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Neural network approach for solving nonlocal boundary value problems

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Abstract

This paper proposes a radial basis function (RBF) network-based method for solving a nonlinear second-order elliptic equation with Dirichlet boundary conditions. The nonlocal term involved in the differential equation needs a completely different approach from the up-to-now-known methods for solving boundary value problems by using neural networks. A numerical integration procedure is developed for computing the local \(L^2\)-inner product. It is known that the non-variational methods are not effective in solving nonlocal problems. In this paper, the weak formulation of the nonlocal problem is reduced to the minimization of a nonlinear functional. Unlike many previous works, we use an integral objective functional for implementing the learning procedure. Well-distributed nodes are used as the centers of the RBF neural network. The weights of the RBF network are determined by a two-point step size gradient method. The neural network method proposed in this paper is an alternative to the finite-element method (FEM) for solving nonlocal boundary value problems in non-Lipschitz domains. A new variable learning rate strategy has been developed and implemented in order to avoid the divergence of the training process. A comparison between the proposed neural network approach and the FEM is illustrated by challenging examples, and the performance of both methods is thoroughly analyzed.

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References

  1. Gaspar FJ, Lisbona FJ, Rodrigo C (2010) Multigrid finite element method on semi-structured grids for the poroelasticity problem. In: Kreiss G, Lutstedt P, Melqvist A, Neytcheva M (eds) Numerical mathematics and advanced applications. Springer, Berlin

    Google Scholar 

  2. Jung M, Todorov TD (2006) Isoparametric multigrid method for reaction–diffusion equations. Appl Numer Math 56:1570–1583

    MathSciNet  MATH  Google Scholar 

  3. Brandts J, Korotov S, Křížek M (2011) Generalization of the Zlamal condition for simplicial finite elements in \({ R}^d\). Appl Math 56:417–424

    MathSciNet  MATH  Google Scholar 

  4. Brandts J, Korotov S, Křížek M (2007) Simplicial finite elements in higher dimensions. Appl Math 52(3):251–265

    MathSciNet  MATH  Google Scholar 

  5. Brandts J, Korotov S, Křížek M, Šolc J (2009) On nonobtuse simplicial partitions. SIAM Rev 51(2):317–335

    MathSciNet  MATH  Google Scholar 

  6. Jianyua L, Siweia L, Yingjiana Q, Yapinga H (2003) Numerical solution of elliptic partial differential equation using radial basis function neural networks. Neural Netw 16:729–734

    Google Scholar 

  7. Craddock RJ, Warwick K (1996) Multi-layer radial basis function networks: an extension to the radial basis function. In: IEEE international conference on neural networks 1996, 3–6 Jun 1996, Washington DC, USA, pp 700–705

  8. Figueiredo MA (2000) On Gaussian radial basis function approximations: interpretation, extensions, and learning strategies. In: International conference on (ICPR) pattern recognition, Barcelona, Spain, p 2618

  9. Takeuchi J, Kosugi Y (1994) Neural network representation of finite element method. Neural Netw 7(2):389–395

    Google Scholar 

  10. Shirvany Y, Hayati M, Moradian R (2008) Numerical solution of the nonlinear Schrödinger equation by feed forward neural networks. Commun Nonlinear Sci Numer Simul 13:2132–2145

    MathSciNet  MATH  Google Scholar 

  11. Ilati M, Dehghan M (2018) Error analysis of a meshless weak form method based on radial point interpolation technique for Sivashinsky equation arising in the alloy solidification problem. J Comput Appl Math 327:314–324

    MathSciNet  MATH  Google Scholar 

  12. Jianyu L, Siwei L, Yingjian Q, Yaping H (2003) Numerical solution of elliptic partial differential equation using radial basis function neural networks. Neural Netw 16(5–6):729–734

    Google Scholar 

  13. Aarts LP, van der Veer P (2001) Neural network method for solving partial differential equations. Neural Process Lett 14:261–271

    MATH  Google Scholar 

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

    Google Scholar 

  15. Acosta G, Armentano MG, Durán RG, Lombardi AL (2007) Finite element approximations in a non-Lipschitz domain. SIAM J Numer Anal 45(1):277–295

    MathSciNet  MATH  Google Scholar 

  16. Acosta G, Armentano MG (2011) Finite element approximations in a non-Lipschitz domain: part II. Math Comput 80(276):1949–1978

