Artificial neural networks for solving ordinary and partial differential equations

Lagaris, Isaac E.; Likas, Aristidis; Fotiadis, D.I.

doi:10.1109/72.712178

Cited by 1,978 publications

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“…As metodologias descritas neste artigo seguem as propostas por Lagaris et al (1998Lagaris et al ( , 2000, estendidas para modelos com condições de contorno mistas e pelo uso do método da penalidade para converter o problema de otimização original de restrito para irrestrito no treinamento das redes MLP. Os resultados são compatíveis com aqueles apresentados em Luize e Biscaia (1991), que foram obtidos com técnicas numéricas já consagradas, como elementos finitos e colocação ortogonal.…”

Section: Introductionunclassified

“…The interpolation capabilities of multilayer perceptron networks (MLP) were used to solve a system of ordinary differential equations that models an axial dispersed non-adiabatic fixed bed reactor. The methodologies described in this paper follow the first ones proposed by Lagaris et al (1998Lagaris et al ( , 2000, but enlarge them to differential models with mix boundary conditions and by the use of the penalty method to convert the original constrained to unconstrained optimization problem in training the MLP networks. The results are in agreement on those in Luize e Biscaia (1991), which were obtained by well-established numerical techniques as finite element and orthogonal collocation methods.…”

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Resolução de um modelo de reator de leito fixo não adiabático com dispersão axial utilizando redes neurais artificiais

Silva¹,

Neitzel²,

Lima³

2003

Acta Sci. Technol.

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RESUMO.As capacidades de interpolação de redes perceptron multicamada (MLP) foram utilizadas para resolver um sistema de equações diferencias ordinárias que modela um reator não-adiabático com leito fixo e dispersão axial. As metodologias descritas neste artigo seguem as propostas por Lagaris et al. (1998Lagaris et al. ( , 2000, estendidas para modelos com condições de contorno mistas e pelo uso do método da penalidade para converter o problema de otimização original de restrito para irrestrito no treinamento das redes MLP. Os resultados são compatíveis com aqueles apresentados em Luize e Biscaia (1991), que foram obtidos com técnicas numéricas já consagradas, como elementos finitos e colocação ortogonal. O método de neuro-interpolação adotado neste artigo é de fácil manuseio se comparado com os métodos clássicos para solução numérica de equações diferenciais, particularmente para sistemas diferenciais não-lineares, e define uma aproximação global, na forma analítica, para a solução de problemas.Palavras chave: redes neurais, equações diferenciais.ABSTRACT. Model resolution of an axial dispersed non-adiabatic fixed bed reactor using artificial neural networks. The interpolation capabilities of multilayer perceptron networks (MLP) were used to solve a system of ordinary differential equations that models an axial dispersed non-adiabatic fixed bed reactor. The methodologies described in this paper follow the first ones proposed by Lagaris et al. (1998Lagaris et al. ( , 2000, but enlarge them to differential models with mix boundary conditions and by the use of the penalty method to convert the original constrained to unconstrained optimization problem in training the MLP networks. The results are in agreement on those in Luize e Biscaia (1991), which were obtained by well-established numerical techniques as finite element and orthogonal collocation methods. The neural interpolation method used in this paper is easier to handle than the classical methods for numerical solution of differential equations, particularly for non-linear differential systems, and defines a global approximation, in analytic form, for problems solution.

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Section: Introductionunclassified

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confidence: 99%

Resolução de um modelo de reator de leito fixo não adiabático com dispersão axial utilizando redes neurais artificiais

Silva¹,

Neitzel²,

Lima³

2003

Acta Sci. Technol.

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“…Then, another approach by Meade & Fernandez [25,26] was proposed for both linear and non-linear differential equations using Splines and feed forward neural network. Artificial neural networks based on Broyden-Fletcher-Goldfarb-Shanno (BF GS) optimization technique for solving ordinary and partial differential equations have been excellently presented by Lagaris et al [27]. Furthermore, Lagaris et al [28] investigated neural network methods for boundary value problems with irregular boundaries.…”

Section: Literature Reviewmentioning

confidence: 99%

Numerical Solution of Sixth-Order Differential Equations Arising in Astrophysics by Neural Network

Khalid¹,

Sultana²,

Zaidi³

2014

IJCA

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In the current paper, a neural network method to solve sixth-order differential equations and their boundary conditions has been presented. The idea this method incorporates is to integrate knowledge about the differential equation and its boundary conditions into neural networks and the training sets. Neural networks are being used incessantly to solve all kinds of problems hailing a wide range of disciplines. Several examples are given to illustrate the efficiency and implementation of the Neural Network method.

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“…For this reason, ANNs are popular in applications such as image identification [19] and speech recognition [20], as a substitute for complex rule-based algorithms which are often difficult to program. ANNs have also been used to solve certain classes of ordinary and partial differential equations [21,22,23].…”

Section: Introductionmentioning

confidence: 99%

An artificial neural network as a troubled-cell indicator

Ray

Hesthaven

2018

Journal of Computational Physics

128

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High-resolution schemes for conservation laws need to suitably limit the numerical solution near discontinuities, in order to avoid Gibbs oscillations. The solution quality and the computational cost of such schemes strongly depend on their ability to correctly identify troubled-cells, namely, cells where the solution loses regularity. Motivated by the objective to construct a universal troubled-cell indicator that can be used for general conservation laws, we propose a new approach to detect discontinuities using artificial neural networks (ANNs). In particular, we construct a multilayer perceptron (MLP), which is trained offline using a supervised learning strategy, and thereafter used as a black-box to identify troubled-cells. The proposed MLP indicator can accurately identify smooth extrema and is independent of problem-dependent parameters, which gives it an advantage over traditional limiter-based indicators. Several numerical results are presented to demonstrate the robustness of the MLP indicator in the framework of Runge-Kutta discontinuous Galerkin schemes, and its performance is compared with the minmod limiter and the minmod-based TVB limiter.

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Artificial neural networks for solving ordinary and partial differential equations

Cited by 1,978 publications

References 10 publications

Resolução de um modelo de reator de leito fixo não adiabático com dispersão axial utilizando redes neurais artificiais

Resolução de um modelo de reator de leito fixo não adiabático com dispersão axial utilizando redes neurais artificiais

Numerical Solution of Sixth-Order Differential Equations Arising in Astrophysics by Neural Network

An artificial neural network as a troubled-cell indicator

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