“In our work, we build upon the ideas of Kolmogorov, Arnold, Namig Guliyev, and Vugar Ismailov,” write V.A. Eskin, A.O. Malkhanov, and M.E. Smorkalov in their article “Are Two Hidden Layers Still Enough for the Physics–Informed Neural Networks?”, published in the prestigious Journal of Computational Physics in May 2025 (https://doi.org/10.1016/j.jcp.2025.114085).

The article is based on the Kolmogorov–Arnold superposition theorem, as well as a result obtained by researchers from the Institute of Mathematics and Mechanics—Head of the Department of Function Theory, Professor Vugar Ismailov, and Leading Researcher of the Department of Functional Analysis, Candidate of Physical and Mathematical Sciences Namig Guliyev—published in 2018 in the journal Neurocomputing.

As early as the 1950s, Kolmogorov and Arnold developed a formula that allows any continuous multivariable function to be represented as a sum and composition of univariate functions. Namig Guliyev and Vugar Ismailov proved that all these univariate functions in the formula can be replaced with the same specially constructed univariate function. As a result, a neural network model with two hidden layers is obtained, which is capable of approximating any continuous multivariable function with a given precision. Moreover, this special function was constructed algorithmically (https://doi.org/10.1016/j.neucom.2018.07.075).

The mention of the names of these researchers alongside such great scientists as Kolmogorov and Arnold in a prestigious international journal testifies to the significance of their scientific achievement. This international recognition proves that Azerbaijani science is well represented in the world and plays an important role in applying theoretical aspects of mathematics to modern fields such as artificial intelligence and neural networks.

It should be noted that, to date, the work of Namig Guliyev and Vugar Ismailov has received around 100 citations from various international experts. You can view them at the following link:
https://scholar.google.com/citations?view_op=view_citation&hl=ru&user=ziQBM_cAAAAJ&citation_for_view=ziQBM_cAAAAJ:L8Ckcad2t8MC

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