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The paper presents a mathematical model that describes the process of obtaining biogas from livestock waste. This model describes the processes occurring in a biogas plant for mesophilic and thermophilic media, as well as for continuous and periodic modes of substrate inflow. The values of the coefficients of this model found earlier for the periodic mode, obtained by solving the problem of model identification from experimental data using a genetic algorithm, are given.

For the model of methanogenesis, an optimal control problem is formulated in the form of a Lagrange problem, whose criterial functionality is the output of biogas over a certain period of time. The controlling parameter of the task is the rate of substrate entry into the biogas plant. An algorithm for solving this problem is proposed, based on the numerical implementation of the Pontryagin maximum principle. In this case, a hybrid genetic algorithm with an additional search in the vicinity of the best solution using the method of conjugate gradients was used as an optimization method. This numerical method for solving an optimal control problem is universal and applicable to a wide class of mathematical models.

In the course of the study, various modes of submission of the substrate to the digesters, temperature environments and types of raw materials were analyzed. It is shown that the rate of biogas production in the continuous feed mode is 1.4–1.9 times higher in the mesophilic medium (1.9–3.2 in the thermophilic medium) than in the periodic mode over the period of complete fermentation, which is associated with a higher feed rate of the substrate and a greater concentration of nutrients in the substrate. However, the yield of biogas during the period of complete fermentation with a periodic mode is twice as high as the output over the period of a complete change of the substrate in the methane tank at a continuous mode, which means incomplete processing of the substrate in the second case. The rate of biogas formation for a thermophilic medium in continuous mode and the optimal rate of supply of raw materials is three times higher than for a mesophilic medium. Comparison of biogas output for various types of raw materials shows that the highest biogas output is observed for waste poultry farms, the least — for cattle farms waste, which is associated with the nutrient content in a unit of substrate of each type.

A cluster method of mathematical modeling of interval-stochastic thermal processes in complex electronic systems (ES), is developed. In the cluster method, the construction of a complex ES is represented in the form of a thermal model, which is a system of clusters, each of which contains a core that combines the heat-generating elements falling into a given cluster, the cluster shell and a medium flow through the cluster. The state of the thermal process in each cluster and every moment of time is characterized by three interval-stochastic state variables, namely, the temperatures of the core, shell, and medium flow. The elements of each cluster, namely, the core, shell, and medium flow, are in thermal interaction between themselves and elements of neighboring clusters. In contrast to existing methods, the cluster method allows you to simulate thermal processes in complex ESs, taking into account the uneven distribution of temperature in the medium flow pumped into the ES, the conjugate nature of heat exchange between the medium flow in the ES, core and shells of clusters, and the intervalstochastic nature of thermal processes in the ES, caused by statistical technological variation in the manufacture and installation of electronic elements in ES and random fluctuations in the thermal parameters of the environment. The mathematical model describing the state of thermal processes in a cluster thermal model is a system of interval-stochastic matrix-block equations with matrix and vector blocks corresponding to the clusters of the thermal model. The solution to the interval-stochastic equations are statistical measures of the state variables of thermal processes in clusters - mathematical expectations, covariances between state variables and variance. The methodology for applying the cluster method is shown on the example of a real ES.

We describe an engineering-psychological experiment that continues the study of ways to adapt a person to the increasing complexity of logical problems by presenting a series of problems of increasing complexity, which is determined by the volume of initial data. Tasks require calculations in an associative or non-associative system of operations. By the nature of the change in the time of solving the problem, depending on the number of necessary operations, we can conclude that a purely sequential method of solving problems or connecting additional brain resources to the solution in parallel mode. In a previously published experimental work, a person in the process of solving an associative problem recognized color images with meaningful images. In the new study, a similar problem is solved for abstract monochrome geometric shapes. Analysis of the result showed that for the second case, the probability of the subject switching to a parallel method of processing visual information is significantly reduced. The research method is based on presenting a person with two types of tasks. One type of problem contains associative calculations and allows a parallel solution algorithm. Another type of problem is the control one, which contains problems in which calculations are not associative and parallel algorithms are ineffective. The task of recognizing and searching for a given object is associative. A parallel strategy significantly speeds up the solution with relatively small additional resources. As a control series of problems (to separate parallel work from the acceleration of a sequential algorithm), we use, as in the previous experiment, a non-associative comparison problem in cyclic arithmetic, presented in the visual form of the game “rock, paper, scissors”. In this problem, the parallel algorithm requires a large number of processors with a small efficiency coefficient. Therefore, the transition of a person to a parallel algorithm for solving this problem is almost impossible, and the acceleration of processing input information is possible only by increasing the speed. Comparing the dependence of the solution time on the volume of source data for two types of problems allows us to identify four types of strategies for adapting to the increasing complexity of the problem: uniform sequential, accelerated sequential, parallel computing (where possible), or undefined (for this method) strategy. The Reducing of the number of subjects, who switch to a parallel strategy when encoding input information with formal images, shows the effectiveness of codes that cause subject associations. They increase the speed of human perception and processing of information. The article contains a preliminary mathematical model that explains this phenomenon. It is based on the appearance of a second set of initial data, which occurs in a person as a result of recognizing the depicted objects.

