Vol 26, No 2 (2024)

The authors construct a groupoid whose elements are associated with multilayer feedforward neural networks. This groupoid is called the complete groupoid of the composition of neural networks. Multilayer feedforward neural networks (hereinafter referred to as neural networks) are modelled by defining a special type of tuple. Its components define layers of neurons and structural mappings that specify weights of synaptic connections, activation functions and threshold values. Using the artificial neuron model (that of McCulloch-Pitts) for each such tuple it is possible to define a mapping that models the operation of a neural network as a computational circuit. This approach differs from defining a neural network using abstract automata and related constructions. Modeling neural networks using the proposed method makes it possible to describe the architecture of the network (that is, the network graph, the synaptic weights, etc.). The operation in the full neural network composition groupoid models the composition of two neural networks. A network, obtained as the product of a pair of neural networks, operates on input signals by sequentially applying original networks and contains information about their structure. It is proved that the constructed groupoid is a free.

This article examines application of computational algorithms with an increased order of accuracy for modeling two-dimensional problems of development of hydrodynamic instabilities. The efficiency of using algorithms to improve the accuracy and reliability of modeling in this area is considered. More specifically, the paper describes a numerical algorithm for solving the problem of development of Richtmayer-Meshkov instability. To construct the algorithm, the authors use the WENO scheme of the fifth order of accuracy Several problems are solved numerically using the developed method. The article models such processes as flows at a time of 4 046 microseconds, a change in the width of the region filled with sulfur hexafluoride, numerical schlieren patterns at a time of 877 microseconds, a change in the width of the region filled with heavy gas. The results are obtained by various methods on grids of different dimensions and compared with experimental data. It is shown that schemes with WENO reconstruction of the 5th order of accuracy demonstrate results closer to full-scale experiments.

The article investigates the nonlocal method of peridynamics, which makes it possible to simulate the brittle fracture of a solid body without using spatial derivatives. The basic motion equation of a particle with a given volume is written in integral form. A model combining the key features of continuum mechanics and of the nonlocal method is considered. To determine the forces of pair interaction, the dependence of the Cauchy stress tensor on the rate-of strain tensor was used. This formulation correctly describes the behavior of the material during damage and allows to get rid of the limitations inherent to simple bond-based model and ordinary state-based model. The maximum value of the tensile stress is used as a criterion of fracture, which describes the process of nucleation and evolution of damage. To test the implemented model, tasks in a two-dimensional formulation were used. Using the example of the elastic problem about uniaxial tension of a thin rod, the convergence of the numerical solution is shown with a decrease of interaction horizon and an increase of particles number. The second task demonstrates the capabilities of the implemented model to describe the nucleation and evolution of a crack under uniaxial load on a plate with an initial horizontal defect.