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

Deadline for manuscript submissions: closed (29 February 2024) | Viewed by 22299

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Special Issue Editors

Prof. Dr. Nicholas Christakis

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Guest Editor

Department of Mathematics and Applied Mathematics, University of Crete, 71003 Heraklion, Greece
Interests: artificial neural networks; computational fluid dynamics; atmospheric physics; complex systems

Prof. Dr. George Kossioris

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Co-Guest Editor

Department of Mathematics and Applied Mathematics, University of Crete, 71003 Heraklion, Greece
Interests: partial differential equations; optimal control; numerical computations; numerical weather prediction; machine learning

Prof. Dr. Mayur Patel

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Co-Guest Editor

Computational Mechanics and Reliability Group (CMRG), Centre for Numerical Modelling and Process Analysis, University of Greenwich, 30 Park Row, London SE10 9LS, UK
Interests: computational modelling; computational fluid dynamics; computational science and engineering; computational fire modelling

Special Issue Information

Dear Colleagues,

The current Special Issue is devoted to the applications of mathematical modeling and neural networks. It has always been of great importance to be able to determine the evolution of physical/engineering/medical/social systems. This is mainly achieved through either mathematical modeling or machine learning. Neural networks have been at the heart of machine learning, since its inception, being one of the most popular techniques, utilized by many supervised learning algorithms. Both approaches have been proven very effective in correctly predicting system behavior and, despite their limitations, are at the forefront of numerical and computational mathematics. Especially today, with the abundance in computational tools and resources, new techniques and algorithms are continuously being developed and tested in a variety of applications. This Special Issue welcomes papers from all fields of applied mathematical modeling and machine learning (in particular, algorithms based on neural networks), where the emphasis is placed on the application of techniques and algorithms to physical/engineering/medical/social systems. Novel applications from all fronts of machine learning will also be considered. Research in areas where both approaches work in synergy, complementing one another in the modeling of complex systems, are particularly welcome.

Prof. Dr. Nicholas Christakis
Prof. Dr. George Kossioris
Prof. Dr. Mayur Patel
Guest Editors

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Keywords

mathematical modeling
neural networks
machine learning
computations
algorithms
applications

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Published Papers (12 papers)

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Research

21 pages, 12612 KiB

Open AccessArticle

Hybrid Deep Learning and Sensitivity Operator-Based Algorithm for Identification of Localized Emission Sources

by Alexey Penenko, Mikhail Emelyanov, Evgeny Rusin, Erjena Tsybenova and Vasily Shablyko

Mathematics 2024, 12(1), 78; https://doi.org/10.3390/math12010078 - 25 Dec 2023

Viewed by 1056

Abstract

Hybrid approaches combining machine learning with traditional inverse problem solution methods represent a promising direction for the further development of inverse modeling algorithms. The paper proposes an approach to emission source identification from measurement data for advection–diffusion–reaction models. The approach combines general-type source identification and post-processing refinement: first, emission source identification by measurement data is carried out by a sensitivity operator-based algorithm, and then refinement is done by incorporating a priori information about unknown sources. A general-type distributed emission source identified at the first stage is transformed into a localized source consisting of multiple point-wise sources. The second, refinement stage consists of two steps: point-wise source localization and emission rate estimation. Emission source localization is carried out using deep learning with convolutional neural networks. Training samples are generated using a sensitivity operator obtained at the source identification stage. The algorithm was tested in regional remote sensing emission source identification scenarios for the Lake Baikal region and was able to refine the emission source reconstruction results. Hence, the aggregates used in traditional inverse problem solution algorithms can be successfully applied within machine learning frameworks to produce hybrid algorithms. Full article

(This article belongs to the Special Issue Applications of Mathematical Modeling and Neural Networks)

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25 pages, 5170 KiB

Open AccessArticle

Prevention of PID Phenomenon for Solar Panel Based on Mathematical Data Analysis Models

by You-Shyang Chen, Ying-Hsun Hung, Yu-Sheng Lin, Jieh-Ren Chang, Chi-Hsiang Lo and Hong-Kai You

Mathematics 2023, 11(19), 4044; https://doi.org/10.3390/math11194044 - 23 Sep 2023

