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A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Mathematics and Computer Science".

Deadline for manuscript submissions: closed (30 June 2024) | Viewed by 10895

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Special Issue Editors

Prof. Dr. Andrey Gorshenin

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Guest Editor

Federal Research Center “Computer Science and Control”, Russian Academy of Science, 119333 Moscow, Russia
Interests: machine learning; neural networks; semiparametric models; stochastic models; mixture distributions; computational statistics; data analysis
Special Issues, Collections and Topics in MDPI journals

Prof. Dr. Mikhail Posypkin

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Guest Editor

Federal Research Center “Computer Science and Control” of the Russian Academy of Sciences, 119333 Moscow, Russia
Interests: discrete optimization; global optimization; parallel programming; multi-objective optimization; complex systems
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Prof. Dr. Vladimir Titarev

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Guest Editor

Federal Research Center “Computer Science and Control” of the Russian Academy of Sciences, 119333 Moscow, Russia
Interests: computational fluid dynamics; numerical analysis; parallel computing; computational physics; rarefied gas dynamics
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Mathematical optimization and machine learning are two highly sophisticated, advanced analytics technologies that are used in a vast array of applications. Both are based on a substantial mathematical background and are convincing examples of how mathematics can be used to solve complex problems. Both technologies have a seemingly endless range of applications, including image and speech recognition, virtual personal assistants, fraud detection, autonomic driving vehicles, production planning, workforce scheduling, electric power distribution, shipment routing, design optimization, robotics, etc.

Optimization and machine learning are tightly coupled with a mature but still sought-after research direction—mathematical modeling. For example, optimization that operates with a detailed mathematical model of a business process, technical construct, or physical phenomenon. Machine learning methods can be effectively employed to estimate the parameters of models when traditional methods fail due to uncertainty, including variance or noise in the specific data values.

This Special Issue of Mathematics is a follow-up to the successful first edition titled “Mathematical Modeling, Optimization and Machine Learning”. This Special Issue series is devoted to topics in mathematical modeling, optimization methods, and various machine learning approaches. Submitted papers should satisfy the general requirements of the Mathematics journal, with a strong focus on new analytic or numerical methods for solving challenging problems. Potential topics include, but are not limited to, the following:

Mathematical foundations of machine learning;
New machine learning algorithms, approaches, and architectures of neural networks;
Mathematical models and machine learning;
Data analysis based on mathematical models, optimization, and machine learning algorithms;
Mathematical models, optimization techniques, and machine learning algorithms in applied sciences;
Statistical models and stochastic processes;
Continuous and discrete optimization, linear and nonlinear optimization, derivative-free optimization;
Deterministic and stochastic optimization algorithms;
Numerical simulation in physical, social, and life sciences;
High-performance computing for mathematical modeling;
Application of machine learning, mathematical modeling, and optimization in science and technology.

Prof. Dr. Andrey Gorshenin
Prof. Dr. Mikhail Posypkin
Prof. Dr. Vladimir Titarev
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

mathematical modeling
mathematical optimization
control theory and applications
high-performance computing
stochastic processes
numerical analysis and simulation
computational fluid dynamics
machine learning
data analytics

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Published Papers (9 papers)

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Research

22 pages, 2742 KiB

Open AccessArticle

Simulation of Shock Waves in Methane: A Self-Consistent Continuum Approach Enhanced Using Machine Learning

by Zarina Maksudova, Liia Shakurova and Elena Kustova

Mathematics 2024, 12(18), 2924; https://doi.org/10.3390/math12182924 - 20 Sep 2024

