Discrete and Continuous Models and Applied Computational Science

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The article is devoted to the construction of fast neurons and neural networks for the implementation of two complete logical bases and modeling of computing devices on their basis. The main idea is to form a fast activation function based on semi-parabolas and its variations that have effective computational support. The constructed activation functions meet the basic requirements that allow configuring logical circuits using the backpropagation method. The main result is obtaining complete logical bases that open the way to constructing arbitrary logical functions. Models of such elements as a trigger, a half adder, and an adder, which form the basis of various specific computing devices, are presented and tested. It is shown that the new activation functions allow obtaining fast solutions with a slight decrease in quality compared to reference outputs. To standardize the outputs, it is proposed to combine the constructed circuits with a unit jump activation function.

In this paper, we examine difference approximations for dynamic systems characterized by polynomial Hamiltonians, specifically focusing on cases where these approximations establish birational correspondences between the initial and final states of the system. Difference approximations are commonly used numerical methods for simulating the evolution of complex systems, and when applied to Hamiltonian dynamics, they present unique algebraic properties due to the polynomial structure of the Hamiltonian. Our approach involves analyzing the conditions under which these approximations preserve key features of the Hamiltonian system, such as energy conservation and phase-space volume preservation. By investigating the algebraic structure of the birational mappings induced by these approximations, we aim to provide insights into the stability and accuracy of numerical simulations in identifying the true behavior of Hamiltonian systems. The results offer a framework for designing efficient and accurate numerical schemes that retain essential properties of polynomial Hamiltonian systems over time.

This paper describes the application of physics-informed neural network (PINN) for solving partial derivative equations. Physics Informed Neural Network is a type of deep learning that takes into account physical laws to solve physical equations more efficiently compared to classical methods. The solution of partial derivative equations (PDEs) is of most interest, since numerical methods and classical deep learning methods are inefficient and too difficult to tune in cases when the complex physics of the process needs to be taken into account. The advantage of PINN is that it minimizes a loss function during training, which takes into account the constraints of the system and th e laws of the domain. In this paper, we consider the solution of ordinary differential equations (ODEs) and PDEs using PINN, and then compare the efficiency and accuracy of this solution method compared to classical methods. The solution is implemented in the Julia programming language. We use NeuralPDE.jl, a package containing methods for solving equations in partial derivatives using physics-based neural networks. The classical method for solving PDEs is implemented through the DifferentialEquations.jl library. As a result, a comparative analysis of the considered solution methods for ODEs and PDEs has been performed, and an evaluation of their performance and accuracy has been obtained. In this paper we have demonstrated the basic capabilities of the NeuralPDE.jl package and its efficiency in comparison with numerical methods.

New possibilities of the 16-spinor realization of the Skyrme-Faddeev chiral model are discussed. Using gauge invariance principle, it is shown that there exist two independent ways for breaking the isotopic invariance symmetry. The first one concerns the interaction with the electromagnetic field (ordinary photons generated by the electric charge) and the second one includes the interaction with the new vector field (shadow/dark photons generated by the special neutrino charge). The neutrino oscillation phenomenon is explained.

A study of ancient written texts and signs showed that the hieroglyphs and structure of the archaic sentence have much in common with the modern Chinese language. In the context of the history and evolution of the Chinese language, its characteristic tonality and melody are emphasized. The main focus of the work is on the study of the sound properties of hieroglyphs (keys / Chinese radicals) found simultaneously in ancient inscriptions as well as in modern text messages. The article uses modern digital methods of sound analysis with their simultaneous visualization. To characterize the sound of hieroglyphs (in accordance with the Pinyin phonetic transcription adopted in China), two (FI, FII), three (FI, FII, FIII) or four (FS, FI, FII, FIII) formants are used, which create a characteristic F-pattern. Our proposed model of four formants for typical hieroglyphs is called the basic one “F-model”, it’s new and original. To visualize the formants, digital audio signal processing programs were used. The data obtained were compared with the corresponding spectrograms for Mandarin (standard) Chinese. Their correspondence to each other has been established. When analyzing F-patterns, an original model was used, which made it possible to characterize spectrograms in the frequency and time domains. The formalized description of basic components of basic “F-model” of a pronunciation of hieroglyphs is given. In conclusion, several areas are noted in which the use of various methods of audiovisual research is promising: advanced innovative technologies (artificial intelligence and virtual reality); television, theatrical video production; evaluation of the quality of audiovisual content; educational process. The present study has shown that described promising research methods can be useful in analyzing similar ancient hieroglyphs.