Article
Computer Science, Artificial Intelligence
Masaki Kobayashi
Summary: This study addresses the issue of low noise tolerance in complex-valued Hopfield neural networks by reducing self-feedback and extending stability conditions through the introduction of noise-robust projection rules.
IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS
(2022)
Article
Mathematics, Interdisciplinary Applications
Yao-Qun Xu, Xin-Xin Zhen, Meng Tang
Summary: This paper investigates the problem of time delay in neural transmission and proposes a new chaotic neuron model with time delay self-feedback. The chaotic characteristics of neurons in the model are analyzed using bifurcation diagrams and Lyapunov exponential diagrams. Experimental results show that the model exhibits rich dynamic behavior and the randomness of chaotic sequences generated by chaotic neurons under different conditions is verified. Additionally, a proposed image encryption scheme is analyzed for its security, and the results demonstrate excellent anti-attack ability.
Article
Computer Science, Interdisciplinary Applications
Victor Churchill, Steve Manns, Zhen Chen, Dongbin Xiu
Summary: Recent research has focused on using deep neural networks (DNNs) for data-driven learning of the evolution of unknown systems, aiming to predict the long-term evolution of these systems. Training a DNN with low generalization error is crucial in this case to minimize the accumulation of error over time. However, due to the randomness in DNN training, particularly in stochastic optimization, there is uncertainty in the resulting prediction, leading to uncertainty in the generalization error. This paper introduces a computational technique that reduces the variance of the generalization error, improving the reliability and consistency of the DNN model's predictions.
JOURNAL OF COMPUTATIONAL PHYSICS
(2023)
Article
Automation & Control Systems
Md Arzoo Jamal, Rakesh Kumar, Santwana Mukhopadhyay, Subir Das
Summary: The present article investigates the fixed-time stability analysis of nonlinear dynamical systems with impulsive effects. Novel criteria are derived to achieve stability in fixed-time under stabilizing and destabilizing impulses. Theoretical results show that the estimated fixed-time in this study is less conservative and more accurate compared to existing theorems. The theoretical findings are also applied to impulsive control of general neural network systems.
JOURNAL OF THE FRANKLIN INSTITUTE-ENGINEERING AND APPLIED MATHEMATICS
(2022)
Article
Automation & Control Systems
Nallappan Gunasekaran, N. Mohamed Thoiyab, Quanxin Zhu, Jinde Cao, P. Muruganantham
Summary: This article presents a new upper bound norm method for delayed dynamical neural networks, which is shown to be effective through numerical examples.
IEEE TRANSACTIONS ON CYBERNETICS
(2022)
Article
Mathematics, Interdisciplinary Applications
Wenzhao Qin, Yukai Shen, Lifang Wang
Summary: This paper analyzes the H-infinity performance of a class of T-S fuzzy systems with Markovian-jump parameters. A state feedback controller is designed for T-S fuzzy systems with Markovian-jump parameters. First, a new modal-dependent Lyapunov function composed of closed-loop functions is constructed, which fully utilizes the system status information. Based on this function, the stability conditions with less conservatism are given by linear matrix inequalities (LMIS). At the same time, a design algorithm for a state feedback controller is proposed for the Markovian-jump-parameters T-S fuzzy systems, which ensures the system's stability under the condition of H-infinity-gamma performance. Simulation results demonstrate the accuracy and practicality of the proposed method.
Article
Automation & Control Systems
Weiming Xiang
Summary: This article discusses a method for runtime safety monitoring of dynamical systems embedded with neural-network components, using a developed runtime safety state estimator to construct lower and upper bounds of system state trajectories.
IEEE TRANSACTIONS ON CYBERNETICS
(2022)
Article
Mathematics, Applied
Jenjira Thipcha, Presarin Tangsiridamrong, Thongchai Botmart, Boonyachat Meesuptong, M. Syed Ali, Pantiwa Srisilp, Kanit Mukdasai
Summary: This paper investigates the robust stability analysis issue for discrete-time neural networks with interval time-varying leakage and discrete and distributed delays using a rebuilt summation inequality. A novel inequality, less conservative than the well-known Jensen inequality, is considered and applied in the context of discrete-time delay systems. Stability and passivity criteria are obtained in terms of linear matrix inequalities (LMIs) using various techniques. Numerical examples are provided to demonstrate the validity and efficiency of the theoretical findings of this research with the assistance of the LMI Control toolbox in Matlab.
Article
Computer Science, Artificial Intelligence
Emel Arslan
Summary: This research article introduces new criteria to address the global robust asymptotic stability problem of dynamical neural networks with multiple constant time delays. By utilizing the Homomorphic transformation theory and a novel Lyapunov functional candidate, the paper establishes sufficient conditions for the existence, uniqueness, and stability of equilibrium points in this type of neural network. The derived robust stability conditions provide novel relationships among network parameters and can be easily applied and tested for delayed-type neural networks.
