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For a non-homogeneous model transport equation with source terms, the stability analysis of a linear hybrid scheme (a combination of upwind and central approximations) is performed. Stability conditions are obtained that depend on the hybridity parameter, the source intensity factor (the product of intensity per time step), and the weight coefficient of the linear combination of source power on the lower- and upper-time layer. In a nonlinear case for the non-equilibrium by velocities and temperatures equations of gas suspension motion, the linear stability analysis was confirmed by calculation. It is established that the maximum permissible Courant number of the hybrid large-particle method of the second order of accuracy in space and time with an implicit account of friction and heat exchange between gas and particles does not depend on the intensity factor of interface interactions, the grid spacing and the relaxation times of phases (K-stability). In the traditional case of an explicit method for calculating the source terms, when a dimensionless intensity factor greater than 10, there is a catastrophic (by several orders of magnitude) decrease in the maximum permissible Courant number, in which the calculated time step becomes unacceptably small.

On the basic ratios of Riemann’s problem in the equilibrium heterogeneous medium, we obtained an asymptotically exact self-similar solution of the problem of interaction of a shock wave with a layer of gas-suspension to which converge the numerical solution of two-velocity two-temperature dynamics of gassuspension when reducing the size of dispersed particles.

The dynamics of the shock wave in gas and its interaction with a limited gas suspension layer for different sizes of dispersed particles: 0.1, 2, and 20 ìm were studied. The problem is characterized by two discontinuities decay: reflected and refracted shock waves at the left boundary of the layer, reflected rarefaction wave, and a past shock wave at the right contact edge. The influence of relaxation processes (dimensionless phase relaxation times) to the flow of a gas suspension is discussed. For small particles, the times of equalization of the velocities and temperatures of the phases are small, and the relaxation zones are sub-grid. The numerical solution at characteristic points converges with relative accuracy $O \, (10^{-4})$ to self-similar solutions.

This work is devoted to the numerical study of high-speed mixing layers of compressible flows. The problem under consideration has a wide range of applications in practical tasks and, despite its apparent simplicity, is quite complex in terms of modeling. Because in the mixing layer, as a result of the instability of the tangential discontinuity of velocities, the flow passes from laminar flow to turbulent mode. Therefore, the obtained numerical results of the considered problem strongly depend on the adequacy of the used turbulence models. In the presented work, this problem is studied based on the two-fluid approach to the problem of turbulence. This approach has arisen relatively recently and is developing quite rapidly. The main advantage of the two-fluid approach is that it leads to a closed system of equations, when, as is known, the long-standing Reynolds approach leads to an open system of equations. The paper presents the essence of the two-fluid approach for modeling a turbulent compressible medium and the methodology for numerical implementation of the proposed model. To obtain a stationary solution, the relaxation method and Prandtl boundary layer theory were applied, resulting in a simplified system of equations. In the considered problem, high-speed flows are mixed. Therefore, it is also necessary to model heat transfer, and the pressure cannot be considered constant, as is done for incompressible flows. In the numerical implementation, the convective terms in the hydrodynamic equations were approximated by the upwind scheme with the second order of accuracy in explicit form, and the diffusion terms in the right-hand sides of the equations were approximated by the central difference in implicit form. The sweep method was used to implement the obtained equations. The SIMPLE method was used to correct the velocity through the pressure. The paper investigates a two-liquid turbulence model with different initial flow turbulence intensities. The obtained numerical results showed that good agreement with the known experimental data is observed at the inlet turbulence intensity of $0.1 < I < 1 \%$. Data from known experiments, as well as the results of the $k − kL + J$ and LES models, are presented to demonstrate the effectiveness of the proposed turbulence model. It is demonstrated that the two-liquid model is as accurate as known modern models and more efficient in terms of computing resources.

The computer code NINE (Newtonian Iteration for Nonlinear Equation) for numerical solution of the boundary problems for nonlinear differential equations on the basis of continuous analogue of the Newton method (CANM) is presented. Numerov’s finite-difference appproximation is applied to provide the fourth accuracy order with respect to the discretization stepsize. Algorithms of calculating the Newtonian iterative parameter are discussed. A convergence of iteration process in dependence on choice of the iteration parameter has been studied. Results of numerical investigation of the particle-like solutions of the scalar field equation are given.

In the paper, it is represented a linearization method for the stress-strain curves of nonlinearly deformable beams and circular plates in order to generalize the pure bending vibration equations. It is considered composite, on average isotropic prismatic beams of a constant rectangular cross-section and circular plates of a constant thickness made of nonlinearly elastic materials. The technique consists in determining the approximate Young’s moduli from the initial stress-strain state of beam and plate subjected to the action of the bending moment.

