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

TL;DR

This paper presents a novel method for constructing Kolmogorov-Arnold models using the Newton-Kaczmarz algorithm, demonstrating efficiency, robustness, and superior performance over neural networks in large-scale data-driven tasks.

Contribution

It introduces a new parameter estimation approach for Kolmogorov-Arnold models based on the Newton-Kaczmarz method, including parallelization and applications to PDEs.

Findings

The Newton-Kaczmarz method is efficient and robust for parameter estimation.

The approach outperforms neural networks in large-scale problems.

The method is effective for data-driven solutions of partial differential equations.

Abstract

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 (MLPs) in this particular problem (4-10 minutes vs. 4-8 hours of training time, as well as higher accuracy, lower CPU usage, and smaller memory footprint).