When training an ANN with a set of input and output data, we wish to adjust the weights in the ANN, to make the ANN give the same outputs as seen in the training data. On the other hand, we do not want to make the ANN too specific, making it give precise results for the training data, but incorrect results for all other data. When this happens, we say that the ANN has been over-fitted.
The training process can be seen as an optimization problem, where we wish to minimize the mean square error of the entire set of training data. This problem can be solved in many different ways, ranging from standard optimization heuristics like simulated annealing, through more special optimization techniques like genetic algorithms to specialized gradient descent algorithms like backpropagation.
The most used algorithm is the backpropagation algorithm, but this algorithm has some limitations concerning, the extent of adjustment to the weights in each iteration. This problem has been solved in more advanced algorithms like RPROP [Riedmiller and Braun, 1993] and quickprop [Fahlman, 1988].