Solving Partial Differential Equation Based on Bernstein Neural Network and Extreme Learning Machine Algorithm

Sun, Hongli; Hou, Muzhou; Yang, Yunlei; Zhang, Tianle; Weng, Futian; Han, Feng

doi:10.1007/s11063-018-9911-8

Cited by 81 publications

(37 citation statements)

References 40 publications

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“…10 and 11, the sampling parameter is m = 10 in this study, and only with n = 9 neurons. By comparison with the method proposed in [49] and BeNN method in [60], the maximum errors are 2.3e-5 and 6.8e-8, the obtained maximum error of LNN method with IELM algorithm is 4.0e-12, which fully validates the superiority of the new proposed method.…”

Section: Example 14supporting

confidence: 58%

“…9(b). By comparison, the maximum error of BeNN method in [60] is 7.3e-9, and the maximum error of the proposed LNN method is 5.2e-12, so it is easy to seen from Fig. 9(b) that LNN method can obtain higher accuracy solution than BeNN method in [60].…”

Section: Comparative Studymentioning

confidence: 92%

“…9(a). By comparison, the maximum error in [51] is 1.9e-2, the maximum error of BeNN method in [60] is 2.7e-3, and the maximum error of the proposed LNN method is 4.9e-10. It is easy to seen from Fig.…”

Section: Comparative Studymentioning

confidence: 99%

“…By comparison, the maximum error of BeNN method in [60] is 7.3e-9, and the maximum error of the proposed LNN method is 5.2e-12, so it is easy to seen from Fig. 9(b) that LNN method can obtain higher accuracy solution than BeNN method in [60]. Considering the method given by [61], the maximum error is 3.5e-2 with m = 50, while, using LNN method, we are able to obtain a higher accuracy with maximum error 2.4e-13 by n = 10 neurons, which also fully validates the superiority of LNN method with IELM algorithm.…”

Section: Comparative Studymentioning

confidence: 99%

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A novel improved extreme learning machine algorithm in solving ordinary differential equations by Legendre neural network methods

2018

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This paper develops a Legendre neural network method (LNN) for solving linear and nonlinear ordinary differential equations (ODEs), system of ordinary differential equations (SODEs), as well as classic Emden-Fowler equations. The Legendre polynomial is chosen as a basis function of hidden neurons. A single hidden layer Legendre neural network is used to eliminate the hidden layer by expanding the input pattern using Legendre polynomials. The improved extreme learning machine (IELM) algorithm is used for network weights training when solving algebraic equation systems, and several algorithm steps are summed up. Convergence was analyzed theoretically to support the proposed method. In order to demonstrate the performance of the method, various testing problems are solved by the proposed approach. A comparative study with other approaches such as conventional methods and latest research work reported in the literature are described in detail to validate the superiority of the method. Experimental results show that the proposed Legendre network with IELM algorithm requires fewer neurons to outperform the numerical algorithm in the latest literature in terms of accuracy and execution time.

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Section: Example 14supporting

confidence: 58%

Section: Comparative Studymentioning

confidence: 92%

Section: Comparative Studymentioning

confidence: 99%

Section: Comparative Studymentioning

confidence: 99%

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A novel improved extreme learning machine algorithm in solving ordinary differential equations by Legendre neural network methods

2018

Self Cite

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“…Neural network, which is one of machine intelligence techniques, has universal function approximation capabilities [20][21][22], and the solution obtained from the neural network is differentiable and in closed analytic form. Neural network has been widely used for solving ordinary differential equations [23,24], partial differential equations [25][26][27], fractional differential equations [28][29][30], and integro-differential equations [31,32]. Chakraverty and Mall [33] analyzed a regression-based neural network algorithm to solve two-point boundary value problems of fourth-order linear ordinary differential equations.…”

Section: Introductionmentioning

confidence: 99%

The LS-SVM algorithms for boundary value problems of high-order ordinary differential equations

Yin

et al. 2019

Adv Differ Equ

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This paper introduces the improved LS-SVM algorithms for solving two-point and multi-point boundary value problems of high-order linear and nonlinear ordinary differential equations. To demonstrate the reliability and powerfulness of the improved LS-SVM algorithms, some numerical experiments for third-order, fourth-order linear and nonlinear ordinary differential equations with two-point and multi-point boundary conditions are performed. The idea can be extended to other complicated ordinary differential equations.

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An extensive critique of sliding mode control and adaptive neuro‐fuzzy inference system for nonlinear system

George

Mani

Stonier³

2021

Asian Journal of Control

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Nonlinear control hypothesis is the region of control hypothesis which manages frameworks that are nonlinear, time‐variant, or both. Nonlinear control structures have extended high prevalence fundamentally because of the wide use of hypothetical intends to deal with credible issues in different zones of engineering and medical aspects. Sliding mode control (SMC) is a nonlinear control strategy applied when the deterministic model of the nonlinear system is discrepant with actual plant in terms of plant parameters and undetermined external disturbances. SMC is a sort of nonlinear control that accentuates zero error convergence modeling error, demonstrating nonlinearities and outside aggravations. Adaptive neuro‐fuzzy inference system (ANFIS) captures neural associations and soft reason guidelines in a single framework. ANFIS causes fewer errors and is more obvious to the customer isolated from the ANN. The paper centers that neural organizations accept a critical capacity in controlling nonlinear frameworks. This review aims an overview of existing methods for nonlinear control and application of SMC for nonlinear system based on ANFIS.

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Solving Partial Differential Equation Based on Bernstein Neural Network and Extreme Learning Machine Algorithm

Cited by 81 publications

References 40 publications

A novel improved extreme learning machine algorithm in solving ordinary differential equations by Legendre neural network methods

A novel improved extreme learning machine algorithm in solving ordinary differential equations by Legendre neural network methods

The LS-SVM algorithms for boundary value problems of high-order ordinary differential equations

An extensive critique of sliding mode control and adaptive neuro‐fuzzy inference system for nonlinear system

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