The lateral-line system that has evolved in many aquatic animals enables them to navigate murky fluid environments, locate and discriminate obstacles. Here, we present a data-driven model that uses artificial neural networks to process flow data originating from a stationary sensor array located away from an obstacle placed in a potential flow. The ability of neural networks to estimate complex underlying relationships between parameters, in the absence of any explicit mathematical description, is first assessed with two basic potential flow problems: single source/sink identification and doublet detection. Subsequently, we address the inverse problem of identifying an obstacle shape from distant measures of the pressure or velocity field. Using the analytical solution to the forward problem, very large training data sets are generated, allowing us to obtain the synaptic weights by means of a gradient-descent based optimization. The resulting neural network exhibits remarkable effectiveness in predicting unknown obstacle shapes, especially at relatively large distances for which classical linear regression models are completely ineffectual. These results have far-reaching implications for the design and development of artificial passive hydrodynamic sensing technology.