Viewed by 790

Abstract

This study presents a self-consistent one-temperature approach for modeling shock waves in single-component methane. The rigorous mathematical model takes into account the complex structure of CH₄ molecules with multiple vibrational modes and incorporates exact kinetic theory-based transport coefficients, including bulk viscosity. The effects of the bulk viscosity on gas-dynamic variables and transport terms are investigated in detail under varying degree of gas rarefaction. It is demonstrated that neglecting bulk viscosity significantly alters the shock front width and peak values of normal stress and heat flux, with the effect being more evident in denser gases. The study also evaluates limitations in the use of a constant specific heat ratio, revealing that this approach fails to accurately predict post-shock parameters in polyatomic gases, even at moderate Mach numbers. To enhance computational efficiency, a simplified approach based on a reduced vibrational spectrum is assessed. The results indicate that considering only the ground state leads to substantial errors in the fluid-dynamic variables across the shock front. Another approach explored involves the application of machine learning techniques to calculate vibrational energy and specific heat. Among the methods tested, the Feedforward Neural Network (FNN) proves to be the most effective, offering significant acceleration in calculations and providing one of the lowest errors. When integrated into the fluid-dynamic solver, the FNN approach yields nearly a three-fold increase in speed in numerical simulations of the shock wave structure. Full article

(This article belongs to the Special Issue Mathematical Modeling, Optimization and Machine Learning, 2nd Edition)

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12 pages, 6818 KiB

Open AccessArticle

Image Noise Reduction and Solution of Unconstrained Minimization Problems via New Conjugate Gradient Methods

by Bassim A. Hassan, Issam A. R. Moghrabi, Thaair A. Ameen, Ranen M. Sulaiman and Ibrahim Mohammed Sulaiman

Mathematics 2024, 12(17), 2754; https://doi.org/10.3390/math12172754 - 5 Sep 2024

Viewed by 525

Abstract

The conjugate gradient (CG) directions are among the important components of the CG algorithms. These directions have proven their effectiveness in many applications—more specifically, in image processing due to their low memory requirements. In this study, we derived a new conjugate gradient coefficient based on the famous quadratic model. The derived algorithm is distinguished by its global convergence and essential descent properties, ensuring robust performance across diverse scenarios. Extensive numerical testing on image restoration and unconstrained optimization problems have demonstrated that the new formulas significantly outperform existing methods. Specifically, the proposed conjugate gradient scheme has shown superior performance compared to the traditional Fletcher–Reeves (FR) conjugate gradient method. This advancement not only enhances computational efficiency on unconstrained optimization problems, but also improves the accuracy and quality of image restoration, making it a highly valuable tool in the field of computational imaging and optimization. Full article

(This article belongs to the Special Issue Mathematical Modeling, Optimization and Machine Learning, 2nd Edition)

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15 pages, 349 KiB

Open AccessFeature PaperArticle

Risk Measures’ Duality on Ordered Linear Spaces

by Christos E. Kountzakis and Damiano Rossello

Mathematics 2024, 12(8), 1165; https://doi.org/10.3390/math12081165 - 12 Apr 2024

Viewed by 683

Abstract

The aim of this paper is to provide a dual representation of convex and coherent risk measures in partially ordered linear spaces with respect to the algebraic dual space. An algebraic robust representation is deduced by weak separation of convex sets by functionals, which are assumed to be only linear; thus, our framework does not require any topological structure of the underlying spaces, and our robust representations are found without any continuity requirement for the risk measures. We also use such extensions to the representation of acceptability indices. Full article

(This article belongs to the Special Issue Mathematical Modeling, Optimization and Machine Learning, 2nd Edition)

23 pages, 12259 KiB

Open AccessArticle

Exploring the Entropy-Based Classification of Time Series Using Visibility Graphs from Chaotic Maps

by J. Alberto Conejero, Andrei Velichko, Òscar Garibo-i-Orts, Yuriy Izotov and Viet-Thanh Pham

