In principle, ANNs can compute any computable function, i.e. they can do everything a normal digital computer do. [I1]
In practice, ANNs are especially useful for classification and function approximation/mapping problems which are tolerant of some imprecision, which have lots of training data available, but to which hard and fast rules (such as those that might be used in an expert system) cannot easily be applied. Almost any mapping between vector spaces can be approximated to arbitrary precision by feedforward ANN.
ANNs are, at least today, difficult to apply successfully to problems that concern manipulation of symbols and memory. And there are no methods for training ANNs that can magically create information that is not contained in the training data. [I1]
Please send Questions, Comments, Suggestions to bencr@hotmail.com