The paper considers the process of flow formation in an outflow of a binary gas mixture through a thin slit into vacuum. An approach to modeling the flows of rarefied gas mixtures in the transient regime is proposed based on the direct solution of the Boltzmann kinetic equation, in which the conservative projection method is used to calculate the collision integrals. Calculation formulas are provided; the calculation procedure is described in detail in relation to the flow of a binary gas mixture. The Lennard–Jones potential is used as an interaction potential of molecules. A software modeling environment has been developed that makes it possible to study the flows of gas mixtures in a transitional regime on systems of cluster architecture. Due to the use of code parallelization technologies, an acceleration of calculations by 50–100 times was obtained. Numerical simulation of a two-dimensional outflow of a binary argon-neon gas mixture from a vessel into vacuum through a thin slit is carried out for various values of the Knudsen number. The graphs of the dependence of gas mixture components output flow on time in the process of establishing the flow are obtained. Non-stationary regions of strong separation of gas mixture components, in which the molecular densities ratio reaches 10 or more, were discovered. The discovered effect can have applications in the problem of gas mixtures separation.

The two significant geometric parameters are proposed that affect the physical quantities interpolation in the smoothed particle hydrodynamics method (SPH). They are: the smoothing coefficient which the particle size and the smoothing radius are connecting and the volume coefficient which determine correctly the particle mass for a given particles distribution in the medium.

In paper proposes a technique for these parameters influence assessing on the SPH method interpolations accuracy when the hydrostatic problem solving. The analytical functions of the relative error for the density and pressure gradient in the medium are introduced for the accuracy estimate. The relative error functions are dependent on the smoothing factor and the volume factor. Designating a specific interpolation form in SPH method allows the differential form of the relative error functions to the algebraic polynomial form converting. The root of this polynomial gives the smoothing coefficient values that provide the minimum interpolation error for an assigned volume coefficient.

In this work, the derivation and analysis of density and pressure gradient relative errors functions on a sample of popular nuclei with different smoothing radius was carried out. There is no common the smoothing coefficient value for all the considered kernels that provides the minimum error for both SPH interpolations. The nuclei representatives with different smoothing radius are identified which make it possible the smallest errors of SPH interpolations to provide when the hydrostatic problem solving. As well, certain kernels with different smoothing radius was determined which correct interpolation do not allow provide when the hydrostatic problem solving by the SPH method.

The utilization of unmanned aerial vehicles (UAVs) in high-rise firefighting operations is the right solution for reaching the fire scene on high floors quickly and effectively. The article proposes a quadrotor-type firefighting UAV model carrying a launcher to launch a missile containing fire extinguishing powders into a fire. The kinematic model describing the flight kinematics of this UAV model is built based on the Newton – Euler method when the device is in normal motion and at the time of launching a firefighting missile. The results from the simulation testing the validity of the kinematic model and the simulation of the motion of the UAV show that the variation of Euler angles, flight angles, and aerodynamic angles during a flight are within an acceptable range and overload guarantee in flight. The UAV flew to the correct position to launch the required fire-extinguishing ammunition. The results of the research are the basis for building a control system of high-rise firefighting drones in Vietnam.