Viewed by 1810

Abstract

In recent years, the problem of potential-induced degradation (PID) phenomenon has been deeply associated with solar power issues because it causes serious power attenuation of solar panels and results in lowering its power generation efficiency. Thus, effectively identifying the PID problem from insights of industry data analysis to reduce production costs and increase the performance of power generation is an interesting and important subject for the solar power industry. Moreover, by the traditional standard rule (IEC62804) and the condition of a 96 h testing time, the costs of testing time and assembling materials against PID are very high and must be improved. Given the above reasons, this study proposes a hybrid procedure to organizes four mathematical methods: the mini-module testing, solar cell testing, a settling time, and a neural network, which are named as Method-1–Method-4, respectively, to efficiently solve the PID problem. Consequently, there are four key outcomes from the empirical results for solar power application: (1) In Method-1 with a 96 h testing time, it was found that the large module with higher costs and the mini module with lower costs have a positive correlation; thus, we can replace the large-module testing by the effective mini module for lower cost on module materials. (2) In Method-2 with a 24 h testing time, it was also found that the mini module and the solar cell are positively correlated; this result provides evidence that we can conduct the PID test by the easier solar cell to lower the costs. (3) In Method-3, the settling time achieves an average accuracy of 94% for PID prediction with a 14 h testing time. (4) In Method-4, the experimental result provides an accuracy of 80% when identifying the PID problem with the mathematical neural network model and are obtained within a 2 h testing time. From the above results, these methods succeed in reducing cost of materials and testing time during the manufacturing process; thus, this study has an industrial application value. Concurrently, Method-3 and Method-4 are rarely seen in the limited literature review for identifying PID problem; therefore, this study also offers a novel contribution for technical application innovation. Full article

(This article belongs to the Special Issue Applications of Mathematical Modeling and Neural Networks)

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17 pages, 2518 KiB

Open AccessArticle

Unsupervised Learning of Particles Dispersion

by Nicholas Christakis and Dimitris Drikakis

Mathematics 2023, 11(17), 3637; https://doi.org/10.3390/math11173637 - 23 Aug 2023

Cited by 3 | Viewed by 1101

Abstract

This paper discusses using unsupervised learning in classifying particle-like dispersion. The problem is relevant to various applications, including virus transmission and atmospheric pollution. The Reduce Uncertainty and Increase Confidence (RUN-ICON) algorithm of unsupervised learning is applied to particle spread classification. The algorithm classifies the particles with higher confidence and lower uncertainty than other algorithms. The algorithm’s efficiency remains high also when noise is added to the system. Applying unsupervised learning in conjunction with the RUN-ICON algorithm provides a tool for studying particles’ dynamics and their impact on air quality, health, and climate. Full article

(This article belongs to the Special Issue Applications of Mathematical Modeling and Neural Networks)

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17 pages, 2149 KiB

Open AccessArticle

Reducing Uncertainty and Increasing Confidence in Unsupervised Learning

by Nicholas Christakis and Dimitris Drikakis

Mathematics 2023, 11(14), 3063; https://doi.org/10.3390/math11143063 - 11 Jul 2023

Cited by 5 | Viewed by 1878

Abstract

This paper presents the development of a novel algorithm for unsupervised learning called RUN-ICON (Reduce UNcertainty and Increase CONfidence). The primary objective of the algorithm is to enhance the reliability and confidence of unsupervised clustering. RUN-ICON leverages the K-means++ method to identify the most frequently occurring dominant centres through multiple repetitions. It distinguishes itself from existing K-means variants by introducing novel metrics, such as the Clustering Dominance Index and Uncertainty, instead of relying solely on the Sum of Squared Errors, for identifying the most dominant clusters. The algorithm exhibits notable characteristics such as robustness, high-quality clustering, automation, and flexibility. Extensive testing on diverse data sets with varying characteristics demonstrates its capability to determine the optimal number of clusters under different scenarios. The algorithm will soon be deployed in real-world scenarios, where it will undergo rigorous testing against data sets based on measurements and simulations, further proving its effectiveness. Full article

(This article belongs to the Special Issue Applications of Mathematical Modeling and Neural Networks)

