. In the present paper, it is suggested to decompose the underlying functions of the representation into continuous basis functions and parameters. It is then proposed to find the parameters using the Newton-Kaczmarz method for solving systems of non-linear equations. The algorithm is then modified to support parallelisation. The paper demonstrates that such approach is also an excellent tool for data-driven solution of partial differential equations. Numerical examples show that for the considered model, the Newton-Kaczmarz method for parameter estimation is efficient and more robust with respect to the section of the initial guess than the straightforward application of the Gauss-Newton method. Finally, the Kolmogorov-Arnold model is compared to the MATLAB’s built-in neural networks on a relatively large-scale problem (25 inputs, datasets of 10 million records), significantly outperforming the multilayer perceptrons in this particular problem (4–10 min vs. 4–8 h of training time, as well as higher accuracy, lower CPU usage, and smaller memory footprint).

AB - It is known that any continuous multivariate function can be represented exactly by a composition functions of a single variable—the so-called Kolmogorov-Arnold representation. It can be a convenient tool for tasks where it is required to obtain a predictive model that maps some vector input of a black box system into a scalar output. In this case, the representation may not be exact, and it is more correct to refer to such structure as the Kolmogorov-Arnold model (or, as more recently popularised, ‘network’). Construction of such model based on the recorded input–output data is a challenging task. In the present paper, it is suggested to decompose the underlying functions of the representation into continuous basis functions and parameters. It is then proposed to find the parameters using the Newton-Kaczmarz method for solving systems of non-linear equations. The algorithm is then modified to support parallelisation. The paper demonstrates that such approach is also an excellent tool for data-driven solution of partial differential equations. Numerical examples show that for the considered model, the Newton-Kaczmarz method for parameter estimation is efficient and more robust with respect to the section of the initial guess than the straightforward application of the Gauss-Newton method. Finally, the Kolmogorov-Arnold model is compared to the MATLAB’s built-in neural networks on a relatively large-scale problem (25 inputs, datasets of 10 million records), significantly outperforming the multilayer perceptrons in this particular problem (4–10 min vs. 4–8 h of training time, as well as higher accuracy, lower CPU usage, and smaller memory footprint).

KW - Discrete Urysohn operator

KW - Generalised additive model

KW - Kaczmarz method

KW - Kolmogorov-Arnold networks

KW - Kolmogorov-Arnold representation

KW - Newton-Kaczmarz method

KW - Ridge function

UR - https://www.scopus.com/pages/publications/105010556857

U2 - 10.1007/s10994-025-06800-6

DO - 10.1007/s10994-025-06800-6

M3 - Article

SN - 0885-6125

VL - 114

JO - Machine Learning

JF - Machine Learning

IS - 8

M1 - 185

ER -