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Neural network algorithm based on Legendre improved extreme learning machine for solving elliptic partial differential equations

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Abstract

To provide a new numerical algorithm for solving elliptic partial differential equations (PDEs), the Legendre neural network (LNN) and improved extreme learning machine (IELM) algorithm are introduced to propose a Legendre improved extreme learning machine (L-IELM) method, which is applied to solving elliptic PDEs in this paper. The product of two Legendre polynomials is chosen as basis functions of hidden neurons. Single hidden layer LNN is used to construct approximate solutions and its derivatives of differential equations. IELM algorithm is used for network weights training, and the algorithm steps of the proposed L-IELM method are summarized. Finally, in order to evaluate the present algorithm, various test examples are selected and solved by the proposed approach to validate the calculation accuracy. Comparative study with the earlier methods in literature is described to verify the superiority of the presented L-IELM method. Experiment results show that the proposed L-IELM algorithm can perform well in terms of accuracy and execution time, which in addition provides a new algorithm for solving elliptic PDEs.

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References

  • Abdulla MB, Costa AL, Sousa RL (2018) Probabilistic identification of subsurface gypsum geohazards using artificial neural networks. Neural Comput Appl 29(12):1377–1391

    Google Scholar 

  • Arndt O, Barth T, Freisleben B, Grauer M (2005) Approximating a finite element model by neural network prediction for facility optimization in groundwater engineering. Eur J Oper Res 166(3):769–781

    MATH  Google Scholar 

  • Beltzer AI, Sato T (2003) Neural classification of finite elements. Comput Struct 81(24–25):2331–2335

    Google Scholar 

  • Chedhou JC, Kyamakya K et al (2009) Solving stiff ordinary and partial differential equations using analog computing based on cellular neural networks. In: 2nd International workshop on nonlinear dynamics and synchronization, Klagenfurt, vol 4, p 213

  • Deng J, Yue ZQ, Tham LG et al (2003) Pillar design by combining finite element methods, neural networks and reliability: a case study of the Feng Huangshan copper mine, China. Int J Rock Mech Min Sci 40(4):585–599

    Google Scholar 

  • Deng W, Zhao H, Yang X et al (2017a) Study on an improved adaptive PSO algorithm for solving multi-objective gate assignment. Appl Soft Comput 59:288–302

    Google Scholar 

  • Deng W, Yao R, Zhao H et al (2017b) A novel intelligent diagnosis method using optimal LS-SVM with improved PSO algorithm. Soft Comput. https://doi.org/10.1007/s00500-017-2940-9

    Article  Google Scholar 

  • Deng W, Zhao H, Zou L et al (2017c) A novel collaborative optimization algorithm in solving complex optimization problems. Soft Comput 21:4387–4398

    Google Scholar 

  • Deng W, Zhang S, Zhao H et al (2018) A novel Fault diagnosis method based on integrating empirical wavelet transform and fuzzy entropy for motor bearing. IEEE Access 6:35042–35056

    Google Scholar 

  • Deng W, Junjie X, Zhao H (2019) An improved ant colony optimization algorithm based on hybrid strategies for scheduling problem. IEEE Access 7:20281–20292

    Google Scholar 

  • Dwivedi AK (2018) Artificial neural network model for effective cancer classification using microarray gene expression data. Neural Comput Appl 29(12):1545–1554

    Google Scholar 

  • Esposito A, Marinaro M, Oricchio D, Scarpetta S (2000) Approximation of continuous and discontinuous mappings by a growing neural RBF-based algorithm. Neural Netw 13(6):651–665

    Google Scholar 

  • Haykin S (2002) Neural networks: a comprehensive foundation. Pearson Education, Singapore

    MATH  Google Scholar 

  • He S, Reif K, Unbehauen R (2000) Multilayer neural networks for solving a class of partial differential equations. Neural Netw 13(3):385–396

    Google Scholar 

  • Huang G-B, Chen L (2007) Convex incremental extreme learning machine. Neurocomputing 70(16):3056–3062

    Google Scholar 

  • Huang G-B, Chen L (2008) Enhanced random search based incremental extreme learning machine. Neurocomputing 71(16):3460–3468

    Google Scholar 

  • Huang G-B, Zhu Q-Y, Siew C-K (2006a) Extreme learning machine: theory and applications. Neurocomputing 70(1):489–501

    Google Scholar 

  • Huang G-B, Chen L, Siew C-K (2006b) Universal approximation using incremental constructive feedforward networks with random hidden nodes. IEEE Trans Neural Netw 17(4):879–892

    Google Scholar 

  • Huang G-B, Zhou H, Ding X et al (2012) Extreme learning machine for regression and multiclass classification. IEEE Trans Syst Man Cybernet Part B Cybernet 42(2):513–529

    Google Scholar 

  • Jianyu L, Siwei L, Yingjian Q, Yaping H (2002) Numerical Solution of differential equations by radial basis function neural networks. In: Proceedings of the 2002 international joint conference on neural networks, vol 1, pp 773–777

  • Jianyu L, Siwei L et al (2003) Numerical solution of elliptic partial differential equation using radial basis function neural networks. Neural Netw 16(5–6):729–734

    Google Scholar 

  • Jilani H, Bahreininejad A, Ahmadi MT (2009) Adaptive finite element mesh triangulation using self-organizing neural networks. Adv Eng Softw 40(11):1097–1103

    MATH  Google Scholar 

  • Jinfu L, Zhi G (2003) Numerical solution of partial differential equation, 2nd edn. Tsinghua University Press, Beijing

    Google Scholar 

  • Junfei Q, Wei Z (2018) Dynamic multi-objective optimization control for wastewater treatment process. Neural Comput Appl 29(11):1261–1271

