Single Layer Chebyshev Neural Network Model for Solving Elliptic Partial Differential Equations

Mall, Susmita; Chakraverty, Snehashish

doi:10.1007/s11063-016-9551-9

Cited by 74 publications

(33 citation statements)

References 30 publications

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“…In [76], samples of three-phase currents for the post- [73]. In ChNN polynomials, functional expansion is used to map original input into higher-dimensional space; the hidden layer is interchanged, leaving only one layer in the network as shown in Figure 5 [74]. Thus only one parameter needs to be tuned in ChNN because of its single-layer structure, making it easy to implement than other ANN models with efficient fault classification results.…”

Section: Fc Based On Fuzzy Interface Systems (Fis)mentioning

confidence: 99%

A Review of Fault Diagnosing Methods in Power Transmission Systems

et al. 2020

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Transient stability is important in power systems. Disturbances like faults need to be segregated to restore transient stability. A comprehensive review of fault diagnosing methods in the power transmission system is presented in this paper. Typically, voltage and current samples are deployed for analysis. Three tasks/topics; fault detection, classification, and location are presented separately to convey a more logical and comprehensive understanding of the concepts. Feature extractions, transformations with dimensionality reduction methods are discussed. Fault classification and location techniques largely use artificial intelligence (AI) and signal processing methods. After the discussion of overall methods and concepts, advancements and future aspects are discussed. Generalized strengths and weaknesses of different AI and machine learning-based algorithms are assessed. A comparison of different fault detection, classification, and location methods is also presented considering features, inputs, complexity, system used and results. This paper may serve as a guideline for the researchers to understand different methods and techniques in this field. transmission system topologies can be minimized by using the interspersed sensors for the collection of voltage and current signals. The second limitation is the lack of computational capability and communication. Synchronized global positioning system (GPS) sampling and high-speed broadband communications for IEDs in power grids are proposed in [8]. These technical advancements assure the quick response to faulty scenarios and the effective functioning of online monitoring mechanisms based on sensor networks. The availability of high-performance computing solutions gives provision to the implementation of higher computation complexity methods [7].Short circuit faults are more likely to appear in power systems (PS) than the series faults, break in the path of current. Shunt faults result in catastrophes and leave hazardous effects on PS. Short circuit faults can be divided into symmetrical and asymmetrical faults and further classification is presented in Figure 1 for the three-phase system [11].

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Section: Fc Based On Fuzzy Interface Systems (Fis)mentioning

confidence: 99%

A Review of Fault Diagnosing Methods in Power Transmission Systems

et al. 2020

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“…Among them, Patra and Pal used the trigonometric function as the tool of the functional expansion. Furthermore, many researchers have studied on how to use the orthogonal basis functions as the function expansion tools . It is well known that Chebyshev orthogonal polynomial, due to nonlinear approximation capacity, is very powerful by the best approximation theory .…”

Section: Fchflnn Modelmentioning

confidence: 99%

“…As another popular expansion, the trigonometric polynomial is an effective means for nonlinear active noise control . Moreover, many orthogonal basis functions, such as Legendre, Laugger, Hermite, and Chebyshev polynomials were also extensively investigated under the FLNN framework. In FLNN, number of network parameters are less than that of MLP structure.…”

Section: Introductionmentioning

confidence: 99%

“…Ma et al proposed correntropy‐induced metric as a new cost function in ChFLNN. In particular, ChFLNN yields good numerical results for ordinary and partial differential equations with integer order . However, when we use Chebyshev polynomial for the fractional differential equations (FDEs), the rate of convergence of the numerical approximations does not become reasonable.…”

Section: Introductionmentioning

confidence: 99%

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Fractional Chebyshev functional link neural network‐optimization method for solving delay fractional optimal control problems with Atangana‐Baleanu derivative

