Neural networks are general nonlinear systems that map from one vector space into another. These trainable mappers have been applied to numerous problems in geosciences and scientific computing. This study investigates the use of neural networks to map the present positions of mesoscale features in the ocean into future positions, thus performing neural-based 'forecasting' of mesoscale dynamics. Specifically, a neural network has been trained to predict the position of the North Wall of the Gulf Stream based on a recent history of Gulf Stream positions. An archive of Gulf Stream positional data covering several years has been assembled. Eigenvector analysis of the archive showed that any given realization of Gulf Stream shape can be reasonable parameterized as a set of eigenvector, or normal mode, coefficients. Gulf Stream dynamics can, therefore, be conceptualized as a time series of these coefficients. A neural network has been trained to take advantage of any existing time coherency in Gulf Stream normal mode coefficients to produce forecast coefficients that describe Gulf Stream shape and position in the future. The neural network performance is compared with persistence and with other forecasting systems. Forecast skill for the neural network is found to be generally superior to other methods considered, but computational requirements are only a fraction of those required by alternate methodologies.