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21 pages, 1873 KiB

Open AccessArticle

ESN-Observer-Based Adaptive Stabilization Control for Delayed Nonlinear Systems with Unknown Control Gain

by Shuxian Lun, Zhaoyi Lv, Xiaodong Lu and Ming Li

Mathematics 2023, 11(13), 2965; https://doi.org/10.3390/math11132965 - 3 Jul 2023

Viewed by 893

Abstract

This paper investigates the observer-based adaptive stabilization control problem for a class of time-delay nonlinear systems with unknown control gain using an echo state network (ESN). In order to handle unknown functions, a new recurrent neural network (RNN) approximation method called ESN is utilized. It improves accuracy, reduces computing cost, and is simple to train. To address the issue of unknown control gain, the Nussbaum function is used, and the Lyapunov–Krasovskii functionals are used to address the delay term. The backstepping strategy and command filtering methodology are then used to create an adaptive stabilization controller. All of the closed-loop system’s signals are predicted to be confined by the Lyapunov stability theory. Finally, a simulation example is used to demonstrate the effectiveness of the suggested control mechanism. Full article

(This article belongs to the Special Issue Applications of Mathematical Modeling and Neural Networks)

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27 pages, 1422 KiB

Open AccessArticle

Transfer Learning-Based Coupling of Smoothed Finite Element Method and Physics-Informed Neural Network for Solving Elastoplastic Inverse Problems

by Meijun Zhou and Gang Mei

Mathematics 2023, 11(11), 2529; https://doi.org/10.3390/math11112529 - 31 May 2023

Cited by 2 | Viewed by 2978

Abstract

In practical engineering applications, there is a high demand for inverting parameters for various materials, and obtaining monitoring data can be costly. Traditional inverse methods often involve tedious computational processes, require significant computational effort, and exhibit slow convergence speeds. The recently proposed Physics-Informed Neural Network (PINN) has shown great potential in solving inverse problems. Therefore, in this paper, we propose a transfer learning-based coupling of the Smoothed Finite Element Method (S-FEM) and PINN methods for the inversion of parameters in elastic-plasticity problems. The aim is to improve the accuracy and efficiency of parameter inversion for different elastic-plastic materials with limited data. High-quality small datasets were synthesized using S-FEM and subsequently combined with PINN for pre-training purposes. The parameters of the pre-trained model were saved and used as the initial state for the PINN model in the inversion of new material parameters. The inversion performance of the coupling of S-FEM and PINN is compared with the coupling of the conventional Finite Element Method (FEM) and PINN on a small data set. Additionally, we compared the efficiency and accuracy of both the transfer learning-based and non-transfer learning-based methods of the coupling of S-FEM and PINN in the inversion of different material parameters. The results show that: (1) our method performs well on small datasets, with an inversion error of essentially less than 2%; (2) our approach outperforms the coupling of conventional FEM and PINN in terms of both computational accuracy and computational efficiency; and (3) our approach is at least twice as efficient as the coupling of S-FEM and PINN without transfer learning, while still maintaining accuracy. Our method is well-suited for the inversion of different material parameters using only small datasets. The use of transfer learning greatly improves computational efficiency, making our method an efficient and accurate solution for reducing computational cost and complexity in practical engineering applications. Full article

(This article belongs to the Special Issue Applications of Mathematical Modeling and Neural Networks)

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28 pages, 16409 KiB

Open AccessArticle

Enhancing Computational Accuracy in Surrogate Modeling for Elastic–Plastic Problems by Coupling S-FEM and Physics-Informed Deep Learning

by Meijun Zhou, Gang Mei and Nengxiong Xu

Mathematics 2023, 11(9), 2016; https://doi.org/10.3390/math11092016 - 24 Apr 2023

