# Spatial Treatment of the Slab-geometry Discrete Ordinates Equations Using Artificial Neural Networks

## Abstract

An artificial neural network (ANN) method is developed for treating the spatial variable of the one-group slab-geometry discrete ordinates (S{sub N}) equations in a homogeneous medium with linearly anisotropic scattering. This ANN method takes advantage of the function approximation capability of multilayer ANNs. The discrete ordinates angular flux is approximated by a multilayer ANN with a single input representing the spatial variable x and N outputs representing the angular flux in each of the discrete ordinates angular directions. A global objective function is formulated which measures how accurately the output of the ANN approximates the solution of the discrete ordinates equations and boundary conditions at specified spatial points. Minimization of this objective function determines the appropriate values for the parameters of the ANN. Numerical results are presented demonstrating the accuracy of the method for both fixed source and incident angular flux problems.

- Authors:

- Publication Date:

- Research Org.:
- Lawrence Livermore National Lab., CA (US)

- Sponsoring Org.:
- US Department of Energy (US)

- OSTI Identifier:
- 15005484

- Report Number(s):
- UCRL-JC-143205

TRN: US0305395

- DOE Contract Number:
- W-7405-ENG-48

- Resource Type:
- Conference

- Resource Relation:
- Conference: International Conference on Mathematical Methods to Nuclear Applications, Salt Lake City, UT (US), 09/09/2001--09/13/2001; Other Information: PBD: 23 Mar 2001

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 99 GENERAL AND MISCELLANEOUS//MATHEMATICS, COMPUTING, AND INFORMATION SCIENCE; BOUNDARY CONDITIONS; DISCRETE ORDINATE METHOD; MINIMIZATION; NEURAL NETWORKS; SLABS

### Citation Formats

```
Brantley, P S.
```*Spatial Treatment of the Slab-geometry Discrete Ordinates Equations Using Artificial Neural Networks*. United States: N. p., 2001.
Web.

```
Brantley, P S.
```*Spatial Treatment of the Slab-geometry Discrete Ordinates Equations Using Artificial Neural Networks*. United States.

```
Brantley, P S. Fri .
"Spatial Treatment of the Slab-geometry Discrete Ordinates Equations Using Artificial Neural Networks". United States. https://www.osti.gov/servlets/purl/15005484.
```

```
@article{osti_15005484,
```

title = {Spatial Treatment of the Slab-geometry Discrete Ordinates Equations Using Artificial Neural Networks},

author = {Brantley, P S},

abstractNote = {An artificial neural network (ANN) method is developed for treating the spatial variable of the one-group slab-geometry discrete ordinates (S{sub N}) equations in a homogeneous medium with linearly anisotropic scattering. This ANN method takes advantage of the function approximation capability of multilayer ANNs. The discrete ordinates angular flux is approximated by a multilayer ANN with a single input representing the spatial variable x and N outputs representing the angular flux in each of the discrete ordinates angular directions. A global objective function is formulated which measures how accurately the output of the ANN approximates the solution of the discrete ordinates equations and boundary conditions at specified spatial points. Minimization of this objective function determines the appropriate values for the parameters of the ANN. Numerical results are presented demonstrating the accuracy of the method for both fixed source and incident angular flux problems.},

doi = {},

journal = {},

number = ,

volume = ,

place = {United States},

year = {Fri Mar 23 00:00:00 EST 2001},

month = {Fri Mar 23 00:00:00 EST 2001}

}