Numerical solution of elliptic partial differential equation using radial basis function neural networks

Neural Networks

Volumn 16, Issue 5-6, 2003, Pages 729-734

  Jianyu, Li ^a,b   Siwei, Luo ^a   Yingjian, Qi ^a   Yaping, Huang ^a  

^a BEIJING JIAOTONG UNIVERSITY   (China)

^b Beijing Broadcasting Institute   (China)

Author keywords

Multi quadrics; Partial differential equation; Radial basis function neural networks

Indexed keywords

APPROXIMATION THEORY; COMPUTATIONAL COMPLEXITY; LEARNING SYSTEMS; PARTIAL DIFFERENTIAL EQUATIONS;

ACTIVATION FUNCTIONS;

RADIAL BASIS FUNCTION NETWORKS;

ALGORITHM; ANALYTICAL PARAMETERS; ARTIFICIAL NEURAL NETWORK; CONFERENCE PAPER; MATHEMATICAL COMPUTING; NONLINEAR SYSTEM; PRIORITY JOURNAL;

EID: 0038383455     PISSN: 08936080     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0893-6080(03)00083-2     Document Type: Conference Paper

References (19)

1. Caffarelli L., Nirenberg L., Spruck J. The Dirichlet problem for non-linear second order elliptic equations (1): Monge-Ampere equations. Communication on Pure and Applied Mathematics. 37:1987;369-402.
2. Cheng S.Y., Yau S.Y. On the regularity of the Monge-Ampere equation. Communication on Pure and Applied Mathematics. 30:1977;41-68.
3. Esposito A., Marinaro M., Oricchio D., Scarpetta S. Approximation of continuous and discontinuous mappings by a growing neural RBF-based algorithm. Neural Networks. 13:2000;651-665.
4. Franke C., Schaback R. Convergence orders of meshless collocation methods using radial basis functions. Advances in Computational Mathematics. 8:(4):1997;381-399.
5. Franke C., Schaback R. Solving partial differential equations by collocation using radial basis functions. Applied Mathematics and Computation. 93:(1):1998;73-82.
6. Hughes T.J.R. The finite element method. 1987;Prentice Hall, New Jersey.
7. Jianyu L., Siwei L., Yingjian Q. Interactive gradient algorithm for radial basis function networks. The Sixth International Conference on Signal Processing. 2002;1187-1190.
8. Kansa E.J. Multi-quadrics - a scattered data approximation scheme with applications to computational fluid dynamics - II. Computers in Mathematical Applications. 19:(8/9):1990;127-145.
9. Kansa E.J. Multi-quadrics - a scattered data approximation scheme with applications to computational fluid dynamics - II. Computers in Mathematical Applications. 19:(8/9):1990;147-161.
10. Lions P.-L., Trudinger N.S., Urbas J.I.E. The Neumann problem for equations of Monge-Ampere type. Communication on Pure and Applied Mathematics. XXXIX:1986;539-563.
11. Madych W.R., Nelson S.A. Multi-variable interpolation and conditionally positive definite functions-II. Mathematical Computations. 54:1990;211-230.
12. Mai-Duy N., Tran-Cong T. Numerical solution of differential equations using multi-quadric radial basis function networks. Neural Networks. 14:2001;185-199.
13. Powell M.J.D. The uniform convergence of TPS interpolation in two dimensions. Numerische Mathematik. 68:(1):1994;107-128.
14. Smith G.D. Numerical solution of partial differential equations: Finite difference methods. 1978;Clarendon Press, Oxford.
15. Stead S. Estimation of gradients from scattered data. Rocky Mountain Journal of Mathematics. 14:1984;265-279.
16. Trudiger N. On the Dirichlet problem for Hessian equations. Acta Mathematica. 175:1995;151-164.
17. Wendland H. Meshless Galerkin methods using radial basis functions. Mathematical Computations. 68:(228):1999;1521-1531.
18. Wu Z. Chen Z., Micchelli C.A., Xu Y. Lectures notes on pure and applied mathematics. Lectures notes on pure and applied mathematics. 1998;202 Marcel Dekker, New York.
19. Wu Z., Schaback R. Shape preserving properties and convergence of univariate multi-quadic quasi-interpolation. Acta Mathematice Applicatae Sinica. 10:(4):1994;441-446.