Cited by 2 | Viewed by 2537

Abstract

Physics-informed neural networks (PINNs) provide a new approach to solving partial differential equations (PDEs), while the properties of coupled physical laws present potential in surrogate modeling. However, the accuracy of PINNs in solving forward problems needs to be enhanced, and solving inverse problems relies on data samples. The smoothed finite element method (S-FEM) can obtain high-fidelity numerical solutions, which are easy to solve for the forward problems of PDEs, but difficult to solve for the inverse problems. To the best of the authors’ knowledge, there has been no prior research on coupling S-FEM and PINN. In this paper, a novel approach that couples S-FEM and PINN is proposed. The proposed approach utilizes S-FEM to synthesize high-fidelity datasets required for PINN inversion, while also improving the accuracy of data-independent PINN in solving forward problems. The proposed approach is applied to solve linear elastic and elastoplastic forward and inverse problems. The computational results demonstrate that the coupling of the S-FEM and PINN exhibits high precision and convergence when solving inverse problems, achieving a maximum relative error of 0.2% in linear elasticity and 5.69% in elastoplastic inversion by using approximately 10,000 data points. The coupling approach also enhances the accuracy of solving forward problems, reducing relative errors by approximately 2–10 times. The proposed coupling of the S-FEM and PINN offers a novel surrogate modeling approach that incorporates knowledge and data-driven techniques, enabling it to solve both forward and inverse problems associated with PDEs with high levels of accuracy and convergence. Full article

(This article belongs to the Special Issue Applications of Mathematical Modeling and Neural Networks)

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27 pages, 2245 KiB

Open AccessArticle

Generic Model of Max Heteroassociative Memory Robust to Acquisition Noise

by Valentín Trujillo-Mora, Marco Moreno-Ibarra, Francisco Marroquín-Gutiérrez and Julio-César Salgado-Ramírez

Mathematics 2023, 11(9), 2015; https://doi.org/10.3390/math11092015 - 24 Apr 2023

Viewed by 1233

Abstract

Associative memories are a significant topic in pattern recognition, and therefore, throughout history, numerous memory models have been designed due to their usefulness. One such model is the associative memory minmax, which is highly efficient at learning and recalling patterns as well as being tolerant of high levels of additive and subtractive noise. However, it is not efficient when it comes to mixed noise. To solve this issue in the associative memory minmax, we present the generic model of heteroassociative memory max robust to acquisition noise (mixed noise). This solution is based on understanding the behavior of acquisition noise and mapping the location of noise in binary images and gray-scale through a distance transform. By controlling the location of the noise, the associative memories minmax become highly efficient. Furthermore, our proposed model allows patterns to contain mixed noise while still being able to recall the learned patterns completely. Our results show that the proposed model outperforms a model that has already solved this type of problem and has proven to overcome existing methods that show some solution to mixed noise. Additionally, we demonstrate that our model is applicable to all associative minmax memories with excellent results. Full article

(This article belongs to the Special Issue Applications of Mathematical Modeling and Neural Networks)

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23 pages, 2689 KiB

Open AccessArticle

Deep Learning Nonhomogeneous Elliptic Interface Problems by Soft Constraint Physics-Informed Neural Networks

by Fujun Cao, Xiaobin Guo, Fei Gao and Dongfang Yuan

Mathematics 2023, 11(8), 1843; https://doi.org/10.3390/math11081843 - 13 Apr 2023

Cited by 3 | Viewed by 2239

Abstract

It is a great challenge to solve nonhomogeneous elliptic interface problems, because the interface divides the computational domain into two disjoint parts, and the solution may change dramatically across the interface. A soft constraint physics-informed neural network with dual neural networks is proposed, which is composed of two separate neural networks for each subdomain, which are coupled by the connecting conditions on the interface. It is beneficial to capture the singularity of the solution across the interface. We formulate the PDEs, boundary conditions, and jump conditions on the interface into the loss function by means of the physics-informed neural network (PINN), and the different terms in the loss function are balanced by optimized penalty weights. To enhance computing efficiency for increasingly difficult issues, adaptive activation functions and the adaptive sampled method are used, which may be improved to produce the optimal network performance, as the topology of the loss function involved in the optimization process changes dynamically. Lastly, we present many numerical experiments, in both 2D and 3D, to demonstrate the proposed method’s flexibility, efficacy, and accuracy in tackling nonhomogeneous interface issues. Full article

(This article belongs to the Special Issue Applications of Mathematical Modeling and Neural Networks)

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27 pages, 18273 KiB

Open AccessArticle

A New Neural Network Training Algorithm Based on Artificial Bee Colony Algorithm for Nonlinear System Identification

by Ebubekir Kaya

Mathematics 2022, 10(19), 3487; https://doi.org/10.3390/math10193487 - 24 Sep 2022

