The last decade has seen increasing interest in the application of Artificial Neural Networks (ANNs) for the modelling of the relationship between rainfall and streamflow. Since multi-layer, feed-forward ANNs have the property of being universal approximators, they are able to capture the essence of most input–output relationships, provided that an underlying deterministic relationship exists. Unfortunately, owing to the standardisation of inputs and outputs that is required to run ANNs, a problem arises in extrapolation: if the training data set does not contain the maximum possible output value, an unmodified network will be unable to synthesise this peak value. The occurrence of high magnitude, low frequency events within short periods of record is largely fortuitous. Therefore, the confidence in the neural network model can be greatly enhanced if some methodology can be found for incorporating domain knowledge about such events into the calibration and verification procedure in addition to the available measured data sets. One possible form of additional domain knowledge is the Estimated Maximum Flood (EMF), a notional event with a small but non-negligible probability of exceedence. This study investigates the suitability of including an EMF estimate in the training set of a rainfall–runoff ANN in order to improve the extrapolation characteristics of the network. A study has been carried out in which EMFs have been included, along with recorded flood events, in the training of ANN models for six catchments in the south west of England. The results demonstrate that, with prior transformation of the runoff data to logarithms of flows, the inclusion of domain knowledge in the form of such extreme synthetic events improves the generalisation capabilities of the ANN model and does not disrupt the training process. Where guidelines are available for EMF estimation, the application of this approach is recommended as an alternative means of overcoming the inherent extrapolation problems of multi-layer, feed-forward ANNs.