The paper proposes two criteria for linearization: the equality of the specific potential energy of deformation and the minimization of the standard deviation in the state equation approximation. The method allows obtaining in the closed form the estimated value of the natural frequencies of layered and structurally heterogeneous, on average isotropic nonlinearly elastic beams and circular plates. This makes it possible to significantly reduce the resources in the vibration analysis and modeling of these structural elements. In addition, the paper shows that the proposed linearization criteria allow to estimate the natural frequencies with the same accuracy.

Since in the general case even isotropic materials exhibit different resistance to tension and compression, it is considered the piecewise-linear Prandtl’s diagrams with proportionality limits and tangential Young’s moduli that differ under tension and compression as the stress-strain curves of the composite material components. As parameters of the stress-strain curve, it is considered the effective Voigt’s characteristics (under the hypothesis of strain homogeneity) for a longitudinally layered material structure; the effective Reuss’ characteristics (under the hypothesis of strain homogeneity) for a transversely layered beam and an axially laminated plate. In addition, the effective Young’s moduli and the proportionality limits, obtained by the author’s homogenization method, are given for a structurally heterogeneous, on average isotropic material. As an example, it is calculated the natural frequencies of two-phase beams depending on the component concentrations.

In this paper, we consider convex stochastic optimization problems arising in machine learning applications (e. g., risk minimization) and mathematical statistics (e. g., maximum likelihood estimation). There are two main approaches to solve such kinds of problems, namely the Stochastic Approximation approach (online approach) and the Sample Average Approximation approach, also known as the Monte Carlo approach, (offline approach). In the offline approach, the problem is replaced by its empirical counterpart (the empirical risk minimization problem). The natural question is how to define the problem sample size, i. e., how many realizations should be sampled so that the quite accurate solution of the empirical problem be the solution of the original problem with the desired precision. This issue is one of the main issues in modern machine learning and optimization. In the last decade, a lot of significant advances were made in these areas to solve convex stochastic optimization problems on the Euclidean balls (or the whole space). In this work, we are based on these advances and study the case of arbitrary balls in the $p$-norms. We also explore the question of how the parameter $p$ affects the estimates of the required number of terms as a function of empirical risk.

In this paper, both convex and saddle point optimization problems are considered. For strongly convex problems, the existing results on the same sample sizes in both approaches (online and offline) were generalized to arbitrary norms. Moreover, it was shown that the strong convexity condition can be weakened: the obtained results are valid for functions satisfying the quadratic growth condition. In the case when this condition is not met, it is proposed to use the regularization of the original problem in an arbitrary norm. In contradistinction to convex problems, saddle point problems are much less studied. For saddle point problems, the sample size was obtained under the condition of $\gamma$-growth of the objective function. When $\gamma = 1$, this condition is the condition of sharp minimum in convex problems. In this article, it was shown that the sample size in the case of a sharp minimum is almost independent of the desired accuracy of the solution of the original problem.

The paper is devoted to the study of wave and relaxation effects during the pulsed outflow of a gas mixture with a high content of solid particles from a cylindrical channel during its initial partial filling. The problem is formulated in a two-speed two-temperature formulation and was solved numerically by the hybrid large-particle method of the second order of approximation. The numerical algorithm is implemented in the form of parallel computing using basic Free Pascal language tools. The applicability and accuracy of the method for wave flows of concentrated gas-particles mixtures is confirmed by comparison with test asymptotically accurate solutions. The calculation error on a grid of low detail in the characteristic flow zones of a two-phase medium was 10^-6 . . . 10^-5.

Based on the wave diagram, the analysis of the physical pattern of the outflow of a gas suspension partially filling a cylindrical channel is performed. It is established that, depending on the degree of initial filling of the channel, various outflow modes are formed. The first mode is implemented with a small degree of loading of the high-pressure chamber, at which the left boundary of the gas-particles mixture crosses the outlet section before the arrival of the rarefaction wave reflected from the bottom of the channel. At the same time, the maximum value of the mass flow rate of the mixture is achieved. Other modes are formed in cases of a larger initial filling of the channel, when the rarefaction waves reflected from the bottom of the channel interact with the gas suspension layer and reduce the intensity of its outflow.

The influence of relaxation properties with changing particle size on the dynamics of a limited layer of a gas-dispersed medium is studied. Comparison of the outflow of a limited gas suspension layer with different particle sizes shows that for small particles (the Stokes number is less than 0.001), an anomalous phenomenon of the simultaneous existence of shock wave structures in the supersonic and subsonic flow of gas and suspension is observed. With an increase in the size of dispersed inclusions, the compaction jumps in the region of the two-phase mixture are smoothed out, and for particles (the Stokes number is greater than 0.1), they practically disappear. At the same time, the shock-wave configuration of the supersonic gas flow at the outlet of the channel is preserved, and the positions and boundaries of the energy-carrying volumes of the gas suspension are close when the particle sizes change.