    Google Scholar 

  • Koroglu S, Sergeant P, Umurkan N (2010) Comparison of analytical, finite element and neural network methods to study magnetic shielding. Simul Model Pract Theory 18(2):206–216

    Google Scholar 

  • Lagaris IE, Likas A, Fotiadis DI (1998) Artificial neural networks for solving ordinary and partial differential equations. IEEE Trans Neural Netw 9(5):987–1000

    Google Scholar 

  • Leshno M, Lin Vladimir Ya et al (1993) Multilayer feedforward networks with a nonpolynomial activation function can approximate any function. Neural Netw 6(6):861–867

    Google Scholar 

  • Li X, Ouyang J, Li Q, Ren J (2010) Integration wavelet neural network for steady convection dominated diffusion problem. In: 3rd international conference on information and computing vol 2, no 2, pp 109–112

  • Mai-Duy N, Tran-Cong T (2001) Numerical solution of differential equations using multiquadric radial basis function networks. Neural Netw 14:185–199

    MATH  Google Scholar 

  • Mall S, Chakraverty S (2017) Single layer Chebyshev neural network model for solving elliptic partial differential equations. Neural Process Lett 45(3):825–840

    Google Scholar 

  • Manevitz L, Bitar A, Givoli D (2005) Neural network time series forecasting of finite-element mesh adaptation. Neurocomputing 63:447–463

    Google Scholar 

  • Manganaro G, Arena P, Fortuna L (1999) Cellular neural networks: chaos. complexity and VLSI processing. Springer, Berlin, pp 44–45

    MATH  Google Scholar 

  • Muzhou H, Xuli H (2010) Constructive approximation to multivariate function by decay RBF neural network. IEEE Trans Neural Netw 21(9):1517–1523

    Google Scholar 

  • Muzhou H, Xuli H (2011) The multidimensional function approximation based on constructive wavelet RBF neural network. Appl Soft Comput 11(2):2173–2177

    Google Scholar 

  • Muzhou H, Xuli H (2012) Multivariate numerical approximation using constructive L-2(R) RBF neural network. Neural Comput Appl 21(1):25–34

    Google Scholar 

  • Muzhou H, Xuli H, Yixuan G (2009) Constructive approximation to real function by wavelet neural networks. Neural Comput Appl 18(8):883–889

    Google Scholar 

  • Muzhou H, Taohua L, Yunlei Y et al (2017) A new hybrid constructive neural network method for impacting and its application on tungsten price prediction. Appl Intell 47(1):28–43

    Google Scholar 

  • Muzhou H, Yunlei Y, Taohua L et al (2018) Forecasting time series with optimal neural networks using multi-objective optimization algorithm based on AICc. Front Comput Sci 12(6):1261–1263

    MATH  Google Scholar 

  • Ramuhalli P, Udpa L, Udpa SS (2005) Finite element neural networks for solving differential equations. IEEE Trans Neural Netw 16(6):1381–1392

    Google Scholar 

  • Ronghua L (2010) Numerical solution of partial differential equation, 2nd edn. Higher Education Press, Beijing

    Google Scholar 

  • Sun H, Hou M, Yang Y et al (2018) Solving partial differential equation based on Bernstein neural network and extreme learning machine algorithm. Neural Process Lett. https://doi.org/10.1007/s11063-018-9911-8

    Article  Google Scholar 

  • Tohidi E (2015) Application of Chebyshev collocation method for solving two classes of non-classical parabolic PDEs. Ain Shams Eng J 6(1):373–379

    MathSciNet  Google Scholar 

  • Tsoulos IG, Gavrilis D, Glavas E (2009) Solving differential equations with constructed neural networks. Neurocomputing 72(10–12):2385–2391

    Google Scholar 

  • Vigo-Aguiar J, Higinio R, Clavero C (2017) A first approach in solving initial-value problems in ODEs by elliptic fitting methods. J Comput Appl Math 318:599–603

    MathSciNet  MATH  Google Scholar 

  • Wang Y, Liu M, Bao Z et al (2018) Stacked sparse autoencoder with PCA and SVM for data-based line trip fault diagnosis in power systems. Neural Comput Appl. https://doi.org/10.1007/s00521-018-3490-5

    Article  Google Scholar 

  • Yuanyuan J, Xiaogang P, Zhenyu Z et al (2018) Learning sparse partial differential equations for vector-valued images. Neural Comput Appl 29(11):1205–1216

    Google Scholar 

  • Zhao H, Sun M, Deng W et al (2017) A new feature extraction method based on EEMD and multi-scale fuzzy entropy for motor bearing. Entropy 19(1):14

    Google Scholar 

  • Zhao H, Yao R, Ling X et al (2018) Study on a novel fault damage degree identification method using high-order differential mathematical morphology gradient spectrum entropy. Entropy 20(9):682

    Google Scholar 

  • Ziemianski L (2003) Hybrid neural network finite element modeling of wave propagation in infinite domains. Comput Struct 81(8–11):1099–1109

    Google Scholar 

  • Zolfaghari R, Shidfar A (2013) Solving a parabolic PDE with nonlocal boundary conditions using the Sinc method. Numer Algorithms 62(3):411–427

    MathSciNet  MATH  Google Scholar 

Download references

Funding

This study was funded by the National Natural Science Foundation of China (Grant numbers 61375063, 61271355, 11301549 and 11271378).

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Correspondence to Muzhou Hou.

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Yang, Y., Hou, M., Sun, H. et al. Neural network algorithm based on Legendre improved extreme learning machine for solving elliptic partial differential equations. Soft Comput 24, 1083–1096 (2020). https://doi.org/10.1007/s00500-019-03944-1

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