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In many machine learning applications, one has access, not only to training data, but also to some high-level a priori knowledge about the desired be (cid: 173) havior of the system. For example, it is known in advance that the output of a character recognizer should be invariant with respect to small spa (cid: 173) tial distortions of the input images (translations, rotations, scale changes, etcetera). We have implemented a scheme that allows a network to learn the deriva (cid: 173) tive of its outputs with respect to distortion operators of our choosing. This not only reduces the learning time and the amount of training data, but also provides a powerful language for specifying what generalizations we wish the network to perform.