Solving the inverse problem of contaminant transport equation using a neural network

Abstract

The artificial neural network is considered to be a universal function approximater. As such, the artificial neural network can be used to solve the inverse contaminant transport equation for parameter estimation. In this study the plausibility of the artificial neural network in solving an inverse problem related to values of longitudinal dispersivities using concentration values derived from a forward model at a fixed time is assessed. A set of published data corresponding to the Macrodispersivity Experiment (MADE) study at the Columbus Air Force Base (CAFB) in northeastern Mississippi was used as the base to build a two-dimensional, unconfined, homogenous, synthetic case to investigate the applicability of this method under a variety of scenarios. In this approach, the dispersivity values were estimated by training a feedforward backpropagation network of 20 neurons in one hidden layer, by using concentration fields generated by the MT3D model as an inputs and the dispersivity values were the outputs. The network was able to estimate the dispersivity values with high accuracy using 1600 concentration measurements for each dispersivity value. The total number of dispersivity values was 210, which was divided into four subsets for the propose of training and estimation. The network was also investigated with different grid size of the domain, different number of training patterns, and different number of concentration measurements. The method demonstrated a good accuracy in estimating the dispersivity values to a certain limit of grid size. Also it showed that the parameter can be estimated with a good accuracy using at least 147 concentration fields for different grid size and varying concentration measurements. The network demonstrated some limitation in estimating the dispersivity values with low dispersivity values at some grid sizes. One of the limitation of artificial neural networks is that they cannot reveal any physics and explain in a meaningful way the basic process by which they arrive at a decision. However, the artificial neural networks can be a good tool in evaluating the products of different forward models. The network was able to generalize its behavior in estimating the parameter beyond the training range and with a different discretization level of the domain. The inverse solution was stable with different percentages of random errors added to the concentration fields needed to estimate their dispersivity values. However, the inverse solution was ill-posed due to non-uniqueness.