Abstract
In this paper, a three-nodes integrated radial basis function networks (RBFNs) method with extended precision is reported for numerical solutions of partial differential problems. RBFNs can be considered as a universal approximation scheme, and have emerged as a powerful approximation tool and become one of the main fields of research in the numerical analysis [1]. Derivative approximations of variable fields are computed through the radial basis functions. They have the properties of universal approximation and mesh-free discretisation. Substantial enhancements in the solution accuracy, matrix condition number, and high convergence rate are achieved.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Haykin, S.: Multilayer perceptrons. In: Neural Networks: A Comprehensive Foundation, pp. 156-255 (1999)
Mai-Duy, N., Tran-Cong, T.: Numerical solution of differential equations using multiquadric radial basis function networks. Neural Netw. 14(2), 185–199 (2001)
Mai-Duy, N., Le-Cao, K., Tran-Cong, T.: A Cartesian grid technique based on one-dimensional integrated radial basis function networks for natural convection in concentric annuli. Int. J. Numer. Meth. Fluids 57, 1709–1730 (2008)
Fornberg, B., Wright, G.: Stable computation of multiquadric interpolants for all values of the shape parameter. Comput. Math. Appl. 48(5), 853–867 (2004)
Fornberg, B., Piret, C.: A stable algorithm for flat radial basis functions on a sphere. SIAM J. Sci. Comput. 30(1), 60–80 (2007)
Le-Cao, K., Mai-Duy, N., Tran, C.D., Tran-Cong, T.: Numerical study of stream-function formulation governing flows in multiply-connected domains by integrated RBFs and Cartesian grids. Comput. Fluids 44(1), 32–42 (2011)
Le-Cao, K., Mai-Duy, N., Tran-Cong, T.: An effective integrated-RBFN Cartesian-grid discretisation to the stream function-vorticity-temperature formulation in non-rectangular domains. Numer. Heat Transf. Part B 55, 480–502 (2009)
Le, T.T.V., Mai-Duy, N., Le-Cao, K., Tran-Cong, T.: A time discretisation scheme based on integrated radial basis functions (IRBFs) for heat transfer and fluid flow problems. Numer. Heat Transf. Part B (2018)
Tien, C.M.T., Thai-Quang, N., Mai-Duy, N., Tran, C.D., Tran-Cong, T.: A three-point coupled compact integrated RBF scheme for second-order differential problems. CMES Compu. Model. Eng. Sci. 104(6), 425–469 (2015)
Tian, Z., Liang, X., Yu, P.: A higher order compact finite difference algorithm for solving the incompressible Navier Stokes equations. Int. J. Numer. Meth. Eng. 88(6), 511–532 (2011)
Acknowledgement
This research is supported by Computational Engineering and Science Research Centre (CESRC), University of Southern Queensland, and Institute of Applied Mechanics and Informatics (IAMI), HCMC Vietnam Academy of Science and Technology (VAST).
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2021 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this chapter
Cite this chapter
Le, T.T.V., Le-Cao, K., Duc-Tran, H. (2021). A Radial Basis Neural Network Approximation with Extended Precision for Solving Partial Differential Equations. In: Phuong, N.H., Kreinovich, V. (eds) Soft Computing: Biomedical and Related Applications. Studies in Computational Intelligence, vol 981. Springer, Cham. https://doi.org/10.1007/978-3-030-76620-7_17
Download citation
DOI: https://doi.org/10.1007/978-3-030-76620-7_17
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-76619-1
Online ISBN: 978-3-030-76620-7
eBook Packages: Intelligent Technologies and RoboticsIntelligent Technologies and Robotics (R0)