Artificial Neural Networks (ANNs) are commonly used in place of expensive models to reduce the computational burden required for uncertainty quantification, reliability and sensitivity analyses. ANN with selected architecture is trained with the back-propagation algorithm from few data representatives of the input/output relationship of the underlying model of interest. However, different performing ANNs might be obtained with the same training data as a result of the random initialization of the weight parameters in each of the network, leading to an uncertainty in selecting the best performing ANN. On the other hand, using cross-validation to select the best performing ANN based on the ANN with the highest R value can lead to biassing in the prediction. This is as a result of the fact that the use of R cannot determine if the prediction made by ANN is biased. Additionally, R does not indicate if a model is adequate, as it is possible to have a low R for a good model and a high R for a bad model. Hence, in this paper, we propose an approach to improve the robustness of a prediction made by ANN. The approach is based on a systematic combination of identical trained ANNs, by coupling the Bayesian framework and model averaging. Additionally, the uncertainties of the robust prediction derived from the approach are quantified in terms of confidence intervals. To demonstrate the applicability of the proposed approach, two synthetic numerical examples are presented. Finally, the proposed approach is used to perform a reliability and sensitivity analyses on a process simulation model of a UK nuclear effluent treatment plant developed by National Nuclear Laboratory (NNL) and treated in this study as a black-box employing a set of training data as a test case. This model has been extensively validated against plant and experimental data and used to support the UK effluent discharge strategy.

On-line monitoring techniques have attracted increasing attention as a promising strategy for improving safety, maintaining availability and reducing the cost of operation and maintenance. In particular, pattern recognition tools such as Artificial neural networks are today largely adopted for sensor validation, plant component monitoring, system control, and fault-diagnostics based on the data acquired during operation. However, classic artificial neural networks do not provide an error context for the model response, whose robustness remains thus difficult to estimate. Indeed, experimental data generally exhibit a time/space-varying behaviour and are hence characterized by an intrinsic level of uncertainty that unavoidably affects the performance of the tools adopted and undermine the accuracy of the analysis. For this reason, the propagation of the uncertainty and the quantification of the so called margins of uncertainty in output are crucial in making risk-informed decision. The current study presents a comparison between two different approaches for the quantification of uncertainty in artificial neural networks. The first technique presented is based on the error estimation by a series association scheme, the second approach couples Bayesian model selection technique and model averaging into a unified framework. The efficiency of these two approaches are analysed in terms of their computational cost and predictive performance, through their application to a nuclear power plant fault diagnosis system.

Abstract. Artificial Neural Networks (ANN) are used in place of expensive models to reduce the computational burden required for reliability analysis. Often, ANNs with selected architecture are trained with the back-propagation algorithm from few data representatives of the input/output relationship of the underlying model of interest. However, different performing ANNs might be obtained from the same training data, leading to an uncertainty in selecting the best performing ANN. On the other hand, using cross-validation to select the best performing ANN based on the highest R 2 value can lead to a biassing in terms of the prediction made by the selected ANN. This is due to the fact that the use of R 2 cannot determine if the prediction made by ANN is biased. Additionally, R 2 does not indicate if a model is adequate, as it is possible to have a low R 2 for a good model and a high R 2 for a bad model. Hence we propose an approach to improve the prediction robustness of an ANN based on coupling Bayesian framework and model averaging technique into a unified framework. The model uncertainties propagated to the robust prediction is quantified in terms of confidence intervals. Two examples are used to demonstrate the applicability of the approach