Mathematics 2024, 12(7), 938; https://doi.org/10.3390/math12070938 - 22 Mar 2024

Cited by 2 | Viewed by 1469

Abstract

The classification of time series using machine learning (ML) analysis and entropy-based features is an urgent task for the study of nonlinear signals in the fields of finance, biology and medicine, including EEG analysis and Brain–Computer Interfacing. As several entropy measures exist, the problem is assessing the effectiveness of entropies used as features for the ML classification of nonlinear dynamics of time series. We propose a method, called global efficiency (GEFMCC), for assessing the effectiveness of entropy features using several chaotic mappings. GEFMCC is a fitness function for optimizing the type and parameters of entropies for time series classification problems. We analyze fuzzy entropy (FuzzyEn) and neural network entropy (NNetEn) for four discrete mappings, the logistic map, the sine map, the Planck map, and the two-memristor-based map, with a base length time series of 300 elements. FuzzyEn has greater GEFMCC in the classification task compared to NNetEn. However, NNetEn classification efficiency is higher than FuzzyEn for some local areas of the time series dynamics. The results of using horizontal visibility graphs (HVG) instead of the raw time series demonstrate the GEFMCC decrease after HVG time series transformation. However, the GEFMCC increases after applying the HVG for some local areas of time series dynamics. The scientific community can use the results to explore the efficiency of the entropy-based classification of time series in “The Entropy Universe”. An implementation of the algorithms in Python is presented. Full article

(This article belongs to the Special Issue Mathematical Modeling, Optimization and Machine Learning, 2nd Edition)

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21 pages, 11523 KiB

Open AccessArticle

Comparison of Statistical Approaches for Reconstructing Random Coefficients in the Problem of Stochastic Modeling of Air–Sea Heat Flux Increments

by Konstantin P. Belyaev, Andrey K. Gorshenin, Victor Yu. Korolev and Anastasiia A. Osipova

Mathematics 2024, 12(2), 288; https://doi.org/10.3390/math12020288 - 16 Jan 2024

Cited by 4 | Viewed by 958

Abstract

This paper compares two statistical methods for parameter reconstruction (random drift and diffusion coefficients of the Itô stochastic differential equation, SDE) in the problem of stochastic modeling of air–sea heat flux increment evolution. The first method relates to a nonparametric estimation of the transition probabilities (wherein consistency is proven). The second approach is a semiparametric reconstruction based on the approximation of the SDE solution (in terms of distributions) by finite normal mixtures using the maximum likelihood estimates of the unknown parameters. This approach does not require any additional assumptions for the coefficients, with the exception of those guaranteeing the existence of the solution to the SDE itself. It is demonstrated that the corresponding conditions hold for the analyzed data. The comparison is carried out on the simulated samples, modeling the case where the SDE random coefficients are represented in trigonometric form, which is related to common climatic models, as well as on the ERA5 reanalysis data of the sensible and latent heat fluxes in the North Atlantic for 1979–2022. It is shown that the results of these two methods are close to each other in a quantitative sense, but differ somewhat in temporal variability and spatial localization. The differences during the observed period are analyzed, and their geophysical interpretations are presented. The semiparametric approach seems promising for physics-informed machine learning models. Full article

(This article belongs to the Special Issue Mathematical Modeling, Optimization and Machine Learning, 2nd Edition)

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26 pages, 5895 KiB

Open AccessArticle

Modeling the Solution of the Pursuit–Evasion Problem Based on the Intelligent–Geometric Control Theory

by Mikhail Khachumov and Vyacheslav Khachumov

Mathematics 2023, 11(23), 4869; https://doi.org/10.3390/math11234869 - 4 Dec 2023

Viewed by 1614

Abstract

An important action-planning problem is considered for participants of the pursuit–evasion game with multiple pursuers and a high-speed evader. The objects of study are mobile robotic systems and specifically small unmanned aerial vehicles (UAVs). The problem is complicated by the presence of significant wind loads that affect the trajectory and motion strategies of the players. It is assumed that UAVs have limited computing resources, which involves the use of computationally fast and real-time heuristic approaches. A novel and rapidly developing intelligent–geometric theory is applied to address the discussed problem. To accurately calculate the points of the participant’s rapprochement, we use a geometric approach based on the construction of circles or spheres of Apollonius. Intelligent control methods are applied to synthesize complex motion strategies of participants. A method for quickly predicting the evader’s trajectory is proposed based on a two-layer neural network containing a new activation function of the “s-parabola” type. We consider a special backpropagation training scheme for the model under study. A simulation scheme has been developed and tested, which includes mathematical models of dynamic objects and wind loads. The conducted simulations on pursuit–evasion games in close to real conditions showed the prospects and expediency of the presented approach. Full article

(This article belongs to the Special Issue Mathematical Modeling, Optimization and Machine Learning, 2nd Edition)

