Genetic programming approaches for solving elliptic partial differential equations
Genetic programming approaches for solving elliptic partial differential equations
In this paper, we propose a technique based on genetic programming (GP) for meshfree solution of elliptic partial differential equations. We employ the least-squares collocation principle to define an appropriate objective function, which is optimized using GP. Two approaches are presented for the repair of the symbolic expression for the field variables evolved by the GP algorithm to ensure that the governing equations as well as the boundary conditions are satisfied. In the case of problems defined on geometrically simple domains, we augment the solution evolved by GP with additional terms, such that the boundary conditions are satisfied by construction. To satisfy the boundary conditions for geometrically irregular domains, we combine the GP model with a radial basis function network. We improve the computational efficiency and accuracy of both techniques with gradient boosting, a technique originally developed by the machine learning community. Numerical studies are presented for operator problems on regular and irregular boundaries to illustrate the performance of the proposed algorithms.
boosting, genetic programming (GP), meshfree collocation, partial differential equations (PDEs), radial basis functions
469-478
Sobester, A.
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Nair, P.B.
da7138d7-da7f-45af-887b-acc1d0e77a6f
Keane, A.J.
26d7fa33-5415-4910-89d8-fb3620413def
August 2008
Sobester, A.
096857b0-cad6-45ae-9ae6-e66b8cc5d81b
Nair, P.B.
da7138d7-da7f-45af-887b-acc1d0e77a6f
Keane, A.J.
26d7fa33-5415-4910-89d8-fb3620413def
Sobester, A., Nair, P.B. and Keane, A.J.
(2008)
Genetic programming approaches for solving elliptic partial differential equations.
IEEE Transactions on Evolutionary Computation, 12 (4), .
(doi:10.1109/TEVC.2007.908467).
Abstract
In this paper, we propose a technique based on genetic programming (GP) for meshfree solution of elliptic partial differential equations. We employ the least-squares collocation principle to define an appropriate objective function, which is optimized using GP. Two approaches are presented for the repair of the symbolic expression for the field variables evolved by the GP algorithm to ensure that the governing equations as well as the boundary conditions are satisfied. In the case of problems defined on geometrically simple domains, we augment the solution evolved by GP with additional terms, such that the boundary conditions are satisfied by construction. To satisfy the boundary conditions for geometrically irregular domains, we combine the GP model with a radial basis function network. We improve the computational efficiency and accuracy of both techniques with gradient boosting, a technique originally developed by the machine learning community. Numerical studies are presented for operator problems on regular and irregular boundaries to illustrate the performance of the proposed algorithms.
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Submitted date: 25 August 2003
e-pub ahead of print date: 22 February 2008
Published date: August 2008
Keywords:
boosting, genetic programming (GP), meshfree collocation, partial differential equations (PDEs), radial basis functions
Organisations:
Computational Engineering and Design
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Local EPrints ID: 64449
URI: http://eprints.soton.ac.uk/id/eprint/64449
PURE UUID: d90136da-5064-41f9-9b70-b554da9754a8
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Date deposited: 24 Dec 2008
Last modified: 16 Mar 2024 03:26
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Author:
P.B. Nair
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