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.
Bibliographical noteFunding Information:
This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP) ( NRF-2013R1A2A1A01013886 ) and a Brain Korea 21 (BK-21) fellowship from the Ministry of Education of Korea .
© 2016 Elsevier Ltd
- Advection dispersion
- Boundary value problems
- Gradient descent
- Neural network
ASJC Scopus subject areas
- Modelling and Simulation
- Computational Theory and Mathematics
- Computational Mathematics