This article reviews both historical achievements and modern results in the field of Markov Decision Process (MDP) and convex optimization. This review is the first attempt to cover the field of reinforcement learning in Russian in the context of convex optimization. The fundamental Bellman equation and the criteria of optimality of policy — strategies based on it, which make decisions based on the known state of the environment at the moment, are considered. The main iterative algorithms of policy optimization based on the solution of the Bellman equations are also considered. An important section of this article was the consideration of an alternative to the $Q$-learning approach — the method of direct maximization of the agent’s average reward for the chosen strategy from interaction with the environment. Thus, the solution of this convex optimization problem can be represented as a linear programming problem. The paper demonstrates how the convex optimization apparatus is used to solve the problem of Reinforcement Learning (RL). In particular, it is shown how the concept of strong duality allows us to naturally modify the formulation of the RL problem, showing the equivalence between maximizing the agent’s reward and finding his optimal strategy. The paper also discusses the complexity of MDP optimization with respect to the number of state–action–reward triples obtained as a result of interaction with the environment. The optimal limits of the MDP solution complexity are presented in the case of an ergodic process with an infinite horizon, as well as in the case of a non-stationary process with a finite horizon, which can be restarted several times in a row or immediately run in parallel in several threads. The review also reviews the latest results on reducing the gap between the lower and upper estimates of the complexity of MDP optimization with average remuneration (Averaged MDP, AMDP). In conclusion, the real-valued parametrization of agent policy and a class of gradient optimization methods through maximizing the $Q$-function of value are considered. In particular, a special class of MDPs with restrictions on the value of policy (Constrained Markov Decision Process, CMDP) is presented, for which a general direct-dual approach to optimization with strong duality is proposed.

As the population of cities grows, the need to plan for the development of transport infrastructure becomes more acute. For this purpose, transport modeling packages are created. These packages usually contain a set of convex optimization problems, the iterative solution of which leads to the desired equilibrium distribution of flows along the paths. One of the directions for the development of transport modeling is the construction of more accurate generalized models that take into account different types of passengers, their travel purposes, as well as the specifics of personal and public modes of transport that agents can use. Another important direction of transport models development is to improve the efficiency of the calculations performed. Since, due to the large dimension of modern transport networks, the search for a numerical solution to the problem of equilibrium distribution of flows along the paths is quite expensive. The iterative nature of the entire solution process only makes this worse. One of the approaches leading to a reduction in the number of calculations performed is the construction of consistent models that allow to combine the blocks of a 4-stage model into a single optimization problem. This makes it possible to eliminate the iterative running of blocks, moving from solving a separate optimization problem at each stage to some general problem. Early work has proven that such approaches provide equivalent solutions. However, it is worth considering the validity and interpretability of these methods. The purpose of this article is to substantiate a single problem, that combines both the calculation of the trip matrix and the modal choice, for the generalized case when there are different layers of demand, types of agents and classes of vehicles in the transport network. The article provides possible interpretations for the gauge parameters used in the problem, as well as for the dual factors associated with the balance constraints. The authors of the article also show the possibility of combining the considered problem with a block for determining network load into a single optimization problem.

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.

Modeling of channel processes in the study of coastal channel deformations requires the calculation of hydrodynamic flow parameters that take into account the existence of secondary transverse currents formed at channel curvature. Three-dimensional modeling of such processes is currently possible only for small model channels; for real river flows, reduced-dimensional models are needed. At the same time, the reduction of the problem from a three-dimensional model of the river flow movement to a two-dimensional flow model in the cross-section assumes that the hydrodynamic flow under consideration is quasi-stationary and the hypotheses about the asymptotic behavior of the flow along the flow coordinate of the cross-section are fulfilled for it. Taking into account these restrictions, a mathematical model of the problem of the a stationary turbulent calm river flow movement in a channel cross-section is formulated. The problem is formulated in a mixed formulation of velocity — “vortex – stream function”. As additional conditions for problem reducing, it is necessary to specify boundary conditions on the flow free surface for the velocity field, determined in the normal and tangential direction to the cross-section axis. It is assumed that the values of these velocities should be determined from the solution of auxiliary problems or obtained from field or experimental measurement data.

To solve the formulated problem, the finite element method in the Petrov – Galerkin formulation is used. Discrete analogue of the problem is obtained and an algorithm for solving it is proposed. Numerical studies have shown that, in general, the results obtained are in good agreement with known experimental data. The authors associate the obtained errors with the need to more accurately determine the circulation velocities field at crosssection of the flow by selecting and calibrating a more appropriate model for calculating turbulent viscosity and boundary conditions at the free boundary of the cross-section.