  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 the 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. Furthermore,
numerical experiments show that the proposed approach performs faster than
neural network training to the same accuracy.