Review
Computer Science, Artificial Intelligence
Ezgi Aktas, Ozlem Faydasicok, Sabri Arik
Summary: This article focuses on the robust stability properties of continuous-time dynamical neural networks with time delay parameters. It presents new results for the global robust stability of dynamical Hopfield neural networks with multiple time delay terms and Lipschitz activation functions. The article also demonstrates that simple modifications in the robust stability conditions for the Hopfield model can directly yield robust stability conditions for the Cohen-Grossberg model with multiple delays. Additionally, a comprehensive review of previously published robust stability research results is provided. The proposed stability results in this paper are shown to generalize almost all previously reported robust stability conditions for multiple delayed neural network models.
ARTIFICIAL INTELLIGENCE REVIEW
(2023)
Article
Engineering, Multidisciplinary
Kevin Linka, Amelie Schafer, Xuhui Meng, Zongren Zou, George Em Karniadakis, Ellen Kuhl
Summary: Understanding real-world dynamical phenomena is challenging, and machine learning has become the go-to technology for analyzing and making decisions based on these phenomena. However, traditional neural networks often ignore the fundamental laws of physics and fail to make accurate predictions. In this study, the combination of neural networks, physics informed modeling, and Bayesian inference is used to integrate data, physics, and uncertainties, improving the predictive potential of neural network models.
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2022)
Article
Computer Science, Artificial Intelligence
R. Shobana, Bhavnesh Jaint, Rajesh Kumar
Summary: A novel fully connected recurrent neural network (FCRNN) structure is proposed for the identification of unknown dynamics of nonlinear systems. The proposed structure, with adjustable weights in internal feedback layers, imparts necessary memory property to handle the dynamical systems. Experimental results show that the FCRNN model outperforms other neural models in terms of identification accuracy and robustness.
Article
Mathematics, Interdisciplinary Applications
Ankit Mandal, Yash Tiwari, Prasanta K. Panigrahi, Mayukha Pal
Summary: It has been demonstrated that synchronizing physical prior with a neural network reduces training requirements for learning non-linear physical systems. Recent research shows that parameterizing Lagrangian and Hamiltonian using neural network weights and biases, and then executing the equations of motion, leads to more efficient prediction of non-linear dynamical systems compared to conventional neural networks.
CHAOS SOLITONS & FRACTALS
(2022)
Article
Computer Science, Artificial Intelligence
Melike Solak, Ozlem Faydasicok, Sabri Arik
Summary: This research article focuses on dynamical neural network models with discrete time delay terms, Lipschitz activation functions, and parameter uncertainties. A new upper bound value and stability conditions are proposed to ensure the robust stability of these models. A new general framework is established for determining the robust stability conditions using the Homeomorphism mapping theory and basic Lyapunov stability theory.
Article
Automation & Control Systems
Hanyong Shao, Guangxia Yuan, Qing-Long Han
Summary: This article discusses the stability of linear impulsive delay systems, addressing uncertainties with a Lyapunov-Krasovskii-like functional and deriving a robust stability criterion. Examples are provided to show the validity of the stability results and compare them to existing ones.
INTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROL
(2021)
Article
Mathematics, Applied
M. S. Bruzon, T. M. Garrido, R. de la Rosa
Summary: We study a family of generalized Zakharov-Kuznetsov modified equal width equations in (2+1)-dimensions involving an arbitrary function and three parameters. By using the Lie group theory, we classify the Lie point symmetries of these equations and obtain exact solutions. We also show that this family of equations admits local low-order multipliers and derive all local low-order conservation laws through the multiplier approach.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Dohee Jung, Changbum Chun
Summary: The paper presents a general approach to enhance the Pade iterations for computing the matrix sign function by selecting an arbitrary three-point family of methods based on weight functions. The approach leads to a multi-parameter family of iterations and allows for the discovery of new methods. Convergence and stability analysis as well as numerical experiments confirm the improved performance of the new methods.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Abhishek Yadav, Amit Setia, M. Thamban Nair
Summary: In this paper, we propose a Galerkin's residual-based numerical scheme for solving a system of Cauchy-type singular integral equations using Chebyshev polynomials. We prove the well-posedness of the system and derive a theoretical error bound and convergence order. The numerical examples validate the theoretical results.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Fernando Chacon-Gomez, M. Eugenia Cornejo, Jesus Medina, Eloisa Ramirez-Poussa
Summary: The use of decision rules allows for reliable extraction of information and inference of conclusions from relational databases, but the concepts of decision algorithms need to be extended in fuzzy environments.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Ilhame Amirali, Gabil M. Amiraliyev
Summary: This paper considers the one-dimensional initial-boundary problem for a pseudoparabolic equation with a time delay. To solve this problem numerically, a higher-order difference method is constructed and the error estimate for its solution is obtained. Based on the method of energy estimates, the fully discrete scheme is shown to be convergent of order four in space and of order two in time. The given numerical results illustrate the convergence and effectiveness of the numerical method.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Tong-tong Shang, Guo-ji Tang, Wen-sheng Jia