Cited by 15 | Viewed by 1842

Abstract

Artificial neural networks (ANNs), one of the most important artificial intelligence techniques, are used extensively in modeling many types of problems. A successful training process is required to create effective models with ANN. An effective training algorithm is essential for a successful training process. In this study, a new neural network training algorithm called the hybrid artificial bee colony algorithm based on effective scout bee stage (HABCES) was proposed. The HABCES algorithm includes four fundamental changes. Arithmetic crossover was used in the solution generation mechanisms of the employed bee and onlooker bee stages. The knowledge of the global best solution was utilized by arithmetic crossover. Again, this solution generation mechanism also has an adaptive step size. Limit is an important control parameter. In the standard ABC algorithm, it is constant throughout the optimization. In the HABCES algorithm, it was determined dynamically depending on the number of generations. Unlike the standard ABC algorithm, the HABCES algorithm used a solution generation mechanism based on the global best solution in the scout bee stage. Through these features, the HABCES algorithm has a strong local and global convergence ability. Firstly, the performance of the HABCES algorithm was analyzed on the solution of global optimization problems. Then, applications on the training of the ANN were carried out. ANN was trained using the HABCES algorithm for the identification of nonlinear static and dynamic systems. The performance of the HABCES algorithm was compared with the standard ABC, aABC and ABCES algorithms. The results showed that the performance of the HABCES algorithm was better in terms of solution quality and convergence speed. A performance increase of up to 69.57% was achieved by using the HABCES algorithm in the identification of static systems. This rate is 46.82% for the identification of dynamic systems. Full article

(This article belongs to the Special Issue Applications of Mathematical Modeling and Neural Networks)

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30 pages, 8021 KiB

Open AccessArticle

Artificial Neuron-Based Model for a Hybrid Real-Time System: Induction Motor Case Study

by Manuel I. Capel

Mathematics 2022, 10(18), 3410; https://doi.org/10.3390/math10183410 - 19 Sep 2022

Cited by 1 | Viewed by 1987

Abstract

Automatic Machine Learning (AML) methods are currently considered of great interest for use in the development of cyber-physical systems. However, in practice, they present serious application problems with respect to fitness computation, overfitting, lack of scalability, and the need for an enormous amount of time for the computation of neural network hyperparameters. In this work, we have experimentally investigated the impact of continuous updating and validation of the hyperparameters, on the performance of a cyber-physical model, with four estimators based on feedforward and narx ANNs, all with the gradient descent-based optimization technique. The main objective is to demonstrate that the optimized values of the hyperparameters can be validated by simulation with MATLAB/Simulink following a mixed approach based on interleaving the updates of their values with a classical training of the ANNs without affecting their efficiency and automaticity of the proposed method. For the two relevant variables of an Induction Motor (IM), two sets of estimators have been trained from the input current and voltage data. In contrast, the training data for the speed and output electromagnetic torque of the IM have been established with the help of a new Simulink model developed entirely. The results have demonstrated the effectiveness of ANN estimators obtained with the Deep Learning Toolbox (DLT) that we used to transform the trained ANNs into blocks that can be directly used in cyber-physical models designed with Simulink. Full article

(This article belongs to the Special Issue Applications of Mathematical Modeling and Neural Networks)

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22 pages, 757 KiB

Open AccessArticle

Sustainable Management in Active Networks

by Gennady Ougolnitsky and Olga Gorbaneva

Mathematics 2022, 10(16), 2848; https://doi.org/10.3390/math10162848 - 10 Aug 2022

Viewed by 1268

Abstract

This paper synthesizes two areas of research: control models in networks and sustainable development of active systems. The concept of an active network (an active system with a network structure) is proposed and formalized. Two common classes, hierarchical and non-hierarchical active networks, are identified and studied; some illustrative examples are given. The sustainable management problem of active networks is stated, and approaches to its solution are described. A model example of opinion control in an active network with sustainable development requirements is numerically simulated. For this purpose, the author’s method of qualitatively representative scenarios is used. Full article

(This article belongs to the Special Issue Applications of Mathematical Modeling and Neural Networks)

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