# neural_network

Artificial neural network model

**Specification**

*Alias:*None*Arguments:*None

**Child Keywords:**

Required/Optional |
Description of Group |
Dakota Keyword |
Dakota Keyword Description |
---|---|---|---|

Optional |
Maximum number of hidden layer nodes |
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Optional |
Range for neural network random weights |
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Optional |
(Inactive) Random weight control |
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Optional |
Exports surrogate model in user-specified format(s) |
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Optional |
Import surrogate model from archive file |

**Description**

Dakota’s artificial neural network surrogate is a stochastic layered perceptron network, with a single hidden layer. Weights for the input layer are chosen randomly, while those in the hidden layer are estimated from data using a variant of the Zimmerman direct training approach [Zim96].

This typically yields lower training cost than traditional neural networks, yet good out-of-sample performance. This is helpful in surrogate-based optimization and optimization under uncertainty, where multiple surrogates may be repeatedly constructed during the optimization process, e.g., a surrogate per response function, and a new surrogate for each optimization iteration.

The neural network is a non parametric surface fitting method. Thus, along with Kriging (Gaussian Process) and MARS, it can be used to model data trends that have slope discontinuities as well as multiple maxima and minima. However, unlike Kriging, the neural network surrogate is not guaranteed to interpolate the data from which it was constructed.

This surrogate can be constructed from fewer than \(n_{c_{quad}}\) data points, however, it is a good rule of thumb to use at least \(n_{c_{quad}}\) data points when possible.

*Known Issue: When using discrete variables, there have been
sometimes significant differences in surrogate behavior observed
across computing platforms in some cases. The cause has not yet been
fully diagnosed and is currently under investigation. In addition,
guidance on appropriate construction and use of surrogates with
discrete variables is under development. In the meantime, users
should therefore be aware that there is a risk of inaccurate results
when using surrogates with discrete variables.*

**Theory**

The form of the neural network model is

where \(\mathbf{x}\) is the evaluation point in \(n\) -dimensional parameter space; the terms \(\mathbf{A}_{0}, \theta_{0}\) are the random input layer weight matrix and bias vector, respectively; and \(\mathbf{A}_{1}, \theta_{1}\) are a weight vector and bias scalar, respectively, estimated from training data. These coefficients are analogous to the polynomial coefficients obtained from regression to training data. The neural network uses a cross validation-based orthogonal matching pursuit solver to determine the optimal number of nodes and to solve for the weights and offsets.