Genetic programming and automatic differentiation algorithms applied to the solution of ordinary and partial differential equations

Lobão, Waldir Jesus Araújo; Dias, Douglas Mota; Pacheco, Marco Aurélio Cavalcanti

doi:10.1109/cec.2016.7748362

Cited by 3 publications

(3 citation statements)

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“…In order to evaluate the unguided complexity, one must first be able to evaluate µ(Ω). In the current work this is accomplished by considering the encoding space E. A common choice [10,5,11] for E is to use a tree structure. This approach has the nice property that all mathematical expressions can be mapped in an isomorphic way to a tree object,…”

Section: Enumerating Search Spacesmentioning

confidence: 99%

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On Affine Symbolic Regression Trees for the Solution of Functional Problems

Champneys¹,

Dervilis²,

Worden³

2021

Conference Proceedings of the Society for Experimental Mechanics Series

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Symbolic regression has emerged from the more general method of Genetic Programming (GP) as a means of solving functional problems in physics and engineering, where a functional problem is interpreted here as a search problem in a function space. A good example of a functional problem in structural dynamics would be to find an exact solution of a nonlinear equation of motion. Symbolic regression is usually implemented in terms of a tree representation of the functions of interest; however, this is known to produce search spaces of high dimension and complexity. The aim of this paper is to introduce a new representation -the affine symbolic regression tree. The search space size for the new representation is derived and the results are compared to those for a standard regression tree. The results are illustrated by the search for an exact solution to several benchmark problems.

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Section: Enumerating Search Spacesmentioning

confidence: 99%

“…for the expression trees (indicative of the approach suggested by [10]). The grammar-space data are generated with m = 3 and basis set,…”

Section: Enumerating Search Spacesmentioning

confidence: 99%

On Affine Symbolic Regression Trees for the Solution of Functional Problems

Champneys¹,

Dervilis²,

Worden³

2021

Conference Proceedings of the Society for Experimental Mechanics Series

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“…Chaquet et al [19] further applied RBF network into the basis function. In [20], the authors adopted GP and automatic differentiation algorithms to determine the approximate solutions of ODEs.…”

Section: Introductionmentioning

confidence: 99%

Solving Ordinary Differential Equations With Adaptive Differential Evolution

2020

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Solving ordinary differential equations (ODEs) is vital in diverse fields. However, it is difficult to obtain the exact analytical solutions of ODEs due to their changeable mathematical forms. Traditional numerical methods can find approximate solutions for specific ODEs. Unfortunately, they often suffer from ODEs' forms and characteristics. To approximate different types of ODEs, this paper proposes a generic method based on adaptive differential evolution. Besides, in order to further reduce the error of the obtained approximate solutions, an improved Fourier periodic expansion function is developed, which is then combined with the least square weight method to formulate the ODEs as an optimization problem. Since the proposed method is not limited to ODEs' forms and constraint conditions, it can be used to approximate any ODEs, including linear ODEs and nonlinear ODEs. The proposed method is evaluated on twenty popular test cases. The results indicate that the proposed method is able to accurately approximate different ODEs with better performance compared with other methods.

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