Andrew Polar

Michael Poluektov, Andrew Polar

This paper introduces a multiple-input discrete Urysohn operator for modelling non-linear control systems and a technique of its identification by processing the observed input and output signals. It is shown that, due to the nature of the discrete Urysohn operator, the identification problem always has an infinity of solutions, which exactly convert the inputs to the output. The suggested iterative identification procedure, however, leads to a unique solution with the minimum norm, requires only few arithmetic operations with the parameter values and is applicable to a real-time identification, running concurrently with the data reading. The efficiency of the proposed modelling and identification approaches is demonstrated using an example of a non-linear mechanical system, which is represented by a differential equation, and an example of a complex real-world dynamic object.

Andrew Polar, Michael Poluektov

Training on disjoint datasets can serve two primary goals: accelerating data processing and enabling federated learning. It has already been established that Kolmogorov-Arnold networks (KANs) are particularly well suited for federated learning and can be merged through simple parameter averaging. While the federated learning literature has mostly focused on achieving training convergence across distributed nodes, the present paper specifically targets acceleration of the training, which depends critically on the choice of an optimisation method and the type of the basis functions. To the best knowledge of the authors, the fastest currently-available combination is the Newton-Kaczmarz method and the piecewise-linear basis functions. Here, it is shown that training on disjoint datasets (or disjoint subsets of the training dataset) can further improve the performance. Experimental comparisons are provided, and all corresponding codes are publicly available.

Andrew Polar, Michael Poluektov

The Kolmogorov-Arnold network (KAN) is a regression model that is based on a representation of an arbitrary continuous multivariate function by a composition of functions of a single variable. Experimentally-obtained datasets for regression models typically include uncertainties, which in some cases, cannot be neglected. The conventional way to account for the latter is to model confidence intervals of the systems' outputs in addition to the expected values of the outputs. However, such information may be insufficient, and in some cases, researchers aim to obtain probability distributions of the outputs. The present paper proposes a method for estimating probability distributions of the outputs in the case of aleatoric uncertainty (i.e. for systems that produce different outputs each time an experiment is executed with the same inputs). The suggested approach covers input-dependent probability distributions of the outputs and is capable of capturing the multi-modality, as well as the variation of the distribution type with the inputs. Although the method is applicable to any regression model, the present paper combines it with KANs, since the specific structure of KANs leads to computationally-efficient models' construction. The source code is available online.