An artificial neural network (ANN) was trained to estimate the partial derivatives of a spatially varying field, and compared against a finite difference approach. For the application of elastography, training data were generated using a wave equation. After the training examples were corrupted by noise and missing data, the network was trained to estimate the analytical solution to the partial derivatives. In simulation, the ANN improved accuracy in noisy data but blurred sharp boundaries relative to a finite difference method. In vivo, using the ANN to compute the curl of the displacement field improved confidence in subsequent property estimates.