    MathSciNet  MATH  Google Scholar 

  17. Grisvard P (2011) Elliptic problems in nonsmooth domains. SIAM, Philadelphia, p 425

    MATH  Google Scholar 

  18. Demlow A, Georgoulis EH (2012) Pointwise a posteriori error control for discontinuous Galerkin methods for elliptic problems. SIAM J Numer Anal 50(5):2159–2181

    MathSciNet  MATH  Google Scholar 

  19. Aksoylu B, Mengesha T (2020) Results on nonlocal boundary value problems. Numer Funct AnalOptim 31(12):1301–1317

    MathSciNet  MATH  Google Scholar 

  20. Corrêa FJS, Menezes SDB (2004) Existence of solutions to nonlocal and singular elliptic problems via Galerkin method. Electron J Differ Equ 2004(19):1–10

    MathSciNet  Google Scholar 

  21. Dehghan M (2002) Numerical solution of a non-local boundary value problem with Neumann’s boundary conditions. Commun Numer Methods Eng 19(1):1–12

    MathSciNet  MATH  Google Scholar 

  22. Themistoclakis W, Vecchio A (2016) On the numerical solution of a nonlocal boundary value problem. J Comput Appl Math 292(15):720–731

    MathSciNet  MATH  Google Scholar 

  23. Alves CO, Correa FJS, Figueiredo GM (2010) On a class of nonlocal elliptic problems with critical growth. Differ Equ Appl 2:409–417

    MathSciNet  MATH  Google Scholar 

  24. Dai G (2012) On positive solutions for a class of nonlocal problems. Electron J Qual Theory Differ Equ 58:1–12

    MathSciNet  Google Scholar 

  25. Kumar M, Kumar P (2009) Computational method for finding various solutions for a quasilinear elliptic equation of Kirchhoff type. Adv Eng Softw 40:1104–1111

    MATH  Google Scholar 

  26. Gudi T (2012) Finite element method for a nonlocal problem of Kirchhoff type. SIAM J Numer Anal 50:657–668

    MathSciNet  MATH  Google Scholar 

  27. Sanni SA (2014) Nonlocal degenerate reaction–diffusion equations with general nonlinear diffusion term. Electron J Differ Equ 2014(124):1–27

    MathSciNet  MATH  Google Scholar 

  28. Amster P, De Napoli P (2006) A nonlinear second order problem with a nonlocal boundary condition. Abstr Appl Anal 2006:1–11

    MathSciNet  MATH  Google Scholar 

  29. Kanca F (2016) Inverse coefficient problem for a second-order elliptic equation with nonlocal boundary conditions. Math Methods Appl Sci 39(11):3152–3158

    MathSciNet  MATH  Google Scholar 

  30. Nie C, Shu S, Yu H, An Q (2014) A high order composite scheme for the second order elliptic problem with nonlocal boundary and its fast algorithm. Appl Math Comput 227:212–221

    MathSciNet  MATH  Google Scholar 

  31. Nie C, Yu H (2013) Some error estimates on the finite element approximation for two-dimensional elliptic problem with nonlocal boundary. Appl Numer Math 68:31–38

    MathSciNet  MATH  Google Scholar 

  32. De Schepper H, Slodivcka M (2000) Recovery of the boundary data for a linear second order elliptic problem with a nonlocal boundary condition. ANZIAM J 42:518–535

    MathSciNet  Google Scholar 

  33. Sajavičius S (2014) Radial basis function method for a multidimensional linear elliptic equation with nonlocal boundary conditions. Comput Math Appl 67:1407–1420

    MathSciNet  MATH  Google Scholar 

  34. Liu W, Wang B (2018) A modified approximate method based on Gaussian radial basis functions. Eng Anal Bound Elem 100:256–264

    MathSciNet  MATH  Google Scholar 

  35. Davydov O, Oanhb DT (2011) On the optimal shape parameter for Gaussian radial basis function finite difference approximation of the Poisson equation. Comput Math Appl 62:2143–2161

    MathSciNet  MATH  Google Scholar 

  36. Baymani M, Effati S, Niazmand H, Kerayechian A (2015) Artificial neural network method for solving the Navier–Stokes equations. Neural Comput Appl 26(4):765–773

