Modeling of Multi-Species Contaminant Transport with Spatially-Dependent Dispersion and Coupled Linear/Non-Linear Reactions

<p>Artificial Neural Networks (ANNs) have been widely applied to model hydrological problems with the increasing availability of data and computing power. ANNs are particularly useful to predict dynamic variables and to learn / discover constitutive relationships between variables. In the hydrology field, a specific example of the relationship takes the form of the governing equations of contaminant transport in porous media flow. Fluid flow in porous media is a spatio-temporal problem and it requires a certain numerical structure to solve. The ANNs, on the other hand, are black-box models that lack interpretability especially in their structure and prediction. Therefore, the discovery of the relationships using ANNs is not apparent. Recently, a distributed spatio-temporal ANN architecture (DISTANA) was proposed. The structure consists of transition kernels that learn the connectivity between one spatial cell and its neighboring cells, and prediction kernels that transform the transition kernels output to predict the quantities of interest at the subsequent time step. Another method, namely the Universal Differential Equation (UDE) for scientific machine learning was also introduced. UDE solves spatio-temporal problems by using a Convolutional Neural Network (CNN) structure to handle the spatial dependency and then approximating the differential operator with an ANN. This differential operator will be solved with Ordinary Differential Equation (ODE) solvers to administer the time dependency. In our work, we combine both methods to design an improved network structure to solve a contaminant transport problem in porous media, governed with the non-linear diffusion-sorption equation. The designed architecture consists of flux kernels and state kernels. Flux kernels are necessary to calculate the connectivity between neighboring cells, and are especially useful for handling different types of boundary conditions (Dirichlet, Neumann, and Cauchy). Furthermore, the state kernels are able to predict both observable states and mass-conserved states (total and dissolved contaminant concentration) separately. Additionally, to discover the constitutive relationship of sorption (i.e. the non-linear retardation factor R), we regularize its training to reflect the known monotonicity of R. As a result, our network is able to approximate R generated with the linear, Freundlich, and Langmuir sorption model, as well as the contaminant concentration with high accuracy.</p>

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A Homotopy-Perturbation analysis of the non-linear contaminant transport problem in one dimension with an initial continuous point source

Nigerian Journal of Technological Research ◽

10.4314/njtr.v8i1.88882 ◽

2013 ◽

Vol 8 (1) ◽

Author(s):

YM Aiyesimi ◽

OR Jimoh

Keyword(s):

Point Source ◽

Contaminant Transport ◽

Perturbation Analysis ◽

Transport Problem ◽

One Dimension ◽

Homotopy Perturbation ◽

Non Linear ◽

Continuous Point

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B. The Non-Linear Calculations for Pulsating Stars

Symposium - International Astronomical Union ◽

10.1017/s0074180900019008 ◽

1967 ◽

Vol 28 ◽

pp. 105-176

Author(s):

Robert F. Christy

Keyword(s):

Major Part ◽

Afternoon Session ◽

Discussion Session ◽

Pulsating Stars ◽

Fourth Symposium ◽

Non Linear ◽

Considerable Discussion ◽

Informal Approach ◽

Preceding Session

(Ed. note: The custom in these Symposia has been to have a summary-introductory presentation which lasts about 1 to 1.5 hours, during which discussion from the floor is minor and usually directed at technical clarification. The remainder of the session is then devoted to discussion of the whole subject, oriented around the summary-introduction. The preceding session, I-A, at Nice, followed this pattern. Christy suggested that we might experiment in his presentation with a much more informal approach, allowing considerable discussion of the points raised in the summary-introduction during its presentation, with perhaps the entire morning spent in this way, reserving the afternoon session for discussion only. At Varenna, in the Fourth Symposium, several of the summaryintroductory papers presented from the astronomical viewpoint had been so full of concepts unfamiliar to a number of the aerodynamicists-physicists present, that a major part of the following discussion session had been devoted to simply clarifying concepts and then repeating a considerable amount of what had been summarized. So, always looking for alternatives which help to increase the understanding between the different disciplines by introducing clarification of concept as expeditiously as possible, we tried Christy's suggestion. Thus you will find the pattern of the following different from that in session I-A. I am much indebted to Christy for extensive collaboration in editing the resulting combined presentation and discussion. As always, however, I have taken upon myself the responsibility for the final editing, and so all shortcomings are on my head.)

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