Numerical solution of unsteady advection dispersion equation arising in contaminant transport through porous media using neural networks

Neha Yadav, Anupam Yadav, Joong Hoon Kim

Research output: Contribution to journalArticle

9 Citations (Scopus)

Abstract

A soft computing approach based on artificial neural network (ANN) and optimization is presented for the numerical solution of the unsteady one-dimensional advection-dispersion equation (ADE) arising in contaminant transport through porous media. A length factor ANN method, based on automatic satisfaction of arbitrary boundary conditions (BCs) was chosen for the numerical solution of ADE. The strength of ANN is exploited to construct a trial approximate solution (TAS) for ADE in a way that it satisfies the initial or BCs exactly. An unsupervised error is constructed in approximating the solution of ADE which is minimized by training ANN using gradient descent algorithm (GDA). Two challenging test problems of ADE are considered in this paper, in which, the first problem has steep boundary layers near . x=1 and many numerical methods create non-physical oscillation near steep boundaries. Also for the second problem many numerical schemes suffer from computational noise and instability issues. The proposed method is advantageous as it does not require temporal discretization for the solution of the ADEs as well as it does not suffer from numerical instability. The reliability and effectiveness of the presented algorithm is investigated by sufficient large number of independent runs and comparison of results with other existing numerical methods. The results show that the present method removes the difficulties arising in the solution of the ADEs and provides solution with good accuracy.

Original languageEnglish
JournalComputers and Mathematics with Applications
DOIs
Publication statusAccepted/In press - 2015 Nov 6

Fingerprint

Contaminant Transport
Advection
Porous Media
Porous materials
Numerical Solution
Artificial Neural Network
Neural Networks
Impurities
Neural networks
Numerical methods
Numerical Methods
Boundary conditions
Numerical Instability
Soft computing
Descent Algorithm
Soft Computing
Gradient Algorithm
Gradient Descent
Numerical Scheme
Test Problems

Keywords

  • Advection dispersion
  • Boundary value problems
  • Contaminant
  • Gradient descent
  • Neural network

ASJC Scopus subject areas

  • Computational Theory and Mathematics
  • Modelling and Simulation
  • Computational Mathematics

Cite this

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abstract = "A soft computing approach based on artificial neural network (ANN) and optimization is presented for the numerical solution of the unsteady one-dimensional advection-dispersion equation (ADE) arising in contaminant transport through porous media. A length factor ANN method, based on automatic satisfaction of arbitrary boundary conditions (BCs) was chosen for the numerical solution of ADE. The strength of ANN is exploited to construct a trial approximate solution (TAS) for ADE in a way that it satisfies the initial or BCs exactly. An unsupervised error is constructed in approximating the solution of ADE which is minimized by training ANN using gradient descent algorithm (GDA). Two challenging test problems of ADE are considered in this paper, in which, the first problem has steep boundary layers near . x=1 and many numerical methods create non-physical oscillation near steep boundaries. Also for the second problem many numerical schemes suffer from computational noise and instability issues. The proposed method is advantageous as it does not require temporal discretization for the solution of the ADEs as well as it does not suffer from numerical instability. The reliability and effectiveness of the presented algorithm is investigated by sufficient large number of independent runs and comparison of results with other existing numerical methods. The results show that the present method removes the difficulties arising in the solution of the ADEs and provides solution with good accuracy.",
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AB - A soft computing approach based on artificial neural network (ANN) and optimization is presented for the numerical solution of the unsteady one-dimensional advection-dispersion equation (ADE) arising in contaminant transport through porous media. A length factor ANN method, based on automatic satisfaction of arbitrary boundary conditions (BCs) was chosen for the numerical solution of ADE. The strength of ANN is exploited to construct a trial approximate solution (TAS) for ADE in a way that it satisfies the initial or BCs exactly. An unsupervised error is constructed in approximating the solution of ADE which is minimized by training ANN using gradient descent algorithm (GDA). Two challenging test problems of ADE are considered in this paper, in which, the first problem has steep boundary layers near . x=1 and many numerical methods create non-physical oscillation near steep boundaries. Also for the second problem many numerical schemes suffer from computational noise and instability issues. The proposed method is advantageous as it does not require temporal discretization for the solution of the ADEs as well as it does not suffer from numerical instability. The reliability and effectiveness of the presented algorithm is investigated by sufficient large number of independent runs and comparison of results with other existing numerical methods. The results show that the present method removes the difficulties arising in the solution of the ADEs and provides solution with good accuracy.

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