    Google Scholar 

  37. Kazem S, Rad JA (2012) Radial basis functions method for solving of a non-local boundary value problem with Neumann’s boundary conditions. Appl Math Model 36:2360–2369

    MathSciNet  MATH  Google Scholar 

  38. Kim Hyun-Gyu, Kim Hye-Young (2009) An adaptive procedure based on combining finite elements with meshless methods. J Mech Sci Technol 23:2224–2235

    Google Scholar 

  39. Todorov TD (2018) Dirichlet problem for a nonlocal \(p\)-Laplacian elliptic equation. Comput Math Appl 76(6):1261–1274

    MathSciNet  MATH  Google Scholar 

  40. Todorov TD (2015) Nonlocal problem for a general second-order elliptic operator. Comput Math Appl 69(5):411–422

    MathSciNet  Google Scholar 

  41. Wu Z (1995) Compactly supported positive definite radial functions. Adv Comput Math 4:283–292

    MathSciNet  MATH  Google Scholar 

  42. Kumar M, Singh AK, Srivastava A (2013) Various Newton-type iterative methods for solving nonlinear equations. J Egypt Math Soc 21:334–339

    MathSciNet  MATH  Google Scholar 

  43. Birgin EG, Martínez JM, Raydan M (2014) Spectral projected gradient methods: review and perspectives. J Stat Softw 60(3):1–21

    Google Scholar 

  44. Liu Chein-Shan, Hong Hong-Ki, Atluri SN (2010) Novel algorithms based on the conjugate gradient method for inverting ill-conditioned matrices, and a new regularization method to solve ill-posed linear systems. Comput Model Eng Sci 60(3):279–308

    MathSciNet  MATH  Google Scholar 

  45. Szabó B, Babuška I (1991) Finite element analysis. Wiley, New York

    MATH  Google Scholar 

  46. Khosravifard A, Hematiyan MR (2010) A new method for meshless integration in 2D and 3D Galerkin meshfree methods. Eng Anal Bound Elem 34:30–40

    MathSciNet  MATH  Google Scholar 

  47. Hammer PC, Marlowe OJ, Stroud AH (1956) Numerical integration over simplexes and cones. Math Tables Aids Comput 10:130–137

    MathSciNet  MATH  Google Scholar 

  48. Li H, Mazzucato A, Nistor V (2010) Analysis of the finite element method for transmission/mixed boundary value problems on general polygonal domains. Electron Trans Numer Anal 37:41–69

    MathSciNet  MATH  Google Scholar 

  49. Andreev AB, Petrov MS, Todorov TD (2005) An optimal order numerical quadrature approximation of a planar isoparametric eigenvalue problem on triangular finite element meshes. Calcolo 42(2):47–69

    MathSciNet  MATH  Google Scholar 

  50. Todorov TD (2013) The optimal refinement strategies for 3-D simplicial meshes. Comput Math Appl 66(7):1272–1283

    MathSciNet  MATH  Google Scholar 

  51. Petrov Miroslav S, Todorov TD (2018) Stable subdivision of 4D polytopes. Numer Algorithms 79(2):633–656

    MathSciNet  MATH  Google Scholar 

  52. Todorov TD (2018) A second order problem with nonlocal boundary conditions. Sci Publ State Univ Novi Pazar Ser A Appl Math Inform Mech 10(2):71–74

    Google Scholar 

  53. Atluri SN, Shen S (2002) The meshless local Petrov–Galerkin (MLPG) method: a simple & less-costly alternative to the finite element and boundary element methods. Comput Model Eng Sci 3(1):11–51

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Engineering and Computing, Coventry University, Priory Street, Coventry, UK

    V. Palade

  2. Department of Technical Mechanics, Technical University of Gabrovo, 5300, Gabrovo, Bulgaria

    M. S. Petrov

  3. Department of Mathematics and Informatics, Technical University of Gabrovo, 5300, Gabrovo, Bulgaria

    T. D. Todorov

Authors
  1. V. Palade
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  2. M. S. Petrov
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  3. T. D. Todorov
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Correspondence to T. D. Todorov.

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Palade, V., Petrov, M.S. & Todorov, T.D. Neural network approach for solving nonlocal boundary value problems. Neural Comput & Applic 32, 14153–14171 (2020). https://doi.org/10.1007/s00521-020-04810-0

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