This work gives results of numerical simulation of a superconducting magnetic focusing system. While modeling this system, special care was taken to achieve approximation accuracy over the condition u(∞)=0 by using Richardson method. The work presents the results of comparison of the magnetic field calculated distribution with measurements of the field performed on a modified magnet SP-40 of “MARUSYA” physical installation. This work also presents some results of numeric analysis of magnetic systems of “MARUSYA” physical installation with the purpose to study an opportunity of designing magnetic systems with predetermined characteristics of the magnetic field.

Deep learning’s power stems from complex architectures; however, these can lead to overfitting, where models memorize training data and fail to generalize to unseen examples. This paper proposes a novel probabilistic approach to mitigate this issue. We introduce two key elements: Truncated Log-Uniform Prior and Truncated Log-Normal Variational Approximation, and Automatic Relevance Determination (ARD) with Bayesian Deep Neural Networks (BDNNs). Within the probabilistic framework, we employ a specially designed truncated log-uniform prior for noise. This prior acts as a regularizer, guiding the learning process towards simpler solutions and reducing overfitting. Additionally, a truncated log-normal variational approximation is used for efficient handling of the complex probability distributions inherent in deep learning models. ARD automatically identifies and removes irrelevant features or weights within a model. By integrating ARD with BDNNs, where weights have a probability distribution, we achieve a variational bound similar to the popular variational dropout technique. Dropout randomly drops neurons during training, encouraging the model not to rely heavily on any single feature. Our approach with ARD achieves similar benefits without the randomness of dropout, potentially leading to more stable training.

To evaluate our approach, we have tested the model on two datasets: the Canadian Institute For Advanced Research (CIFAR-10) for image classification and a dataset of Macroscopic Images of Wood, which is compiled from multiple macroscopic images of wood datasets. Our method is applied to established architectures like Visual Geometry Group (VGG) and Residual Network (ResNet). The results demonstrate significant improvements. The model reduced overfitting while maintaining, or even improving, the accuracy of the network’s predictions on classification tasks. This validates the effectiveness of our approach in enhancing the performance and generalization capabilities of deep learning models.

Considered the application of physics-informed neural networks using multi layer perceptrons to solve Cauchy initial value problems in which the right-hand sides of the equation are continuous monotonically increasing, decreasing or oscillating functions. With the use of the computational experiments the influence of the construction of the approximate neural network solution, neural network structure, optimization algorithm and software implementation means on the learning process and the accuracy of the obtained solution is studied. The analysis of the efficiency of the most frequently used machine learning frameworks in software development with the programming languages Python and C# is carried out. It is shown that the use of C# language allows to reduce the time of neural networks training by 20–40%. The choice of different activation functions affects the learning process and the accuracy of the approximate solution. The most effective functions in the considered problems are sigmoid and hyperbolic tangent. The minimum of the loss function is achieved at the certain number of neurons of the hidden layer of a single-layer neural network for a fixed training time of the neural network model. It’s also mentioned that the complication of the network structure increasing the number of neurons does not improve the training results. At the same time, the size of the grid step between the points of the training sample, providing a minimum of the loss function, is almost the same for the considered Cauchy problems. Training single-layer neural networks, the Adam method and its modifications are the most effective to solve the optimization problems. Additionally, the application of twoand three-layer neural networks is considered. It is shown that in these cases it is reasonable to use the LBFGS algorithm, which, in comparison with the Adam method, in some cases requires much shorter training time achieving the same solution accuracy. The specificity of neural network training for Cauchy problems in which the solution is an oscillating function with monotonically decreasing amplitude is also investigated. For these problems, it is necessary to construct a neural network solution with variable weight coefficient rather than with constant one, which improves the solution in the grid cells located near by the end point of the solution interval.

Numerical modeling of turbulent flows requires finding the balance between accuracy and computational efficiency. For example, DNS and LES models allow to obtain more accurate results, comparing to RANS models, but are more computationally expensive. Because of this, modern applied simulations are mostly performed with RANS models. But even RANS models can be computationally expensive for complex geometries or series simulations due to the necessity of resolving the boundary layer. Some methods, such as wall functions and near-wall domain decomposition, allow to significantly improve the speed of RANS simulations. However, they inevitably lose precision due to using a simplified model in the near-wall domain. To obtain a model that is both accurate and computationally efficient, it is possible to construct a surrogate model based on previously made simulations using the precise model.

In this paper, an operator is constructed that allows reconstruction of the flow field obtained by an accurate model based on the flow field obtained by the simplified model. Spalart–Allmaras model with approximate nearwall domain decomposition and Spalart–Allmaras model resolving the near-wall region are taken as the simplified and the base models respectively. The operator is constructed using a local approach, i. e. to reconstruct a point in the flow field, only features (flow variables and their derivatives) at this point in the field are used. The operator is constructed using the Random Forest algorithm. The efficiency and accuracy of the obtained surrogate model are demonstrated on the supersonic flow over a compression corner with different values for angle $\alpha$ and Reynolds number. The investigation has been conducted into interpolation and extrapolation both by $Re$ and $\alpha$.