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12 pages, 1089 KiB

Open AccessArticle

Hybrid Lattice Boltzmann Model for Nonlinear Diffusion and Image Denoising

by Oleg Ilyin

Mathematics 2023, 11(22), 4601; https://doi.org/10.3390/math11224601 - 10 Nov 2023

Viewed by 1012

Abstract

In the present paper, a novel approach for image denoising based on the numerical solution to the nonlinear diffusion equation is proposed. The Perona–Malik-type equation is solved by employing a hybrid lattice Boltzmann model with five discrete velocities. In this method, the regions with large values of the diffusion coefficient are modeled with the lattice Boltzmann scheme for which hyper-viscous defects are reduced, while other regions are modeled with the conventional lattice Boltzmann model. The new method allows us to solve Perona–Malik-type equations with relatively large time steps and good accuracy. In numerical experiments, the removal of salt and pepper, speckle and Gaussian noise is considered. For salt and pepper noise, the novel scheme yields better peak signal-to-noise ratios in image denoising problems compared to the standard lattice Boltzmann approach. For certain non-small values of time steps, the novel model shows better results for speckle and Gaussian noise on average. Full article

(This article belongs to the Special Issue Mathematical Modeling, Optimization and Machine Learning, 2nd Edition)

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26 pages, 3879 KiB

Open AccessArticle

Stochastic Time Complexity Surfaces of Computing Node

by Andrey Borisov and Alexey Ivanov

Mathematics 2023, 11(20), 4379; https://doi.org/10.3390/math11204379 - 21 Oct 2023

Cited by 1 | Viewed by 1165

Abstract

The paper is devoted to the formal description of the running time of the user task on some virtual nodes in the computing network. Based on the probability theory framework, this time represents a random value with a finite mean and variance. For any class of user task, these moments are the functions of the node resources, task numerical characteristics, and the parameters of the current node state. These functions of the vector arguments can be treated as some surfaces in the multidimensional Euclidean spaces, so the proposed models are called the stochastic time complexity surfaces. The paper also presents a class of functions suitable for the description of both the mean and variance. They contain unknown parameters which should be estimated. The article includes the statement of the parameter identification problem given the statistical results of the node stress testing, recommendations concerning the test planning, and preprocessing of the raw experiment data. To illustrate the performance of the proposed model, the authors design it for an actual database application—the prototype of the passengers’ personal data anonymization system. Its application functions are classified into two user task classes: the data anonymization procedures and fulfillment of the statistical queries. The authors identify the stochastic time complexity surfaces for both task types. The additional testing experiments confirm the high performance of the suggested model and its applicability to the solution of the practical providers’ problems. Full article

(This article belongs to the Special Issue Mathematical Modeling, Optimization and Machine Learning, 2nd Edition)

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23 pages, 2611 KiB

Open AccessArticle

Fuzzy CNN Autoencoder for Unsupervised Anomaly Detection in Log Data

by Oleg Gorokhov, Mikhail Petrovskiy, Igor Mashechkin and Maria Kazachuk

Mathematics 2023, 11(18), 3995; https://doi.org/10.3390/math11183995 - 20 Sep 2023

Cited by 1 | Viewed by 1674

Abstract

Currently, the task of maintaining cybersecurity and reliability in various computer systems is relevant. This problem can be solved by detecting anomalies in the log data, which are represented as a stream of textual descriptions of events taking place. For these purposes, reduction to a One-class classification problem is used. Standard One-class classification methods do not achieve good results. Deep learning approaches are more effective. However, they are not robust to outliers and require a lot of computational effort. In this paper, we propose a new robust approach based on a convolutional autoencoder using fuzzy clustering. The proposed approach uses a parallel convolution operation to feature extraction, which makes it more efficient than the currently popular Transformer architecture. In the course of the experiments, the proposed approach showed the best results for both the cybersecurity and the reliability problems compared to existing approaches. It was also shown that the proposed approach is robust to outliers in the training set. Full article

(This article belongs to the Special Issue Mathematical Modeling, Optimization and Machine Learning, 2nd Edition)

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