Summary: The goal of this paper is to investigate a class of linear complementarity problems over tensor-spaces, denoted by TLCP, which is an extension of the classical linear complementarity problem. First, two classes of structured tensors over tensor-spaces (i.e., T-R tensor and T-RO tensor) are introduced and some equivalent characterizations are discussed. Then, the lower bound and upper bound of the solutions in the sense of the infinity norm of the TLCP are obtained when the problem has a solution.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Fabio Difonzo, Pawel Przybylowicz, Yue Wu
Summary: This paper focuses on the existence, uniqueness, and approximation of solutions of delay differential equations (DDEs) with Caratheodory type right-hand side functions. It presents the construction of the randomized Euler scheme for DDEs and investigates its error. Furthermore, the paper reports the results of numerical experiments.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Priyanka Roy, Geetanjali Panda, Dong Qiu
Summary: In this article, a gradient based descent line search scheme is proposed for solving interval optimization problems under generalized Hukuhara differentiability. The innovation and importance of these concepts are presented from practical and computational perspectives. The necessary condition for existence of critical point is presented in inclusion form of interval-valued gradient. Suitable efficient descent direction is chosen based on the monotonic property of the interval-valued function and specific interval ordering. Mathematical convergence of the scheme is proved under the assumption of Inexact line search. The theoretical developments are implemented with a set of interval test problems in different dimensions. A possible application in finance is provided and solved by the proposed scheme.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Zhongqian Wang, Changqing Ye, Eric T. Chung
Summary: In this paper, the constrained energy minimizing generalized multiscale finite element method (CEM-GMsFEM) with mixed boundary conditions for elasticity equations in high contrast media is developed. The method offers advantages such as independence of target region's contrast from precision and significant impact of oversampling domain sizes on numerical accuracy. Furthermore, this is the first proof of convergence of CEM-GMsFEM with mixed boundary conditions for elasticity equations. Numerical experiments demonstrate the method's performance.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Samaneh Soradi-Zeid, Maryam Alipour
Summary: The Laguerre polynomials are a new set of basic functions used to solve a specific class of optimal control problems specified by integro-differential equations, namely IOCP. The corresponding operational matrices of derivatives are calculated to extend the solution of the problem in terms of Laguerre polynomials. By considering the basis functions and using the collocation method, the IOCP is simplified into solving a system of nonlinear algebraic equations. The proposed method has been proven to have an error bound and convergence analysis for the approximate optimal value of the performance index. Finally, examples are provided to demonstrate the validity and applicability of this technique.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Almudena P. Marquez, Maria L. Gandarias, Stephen C. Anco
Summary: A generalization of the KP equation involving higher-order dispersion is studied. The Lie point symmetries and conservation laws of the equation are obtained using Noether's theorem and the introduction of a potential. Sech-type line wave solutions are found and their features, including dark solitary waves on varying backgrounds, are discussed.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Susanne Saminger-Platz, Anna Kolesarova, Adam Seliga, Radko Mesiar, Erich Peter Klement
Summary: In this article, we study real functions defined on the unit square satisfying basic properties and explore the conditions for generating bivariate copulas using parameterized transformations and other constructions.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Lulu Tian, Nattaporn Chuenjarern, Hui Guo, Yang Yang
Summary: In this paper, a new local discontinuous Galerkin (LDG) algorithm is proposed to solve the incompressible Euler equation in two dimensions on overlapping meshes. The algorithm solves the vorticity, velocity field, and potential function on different meshes. The method employs overlapping meshes to ensure continuity of velocity along the interfaces of the primitive meshes, allowing for the application of upwind fluxes. The article introduces two sufficient conditions to maintain the maximum principle of vorticity.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Cheng Wang, Jilu Wang, Steven M. Wise, Zeyu Xia, Liwei Xu
Summary: In this paper, a temporally second-order accurate numerical scheme for the Cahn-Hilliard-Magnetohydrodynamics system of equations is proposed and analyzed. The scheme utilizes a modified Crank-Nicolson-type approximation for time discretization and a mixed finite element method for spatial discretization. The modified Crank-Nicolson approximation allows for mass conservation and energy stability analysis. Error estimates are derived for the phase field, velocity, and magnetic fields, and numerical examples are presented to validate the proposed scheme's theoretical results.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Mingyu He, Wenyuan Liao
Summary: This paper presents a numerical method for solving reaction-diffusion equations in spatially heterogeneous domains, which are commonly used to model biological applications. The method utilizes a fourth-order compact alternative directional implicit scheme based on Pade approximation-based operator splitting techniques. Stability analysis shows that the method is unconditionally stable, and numerical examples demonstrate its high efficiency and high order accuracy in both space and time.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)