Kheyrinataj

Nazemi

2020

Optim Control Appl Methods

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In this article, we propose a higher order neural network, namely the functional link neural network (FLNN), for the model of linear and nonlinear delay fractional optimal control problems (DFOCPs) with mixed control-state constraints. We consider DFOCPs using a new fractional derivative with nonlocal and nonsingular kernel that was recently proposed by Atangana and Baleanu.The derivative possesses more important characteristics that are very useful in modelling. In the proposed method, a fractional Chebyshev FLNN is developed. At the first step, the delay problem is transformed to a nondelay problem, using a Padé approximation. The necessary optimality condition is stated in a form of fractional two-point boundary value problem. By applying the fractional integration by parts and by constructing an error function, we then define an unconstrained minimization problem. In the optimization problem, trial solutions for state, co-state and control functions are utilized where these trial solutions are constructed by using single-layer fractional Chebyshev neural network model. We then minimize the error function using an unconstrained optimization scheme based on the gradient descent algorithm for updating the network parameters (weights and bias) associated with all neurons. To show the effectiveness of the proposed neural network, some numerical results are provided. K E Y W O R D SAtangana-Baleanu derivative, delay fractional optimal control problems, fractional Chebyshev polynomials, functional link neural network, optimization, Padé approximation 1 808 /journal/oca Optim Control Appl Meth. 2020;41:808-832.KHEYRINATAJ and NAZEMI 809 cannot be used in modeling of the complex control systems. To bypass these difficulties, Caputo and Fabrizio proposed a so-called Caputo-Fabrizio derivative based on the exponential kernel 10 instead of the power principle. After that, Atangana and Baleanu (AB) 2 generalized the Caputo-Fabrizio derivative using the Mittag-Leffler function as one of the best candidates among existing kernels which is both nonsingular and nonlocal. The control of systems with time delay and obtaining their approximate solutions are very important issues in control theory. Optimal control problems with delay arise in many important applications in science and engineering. [11][12][13] Some researchers have extended this model to the scope of fractional calculus. From these works, we can mention, Chelyshkov wavelets, 14 Legendre operational method, 15 Müntz-Legendre spectral method 16 , block-pulse function, 17 Bernoulli wavelet 18 , Bernstein polynomials 19 , discretization-based methods, 20,21 and embedding approach. 22 In the above-mentioned works, the functions are approximated locally, using the discretization of domain into the number of finite elements. Despite providing good approximations to the solution of problem, these methods have difficulties. Due to discretization of domain via meshing, these methods need a high computational time when the control problems have high-order dimension. Also, d...

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“…They used functional link artificial neural networks to handle ordinary differential equations with Chebyshev basis polynomials. They solved famous Lane‐Emden, Emden Fowler, and elliptic differential equations using this Chebyshev‐based artificial neural network . Recently, some deep learning paradigms are developed to solve partial differential equations.…”

Section: Introductionmentioning

confidence: 99%

Simulation of nonlinear fractional dynamics arising in the modeling of cognitive decision making using a new fractional neural network

Rasanan

Bajalan

Parand

et al. 2019

Math Methods in App Sciences

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By the rapid growth of available data, providing data-driven solutions for nonlinear (fractional) dynamical systems becomes more important than before. In this paper, a new fractional neural network model that uses fractional order of Jacobi functions as its activation functions for one of the hidden layers is proposed to approximate the solution of fractional differential equations and fractional partial differential equations arising from mathematical modeling of cognitive-decision-making processes and several other scientific subjects. This neural network uses roots of Jacobi polynomials as the training dataset, and the Levenberg-Marquardt algorithm is chosen as the optimizer. The linear and nonlinear fractional dynamics are considered as test examples showing the effectiveness and applicability of the proposed neural network. The numerical results are compared with the obtained results of some other networks and numerical approaches such as meshless methods. Numerical experiments are presented confirming that the proposed model is accurate, fast, and feasible.

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Single Layer Chebyshev Neural Network Model for Solving Elliptic Partial Differential Equations

Cited by 74 publications

References 30 publications

A Review of Fault Diagnosing Methods in Power Transmission Systems

A Review of Fault Diagnosing Methods in Power Transmission Systems

Fractional Chebyshev functional link neural network‐optimization method for solving delay fractional optimal control problems with Atangana‐Baleanu derivative

Simulation of nonlinear fractional dynamics arising in the modeling of cognitive decision making using a new fractional neural network

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