difficult to determine the optimum value for the performance ratio parameter. It is desirable to determine the optimal regularization parameters automatically. One approach to this process is the Bayesian regularization of David MacKay [@MacKay:bayesianinterpolation:1992]. The Bayesian regularization algorithm updates the weight and bias values according to *Levenberg-Marquardt* [@HAGAN:backpropagation:1994][@Marquardt:LM:1963] optimization. It minimizes a linear combination of squared errors and weights and it also modifies the regularization parameters of the linear combination to generate a network that generalizes well. See [@MacKay:bayesianinterpolation:1992][@FORESEE:1997] for more detailed discussions of Bayesian regularization.

For further and general background on the ANN and how to prevent overfitting and improve generalization refer to [@BISHOP:ANN:1995][@HAYKIN:ANN:1999].

ANN Design {#sec:ANNDesign}
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The topological structure of ANNs used in this study is presented in Figure \[fig:ANN\_architecture\]. The designed ANNs contain one input layer with two neurons, one hidden layer with eight neurons and one output layer with one neuron. The inputs were the basis space parameters: the HO energy, $\hbar\Omega$, and the basis truncation parameter, $N_{\rm max}$, described in Section \[sec:TheoreticalFramework\]. The desired outputs were the gs energy and the gs point proton rms radius of $^6\mathrm{Li}$. An ANN was designed for each desired output: one ANN for gs energy and another ANN 